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Stochastic ODEs and PDEs for interacting multi-type populations 2009

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Stochastic ODEs and PDEs for interacting multi-type populations by Sandra Martina Kliem A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Mathematics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) September, 2009 c© Sandra Martina Kliem, 2009 ii Abstract This thesis consists of the manuscripts of three research papers studying stochas- tic ODEs (ordinary differential equations) and PDEs (partial differential equa- tions) that arise in biological models of interacting multi-type populations. In the first paper I prove uniqueness of the martingale problem for a de- generate SDE (stochastic differential equation) modelling a catalytic branching network. This work is an extension of a paper by Dawson and Perkins to ar- bitrary networks. The proof is based upon the semigroup perturbation method of Stroock and Varadhan. In the proof estimates on the corresponding semi- group are given in terms of weighted Hölder norms, which are equivalent to a semigroup norm in this generalized setting. An explicit representation of the semigroup is found and estimates using cluster decomposition techniques are derived. In the second paper I investigate the long-term behaviour of a special class of the SDEs considered above, involving catalytic branching and mutation between types. I analyse the behaviour of the overall sum of masses and the relative dis- tribution of types in the limit using stochastic analysis. For the latter existence, uniqueness and convergence to a stationary distribution are proved by the rea- soning of Dawson, Greven, den Hollander, Sun and Swart. One-dimensional diffusion theory allows for a complete analysis of the two-dimensional case. In the third paper I show that one can construct a sequence of rescaled per- turbations of voter processes in d = 1 whose approximate densities are tight. This is an extension of the results of Mueller and Tribe for the voter model. We combine critical long-range and fixed kernel interactions in the perturbations. In the long-range case, the approximate densities converge to a continuous den- sity solving a class of SPDEs (stochastic PDEs). For integrable initial condi- tions, weak uniqueness of the limiting SPDE is shown by a Girsanov theorem. A special case includes a class of stochastic spatial competing species models in mathematical ecology. Tightness is established via a Kolmogorov tightness criterion. Here, estimates on the moments of small increments for the approxi- mate densities are derived via an approximate martingale problem and Green’s function representation. iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Overview of the Manuscripts . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Degenerate stochastic differential equations for catalytic branching networks . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Long-term behaviour of a cyclic catalytic branching sys- tem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.3 Convergence of rescaled competing species processes to a class of SPDEs . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . 5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Degenerate Stochastic Differential Equations for Catalytic Branching Networks . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.1 Catalytic branching networks . . . . . . . . . . . . . . . . 9 2.1.2 Comparison with Dawson and Perkins [7] . . . . . . . . . 11 2.1.3 The model . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.4 Statement of the main result . . . . . . . . . . . . . . . . 13 2.1.5 Outline of the proof . . . . . . . . . . . . . . . . . . . . . 14 2.1.6 Weighted Hölder norms and semigroup norms . . . . . . 18 2.1.7 Outline of the paper . . . . . . . . . . . . . . . . . . . . . 20 2.2 Properties of the Semigroup . . . . . . . . . . . . . . . . . . . . 21 2.2.1 Representation of the semigroup . . . . . . . . . . . . . . 21 2.2.2 Decomposition techniques . . . . . . . . . . . . . . . . . 26 2.2.3 Existence and representation of derivatives of the semi- group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.2.4 L∞ bounds of certain differentiation operators applied to Ptf and equivalence of norms . . . . . . . . . . . . . . . 32 iv 2.2.5 Weighted Hölder bounds of certain differentiation opera- tors applied to Ptf . . . . . . . . . . . . . . . . . . . . . 40 2.3 Proof of Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . 52 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3 Long-term Behaviour of a Cyclic Catalytic Branching System 58 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.1.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.1.2 Main results and outline of the paper . . . . . . . . . . . 59 3.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.2.1 Existence and nonnegativity . . . . . . . . . . . . . . . . 60 3.2.2 The overall sum and uniqueness . . . . . . . . . . . . . . 62 3.2.3 The normalized processes . . . . . . . . . . . . . . . . . . 68 3.2.4 Properties of a stationary distribution to the system (3.15) of normalized processes . . . . . . . . . . . . . . . . . . . 71 3.2.5 Stationary distribution . . . . . . . . . . . . . . . . . . . 73 3.2.6 Extension to arbitrary networks . . . . . . . . . . . . . . 74 3.2.7 Complete analysis of the case d = 2 . . . . . . . . . . . . 74 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4 Convergence of Rescaled Competing Species Processes to a Class of SPDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.1.1 The voter model and the Lotka-Volterra model . . . . . . 78 4.1.2 Spatial versions of the Lotka-Volterra model . . . . . . . 80 4.1.3 Long-range limits . . . . . . . . . . . . . . . . . . . . . . 80 4.1.4 Overview of results . . . . . . . . . . . . . . . . . . . . . 82 4.1.5 Outline of the paper . . . . . . . . . . . . . . . . . . . . . 83 4.2 Main Results of the Paper . . . . . . . . . . . . . . . . . . . . . 83 4.2.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.2.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.2.3 Reformulation . . . . . . . . . . . . . . . . . . . . . . . . 88 4.3 An Approximate Martingale Problem . . . . . . . . . . . . . . . 94 4.4 Green’s Function Representation . . . . . . . . . . . . . . . . . . 101 4.5 Tightness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.6 Characterizing Limit Points . . . . . . . . . . . . . . . . . . . . 118 4.7 Uniqueness in Law . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.1 Overview of Results and Future Perspectives of the Manuscripts 125 5.1.1 Degenerate stochastic differential equations for catalytic branching networks . . . . . . . . . . . . . . . . . . . . . 125 5.1.2 Long-term behaviour of a cyclic catalytic branching sys- tem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 v5.1.3 Convergence of rescaled competing species processes to a class of SPDEs . . . . . . . . . . . . . . . . . . . . . . . . 127 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Appendices A Appendix for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . 130 A.1 ã is non-singular . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 A.2 Proof of Proposition 3.2.18 . . . . . . . . . . . . . . . . . . . . . 132 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 B Appendix for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . 140 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 vi List of Figures 2.1 Decomposition from the catalyst’s point of view . . . . . . . . . . 11 2.2 Definition of NR, NC and R̄i . . . . . . . . . . . . . . . . . . . . 15 2.3 Decomposition of the system of SDEs . . . . . . . . . . . . . . . 19 3.1 The definition of t2n−1 and t2n. . . . . . . . . . . . . . . . . . . . 66 vii Acknowledgements I am tremendously grateful to my Ph.D. supervisor Ed Perkins for his help. He introduced me to the problems at hand and provided me with helpful com- ments and suggestions throughout. Particular thanks go to him for his valuable feedback after reading my manuscript. I also wish to thank Carl Mueller for a discussion on survival and coexistence questions related to the limiting SPDEs I obtained in my third manuscript. A special thanks goes to the administration and the computer support of my department that made my life easier in many ways. Finally, a big thanks goes to all the people I met during my studies at UBC, my parents and my other friends for providing me with help and encouragement throughout my studies. 1Chapter 1 Introduction In the following three Chapters I investigate degenerate stochastic ODEs (or- dinary differential equations) and SPDEs (stochastic partial differential equa- tions) that arise in biological models of interacting multi-type populations. In the first two Chapters I investigate the behaviour of the respective masses of a finite number of interacting populations in a non-spatial setting, and in the last paper I study two interacting populations only but add a spatial component. The former will result in the consideration of SDEs (stochastic differential equa- tions), the latter in the consideration of limiting SPDEs. The former models can arise from a network of cooperating branching populations which require the presence of other types (catalysts) to reproduce, while the latter class includes scaling limits of ecological models for two types competing for resources. When investigating biological models for the evolution of populations over time, the common question to answer is that for survival, extinction and coex- istence of types. A well-known model for the evolution of the mass of one type of population is Feller’s branching diffusion with parameter γ and linear drift, i.e. the unique solution to the SDE dxt = bxtdt+ √ 2γxtdBt with constants b ∈ R, γ ∈ R+ and Bt a Brownian motion. Such diffusions can be obtained as the limit of a sequence of rescaled Galton-Watson branching processes at criticality (that is, branching processes with an average number of descendents approaching one). By adding a spatial component, one obtains super-Brownian motion with linear drift instead, which is the unique in law solution u(t, x) to the following SPDE ∂u ∂t = ∆u+ bu+ √ γuẆ , where Ẇ = Ẇ (t, x) is space time white noise. Here, ∆u models the spatial motion and dispersion of the population and √ γuẆ models the stochastic fluc- tuations in the population size. ut = u(t, ·) can be interpreted as the continuous spatial density of the population at time t. For both models, the degeneracy in the fluctuation term and its lack of Lipschitz continuity leads to difficulties in establishing uniqueness. In the above, the additivity properties inherent to both models (for example the sum of two in- dependent (b, γ)-Feller-diffusions starting at x1 respectively x2 is a (b, γ)-Feller- diffusion starting at x1 + x2) can be successfully employed to investigate the 2long-term behaviour of the above SDE, respectively SPDE. For extensive liter- ature on the above the interested reader is referred to Perkins [14]. As a next step, one can consider an equation for each type of population and introduce interactions between types by interlinking the equations. Thereby one can model competition of types for resources but also mutual help between types. As a result, the analysis of the resulting equations becomes more complicated as additivity properties are not present anymore in this context. In the first and third paper of my thesis I shall therefore employ different perturbation methods to derive results on the systems at hand from results of more accessible models. For instance, the first paper uses a perturbation method of Stroock and Varadhan to obtain the new system of SDEs as a perturbation of a system of independent Feller diffusions with constant coefficients. The last paper considers perturbations of the biased voter model for which the long-range limit was obtained in Mueller and Tribe [12]. In what follows, I shall give a short overview of the models, objectives and underlying literature of each of the three manuscripts of this thesis. 1.1 Overview of the Manuscripts 1.1.1 Degenerate stochastic differential equations for catalytic branching networks In the first paper I investigate weak uniqueness of solutions to the following system of SDEs: For j ∈ R ⊂ {1, . . . , d} and Cj ⊂ {1, . . . , d}\{j}: dx (j) t = bj(xt)dt+ √√√√√2γj(xt)  ∑ i∈Cj x (i) t  x(j)t dBjt (1.1) and for j /∈ R dx (j) t = bj(xt)dt+ √ 2γj(xt)x (j) t dB j t . (1.2) Here xt ∈ Rd+ and bj , γj , j = 1, . . . , d are Hölder-continuous functions on Rd+ with γj(x) > 0, and bj(x) ≥ 0 if xj = 0. The Bjt , j ∈ {1, . . . , d} are independent Brownian motions. This system of SDEs models catalytic branching networks, where types i ∈ Cj catalyze the replication of type j, j ∈ R (the so-called reactants). Such systems can be obtained as a limit of near-critical branching particle systems. The growth rate of types corresponds to the branching rate in this stochas- tic setting, i.e. type j, j ∈ R in state xt branches at a rate γj(xt) ∑ i∈Cj x (i) t proportional to the sum of masses of types i, i ∈ Cj at time t. The degeneracies in the covariance coefficients of this system and their lack of Lipschitz continuity make the investigation of uniqueness a challenging ques- tion. The former rules out the classic Stroock-Varadhan approach of perturbing 3Brownian motion and the latter prevents application of Itô’s pathwise unique- ness arguments. Similar results have been proven in Athreya, Barlow, Bass and Perkins [1] and Bass and Perkins [2] but without the additional singularity∑ i∈Cj x (i) t in the covariance coefficients of the diffusion. The question of uniqueness of equations with non-constant coefficients arises already in the case d = 2 in the renormalization analysis of hierarchically inter- acting two-type branching models treated in Dawson, Greven, den Hollander, Sun and Swart [6]. In [7], Dawson and Perkins proved weak uniqueness for the following system of SDEs: For j ∈ R and Cj = {cj}, cj 6= j, dx (j) t = bj(xt)dt+ √ 2γj(xt)x (cj ) t x (j) t dB j t , and (1.2) as above. This restriction to at most one catalyst per reactant is sufficient for the renormalization analysis for d = 2 types, but for more than 2 types one will encounter models where one type may have two catalysts. The goal of my first paper was to overcome this restriction and to allow consideration of general multi-type branching networks as envisioned in the section on future challenges in [6]. In my first paper, I extend the techniques of [7] to the setting of general catalytic networks, i.e. to (1.1) and (1.2). My work further includes natural settings such as competing hypercycles (cf. Eigen and Schuster [8], p.55 respectively Hofbauer and Sigmund [10], p.106). This latter work proposed an analogous system of ODEs as a model for the emergence of long polynucleotides in prebiotic evolution. 1.1.2 Long-term behaviour of a cyclic catalytic branching system As an application of the above I investigate the following special case in my second paper, involving cyclic catalytic branching and mutation between types. As I shall point out in this paper, the cyclic setup can easily be extended to arbitrary networks. Questions for survival and coexistence of types in the long time limit arise. Such questions naturally arise in biological competition models. For instance, Fleischmann and Xiong [9] investigated a cyclically catalytic super- Brownian motion in one spatial dimension. They showed global segregation (noncoexistence) of neighbouring types in the limit and other results on the finite time survival-extinction but they were not able to determine, if the overall sum dies out in the limit or not. In [10] (p. 86) multi-type branching processes with independent replication and mutation between types were rejected as a model since typically one type would take over, contrary to the observed diversity which emerged from the primordial soup. Let the following system of SDEs for d ≥ 2 be given: dxit = √ 2γixitx i+1 t dB i t + d∑ j=1 xjtqjidt, i ∈ {1, . . . , d}, 4where xd+1t ≡ x1t . I assumed the γi and qji, i 6= j to be given positive constants and the xi0 ≥ 0, i ∈ {1, . . . , d} to be given initial conditions. (qji) is a Q- matrix modelling the mutations or migrations from type j to type i. I shall investigate in particular the behaviour of the sum of types st = ∑d i=1 x i t and of the normalized processes yit = x i t/st in the time-limit. The latter addresses the diversity issue in [10]. 1.1.3 Convergence of rescaled competing species processes to a class of SPDEs The objectives of this paper were threefold. To better understand them, I shall first introduce the three papers that provide motivation. Firstly, in [12], Mueller and Tribe construct a sequence of rescaled compet- ing species processes ξNt ∈ {0, 1}Z/N in dimension d = 1 and show that its approximate densities A(ξNt )(x) ≡ 1 |{y ∈ Z/N : 0 < |y − x| ≤ 1/√N}| ∑ y∈Z/N, 0<|y−x|≤1/√N ξNt (y), x ∈ N−1Z converge in distribution to a continuous space time density that solves an SPDE. Here, ξNt ∈ {0, 1}Z/N denotes the configuration at time t of a “voter process” with bias τ = θN . That is, each type (0 or 1) invades a randomly chosen “neighbouring site” with constant rate, where θ > 0 would slightly favour 1’s by giving them a slightly larger invasion rate. ξNt (x) = i if site x ∈ Z/N is occupied by type i, i = 0, 1 and hence ut can be interpreted as the limiting continuous space time density of type 1 and 1− ut as the density of type 0. [12] fix θ ≥ 0, i.e. consider the case where the opinion of type 1 is slightly dominant. They show that ut is a solution of the following SPDE, the heat equation with drift, driven by Fisher-Wright noise, namely ∂u ∂t = ∆u 6 + 2θ(1− u)u+ √ 4u(1− u)Ẇ . (1.3) Observe that [12] scale space by 1/N and use {y ∈ Z/N : 0 < |y − x| ≤ 1/ √ N}, the set of neighbours of x, to calculate approximate densities. Hence, the number of neighbours of x ∈ Z/N is increasing proportionally to 2N 1/2. They thus obtain long-range interactions. Finally, they also speed up time to obtain appropriate limits. After rescaling (1.3) appropriately, one obtains the Kolmogorov-Petrovskii- Piscuinov (KPP) equation driven by Fisher-Wright noise. The behaviour of this SPDE has already been investigated in Mueller and Sowers [11] in detail, where the existence of travelling waves was shown. Secondly, in Cox and Perkins [4] it was shown that stochastic spatial Lotka- Volterra models, suitably rescaled in space and time, converge weakly to super- Brownian motion with linear drift. As they choose the parameters in their models to approach 1, the models can also be interpreted as small perturbations 5of the voter model. [4] extended the main results of Cox, Durrett and Perkins [3], which proved similar results for long-range voter models. Both papers treat the low density regime, i.e. where only a finite number of individuals of type 1 is present. Also note that both papers use a different scaling in comparison to [12]. [12] is at the threshold of the results in [3], but not included, and therefore [12] obtains a non-linear drift term in the limiting SPDE as a result. [4] considers fixed kernel models in dimensions d ≥ 3 and long-range kernel models in arbitrary dimension separately. Finally, in Cox and Perkins [5], the results of [4] for d ≥ 3 are used to relate the limiting super-Brownian motions in the fixed kernel case to questions of coexistence and survival of a rare type in the original Lotka-Volterra model. Thirdly, spatial versions of the Lotka-Volterra model with finite range were introduced and investigated in Neuhauser and Pacala [13]. The model from [13] incorporates a fecundity parameter and models both intra- and interspecific competition. The paper shows that short-range interactions alter the predictions of the mean-field model. In my paper I try to extend the approach of [12] for voter models to small perturbations of voter models similar to the perturbations in [4]. I work at criticality in the hope to obtain continuous densities in the limit that solve a class of SPDEs, similar to (1.3) but with more diverse drifts. My second goal is to thereby include spatial versions of Lotka-Volterra mod- els for competition and fecundity parameters near one as introduced in [13] as the approximating models and to determine their limits. As an additional ex- tension to [12] I shall investigate the weak uniqueness of the limiting class of SPDEs as weak uniqueness of the solutions to the SPDE will yield in turn weak uniqueness of the limits of the approximating densities. The last objective of this paper was to combine both long-range models at criticality and fixed kernel models in the perturbations. I investigate if the additional fixed kernel perturbation impacts statements on tightness (equiva- lent to relative compactness in my Polish spaces) of the approximating models. Thereby, results of [4] are extended. As a special case I would then be able to consider rescaled Lotka-Volterra models with long-range dispersal and short- range competition. 1.2 Concluding Remarks Tightness of approximating particle systems can be used to prove existence of limiting points of the approximating particle systems. Often, all limits can be shown to have certain properties in common. For instance, if all limits satisfy an SDE or SPDE as in my third paper for the case of long-range interactions only, weak uniqueness for the limiting systems of SDEs or SPDEs then yields uniqueness of the limits. Additionally, weak uniqueness of the solutions makes available certain tools that are used to investigate the behaviour of the systems at hand. Therefore, the proof of weak uniqueness in the first paper was funda- mental to the analysis of the model for cyclic catalytic branching and mutation 6between types of the second paper. All three papers of my thesis have in common that they investigate multi- type interaction models with a degeneracy in the component modelling fluctua- tions that stems from catalytic branching (the Fisher-Wright noise term can be seen as an application of a 2-cyclic model). Additionally all three models are parameter-dependent, where the parameters can be used to answer questions of survival and coexistence of types. 7Bibliography [1] Athreya, S.R. and Barlow, M.T. and Bass, R.F. and Perkins, E.A. Degenerate stochastic differential equations and super-Markov chains. Probab. Theory Related Fields (2002) 123, 484–520. MR1921011 [2] Bass, R.F. and Perkins, E.A. Degenerate stochastic differential equa- tions with Hölder continuous coefficients and super-Markov chains. Trans. Amer. Math. Soc. (2003) 355, 373–405 (electronic). MR1928092 [3] Cox, J.T. and Durrett, R. and Perkins, E.A. Rescaled voter mod- els converge to super-Brownian motion. Ann. Probab. (2000) 28, 185–234. MR1756003 [4] Cox, J.T. and Perkins, E.A. Rescaled Lotka-Volterra Models converge to Super-Brownian Motion. Ann. Probab. (2005) 33, 904–947. MR2135308 [5] Cox, J.T. and Perkins, E.A. Survival and coexistence in stochastic spa- tial Lotka-Volterra models. Probab. Theory Related Fields (2007) 139, 89– 142. MR2322693 [6] Dawson, D.A. and Greven, A. and den Hollander, F. and Sun, R. and Swart, J.M. The renormalization transformation for two-type branch- ing models. Ann. Inst. H. Poincaré Probab. Statist. (2008) 44, 1038–1077. MR2469334 [7] Dawson, D.A. and Perkins, E.A. On the uniqueness problem for cat- alytic branching networks and other singular diffusions. Illinois J. Math. (2006) 50, 323–383 (electronic). MR2247832 [8] Eigen, M. and Schuster, P. The Hypercycle: a principle of natural self- organization. Springer, Berlin, 1979. [9] Fleischmann, K. and Xiong, J. A cyclically catalytic super-brownian motion. Ann. Probab. (2001) 29, 820–861. MR1849179 [10] Hofbauer, J. and Sigmund, K. The Theory of Evolution and Dynamical Systems. London Math. Soc. Stud. Texts, vol. 7, Cambridge Univ. Press, Cambridge, 1988. MR1071180 [11] Mueller, C. and Sowers, R.B. Random Travelling Waves for the KPP Equation with Noise. J. Funct. Anal. (1995) 128, 439–498. MR1319963 [12] Mueller, C. and Tribe, R. Stochastic p.d.e.’s arising from the long range contact and long range voter processes. Probab. Theory Related Fields (1995) 102, 519–545. MR1346264 [13] Neuhauser, C. and Pacala, S.W. An explicitly spatial version of the Lotka-Volterra model with interspecific competition. Ann. Appl. Probab. (1999) 9, 1226–1259. MR1728561 8[14] Perkins, E.A. Dawson-Watanabe superprocesses and measure-valued dif- fusions. Lectures on Probability Theory and Statistics (Saint-Flour, 1999), 125–324, Lecture Notes in Math., 1781, Springer, Berlin, 2002. MR1915445 9Chapter 2 Degenerate Stochastic Differential Equations for Catalytic Branching Networks1 2.1 Introduction 2.1.1 Catalytic branching networks In this paper we investigate weak uniqueness of solutions to the following system of stochastic differential equations (SDEs): For j ∈ R ⊂ {1, . . . , d} and Cj ⊂ {1, . . . , d}\{j}: dx (j) t = bj(xt)dt+ √√√√√2γj(xt)  ∑ i∈Cj x (i) t  x(j)t dBjt (2.1) and for j /∈ R dx (j) t = bj(xt)dt+ √ 2γj(xt)x (j) t dB j t . (2.2) Here xt ∈ Rd+ and bj , γj , j = 1, . . . , d are Hölder-continuous functions on Rd+ with γj(x) > 0, and bj(x) ≥ 0 if xj = 0. The degeneracies in the covariance coefficients of this system make the inves- tigation of uniqueness a challenging question. Similar results have been proven in [1] and [4] but without the additional singularity ∑ i∈Cj x (i) t in the covariance coefficients of the diffusion. Other types of singularities, for instance replacing the additive form by a multiplicative form ∏ i∈Cj x (i) t , are possible as well, under additional assumptions on the structure of the network (cf. Remark 2.1.9 at the end of Subsection 2.1.5). The given system of SDEs can be understood as a stochastic analogue to a system of ODEs for the concentrations yj , j = 1, . . . , d of a type Tj . Then 1A version of this chapter has been accepted for publication. Kliem, S.M. (2009) Degenerate Stochastic Differential Equations for Catalytic Branching Networks. Ann. Inst. H. Poincaré Probab. Statist. 10 yj/ẏj corresponds to the rate of growth of type Tj and one obtains the following ODEs (see [9]): for independent replication ẏj = bjyj , autocatalytic replication ẏj = γjy 2 j and catalytic replication ẏj = γj (∑ i∈Cj yi ) yj . In the catalytic case the types Ti, i ∈ Cj catalyze the replication of type j, i.e. the growth of type j is proportional to the sum of masses of types i, i ∈ Cj present at time t. An important case of the above system of ODEs is the so-called hypercy- cle, firstly introduced by Eigen and Schuster (see [8]). It models hypercyclic replication, i.e. ẏj = γjyj−1yj and represents the simplest form of mutual help between different types. The system of SDEs can be obtained as a limit of branching particle systems. The growth rate of types in the ODE setting now corresponds to the branching rate in the stochastic setting, i.e. type j branches at a rate proportional to the sum of masses of types i, i ∈ Cj at time t. The question of uniqueness of equations with non-constant coefficients arises already in the case d = 2 in the renormalization analysis of hierarchically inter- acting two-type branching models treated in [6]. The consideration of successive block averages leads to a renormalization transformation on the diffusion func- tions of the SDE dx (i) t = c ( θi − x(i)t ) dt+ √ 2gi(xt)dB i t , i = 1, 2 with θi ≥ 0, i = 1, 2 fixed. Here g = (g1, g2) with gi(x) = xiγi(x) or gi(x) = x1x2γi(x), i = 1, 2 for some positive continuous function γi on R 2 +. The renor- malization transformation acts on the diffusion coefficients g and produces a new set of diffusion coefficients for the next order block averages. To be able to iterate the renormalization transformation indefinitely a subclass of diffusion functions has to be found that is closed under the renormalization transforma- tion. To even define the renormalization transformation one needs to show that the above SDE has a unique weak solution and to iterate it we need to establish uniqueness under minimal conditions on the coefficients. This paper is an extension of the work done in Dawson and Perkins [7]. The latter, motivated by the stochastic analogue to the hypercycle and by [6], proved weak uniqueness in the above mentioned system of SDEs (2.1) and (2.2), where (2.1) is restricted to dx (j) t = bj(xt)dt+ √ 2γj(xt)x (cj ) t x (j) t dB j t , i.e. Cj = {cj} and (2.2) remains unchanged. This restriction to at most one catalyst per reactant is sufficient for the renormalization analysis for d = 2 types, but for more than 2 types one will encounter models where one type may have two catalysts. The present work overcomes this restriction and allows consideration of general multi-type branching networks as envisioned in [6], including further natural settings such as competing hypercycles (cf. [8] page 55 resp. [9], p. 106). In particular, the techniques of [7] will be extended to the setting of general catalytic networks. Intuitively it is reasonable to conjecture uniqueness in the general setting as there is less degeneracy in the diffusion coefficients; x (cj) t changes to ∑ i∈Cj x (i) t , 11 Figure 2.1: Decomposition from the catalyst’s point of view: Arrows point from vertices i ∈ NC to vertices j ∈ Ri (for the definition of NC , Ri and N2 see Subsections 2.1.3 and 2.1.5 to follow). Separate points signify vertices j ∈ N2. The dotted arrows signify arrows which are only allowed in the generalized setting and thus make a decomposition of the kind used in [7] inaccessible. all coordinates i ∈ Cj have to become zero at the same time to result in a singularity. For d = 2 weak uniqueness was proven for a special case of a mutually catalytic model (γ1 = γ2 = const.) via a duality argument in [10]. Unfortunately this argument does not extend to the case d > 2. 2.1.2 Comparison with Dawson and Perkins [7] The generalization to arbitrary networks results in more involved calculations. The most significant change is the additional dependency among catalysts. In [7] the semigroup of the process under consideration could be decomposed into groups of single vertices and groups of catalysts with their corresponding reac- tants (see Figure 2.1). Hence the main part of the calculations in [7], where bounds on the semigroup are derived, i.e. Section 2 of [7] (“Properties of the basic semigroups”), could be reduced to the setting of a single vertex or a single catalyst with a finite number of reactants. In the general setting this strategy is no longer available as one reactant is now allowed to have multiple catalysts (see again Figure 2.1). As a consequence we shall treat all vertices in one step only. This results in more work in Section 2, where bounds on the given semigroup are now derived directly. We also employ a change of perspective from reactants to catalysts. In [7] every reactant j had one catalyst cj only (and every catalyst i a set of reactantsRi). For the general setting it turns out to be more efficient to consider every catalyst i with the set Ri of its reactants. In particular, the restriction from Ri to R̄i, including only reactants whose catalysts are all zero, turns out to be crucial for later definitions and calculations. It plays a key role in the extension of the definition of the weighted Hölder norms to general networks 12 (see Subsection 2.1.6). Changes in one catalyst indirectly impact other catalysts now via common reactants, resulting for instance in new mixed partial derivatives. As a first step a representation for the semigroup of the generalized process had to be found (see (2.15)). In [7], (12) the semigroup could be rewritten in a product form of semigroups of each catalyst with its reactants. Now a change in one catalyst resp. coordinate of the semigroup impacts in particular the local covariance of all its reactants. As the other catalysts of this reactant also appear in this coefficient, a decomposition becomes impossible. Instead the triangle inequality has to be often used to express resulting multi-dimensional coordinate changes of the function G, which is closely related with the semigroup representation (see (2.16)), via one-dimensional ones. As another important tool Lemma 2.2.6 was developed in this context. The ideas of the proofs in [7] often had to be extended. Major changes can be found in the critical Proposition 2.2.25 and its associated Lemmas (especially Lemma 2.2.29). The careful extension of the weighted Hölder norms to arbitrary networks had direct impact on the proofs of Lemma 2.2.19 and Theorem 2.2.20. 2.1.3 The model Let a branching network be given by a directed graph (V, E) with vertices V = {1, . . . , d} and a set of directed edges E = {e1, . . . , ek}. The vertices represent the different types, whose growth is under investigation, and (i, j) ∈ E means that type i “catalyzes” the branching of type j. As in [7] we continue to assume: Hypothesis 2.1.1. (i, i) /∈ E for all i ∈ V . Let C denote the set of catalysts, i.e. the set of vertices which appear as the 1st element of an edge and R denote the set of reactants, i.e. the set of vertices that appear as the 2nd element of an edge. For j ∈ R, let Cj = {i : (i, j) ∈ E} be the set of catalysts of j and for i ∈ C, let Ri = {j : (i, j) ∈ E} be the set of reactants, catalyzed by i. If j /∈ R let Cj = ∅ and if i /∈ C, let Ri = ∅. We shall consider the following system of SDEs: For j ∈ R: dx (j) t = bj(xt)dt+ √√√√√2γj(xt)  ∑ i∈Cj x (i) t  x(j)t dBjt and for j /∈ R dx (j) t = bj(xt)dt+ √ 2γj(xt)x (j) t dB j t . Our goal will be to show the weak uniqueness of the given system of SDEs. 13 2.1.4 Statement of the main result In what follows we shall impose additional regularity conditions on the coeffi- cients of our diffusions, similar to the ones in Hypothesis 2 of [7], which will remain valid unless indicated to the contrary. |x| is the Euclidean length of x ∈ Rd and for i ∈ V let ei denote the unit vector in the ith direction. Hypothesis 2.1.2. For i ∈ V , γi : R d + → (0,∞), bi : R d + → R are taken to be Hölder continuous of some positive index on compact subsets of R d + such that |bi(x)| ≤ c(1 + |x|) on Rd+, and{ bi(x) ≥ 0 if xi = 0. In addition, bi(x) > 0 if i ∈ C ∪ R and xi = 0. Definition 2.1.3. If ν is a probability on Rd+, a probability P on C(R+,Rd+) is said to solve the martingale problem MP(A,ν) if under P , the law of x0(ω) = ω0 (xt(ω) = ω(t)) is ν and for all f ∈ C2b (Rd+), Mf (t) = f(xt)− f(x0)− ∫ t 0 Af(xs)ds is a local martingale under P with respect to the canonical right-continuous filtration (Ft). Remark 2.1.4. The weak uniqueness of a system of SDEs is equivalent to the uniqueness of the corresponding martingale problem (see for instance, [12], V.(19.7)). For f ∈ C2b (Rd+), the generator corresponding to our system of SDEs is Af(x) = A(b,γ)f(x) = ∑ j∈R γj(x)  ∑ i∈Cj xi  xjfjj(x) +∑ j /∈R γj(x)xjfjj(x) + ∑ j∈V bj(x)fj(x). Here fij is the second partial derivative of f w.r.t. xi and xj . As a state space for the generator A we shall use S =  x ∈ Rd+ : ∏ j∈R  ∑ i∈Cj xi + xj   > 0   . (2.3) We first note that S is a natural state space for A. 14 Lemma 2.1.5. If P is a solution to MP(A,ν), where ν is a probability on Rd+, then xt ∈ S for all t > 0 P -a.s. Proof. The proof follows as for Lemma 5, [7] on p. 377 via a comparison argu- ment with a Bessel process, using Hypothesis 2.1.2. We shall now state the main theorem which, together with Remark 2.1.4 provides weak uniqueness of the given system of SDEs for a branching network. Theorem 2.1.6. Assume Hypothesis 2.1.1 and 2.1.2 hold. Then for any prob- ability ν, on S, there is exactly one solution to MP(A,ν). 2.1.5 Outline of the proof Our main task in proving Theorem 2.1.6 consists in establishing uniqueness of solutions to the martingale problem MP(A,ν). Existence can be proven as in Theorem 1.1 of [1]. The main idea in proving uniqueness consists in understand- ing our diffusion as a perturbation of a well-behaved diffusion and applying the Stroock-Varadhan perturbation method (refer to [13]) to it. This approach can be devided into three steps. Step 1: Reduction of the problem. We can assume w.l.o.g. that ν = δx0 . Furthermore it is enough to consider uniqueness for families of strong Markov solutions. Indeed, the first reduction follows by a standard conditioning argu- ment (see p. 136 of [3]) and the second reduction follows by using Krylov’s Markov selection theorem (Theorem 12.2.4 of [13]) together with the proof of Proposition 2.1 in [1]. Next we shall use a localization argument of [13] (see e.g. the argument in the proof of Theorem 1.2 of [4]), which basically states that it is enough if for each x0 ∈ S the martingale problem MP (Ã, δx0) has a unique solution, where bi = b̃i and γi = γ̃i agree on some B(x 0, r0) ∩ Rd+. Here we used in particular that a solution never exits S as shown in Lemma 2.1.5. Finally, if the covariance matrix of the diffusion is non-degenerate, unique- ness follows by a perturbation argument as in [13] (use e.g. Theorem 6.6.1 and Theorem 7.2.1). Hence consider only singular initial points, i.e. where either{ x (j) 0 = 0 or ∑ i∈Cj x (i) 0 = 0 for some j ∈ R } or { x (j) 0 = 0 for some j /∈ R. } Step 2: Perturbation of the generator. Fix a singular initial point x0 ∈ S and set (for an example see Figure 2.2) NR =  j ∈ R : ∑ i∈Cj x0i = 0   ; NC = ∪j∈NRCj ; N2 = V \ (NR ∪NC) ; R̄i = Ri ∩NR, 15 PSfrag replacements 6 7 2 : x02 = 0 5 : x05 = 0 4 : x04 > 0 3 : x03 = 0 ∗2 ∈ NC ∗3 ∈ NC ∗4 /∈ NC ∗5 /∈ NC ∗6 ∈ R̄i, i = 1, 2, 3 ∗x06 > 0 ∗6 ∈ NR ∗6 /∈ R̄3 ∗6 ∈ R3 ∗x06 ≥ 0 ∗7 /∈ NR ∗1 ∈ NC 1 : x01 = 0 Figure 2.2: Definition of NR, NC and R̄i. The ∗’s are the implications deduced from the given setting. i.e. in contrast to the setting in [7], p. 327, N2 can also include zero catalysts, but only those whose reactants have at least one more catalyst being non-zero. Let Z = Z(x0) = {i ∈ V : x0i = 0} (if i /∈ Z, then x0i > 0 and so x(i)s > 0 for small s a.s. by continuity). Moreover, if x0 ∈ S, then NR ∩ Z = ∅ and NR ∪NC ∪N2 = V is a disjoint union. Notation 2.1.7. In what follows let R A ≡ {f, f : A→ R} resp. RA+ ≡ {f, f : A→ R+}. for arbitrary A ⊂ V . Next we shall rewrite our system of SDEs with corresponding generator A as a perturbation of a well-understood system of SDEs with corresponding generator A0, which has a unique solution. The state space of A0 will be S(x0) = S0 = {x ∈ Rd : xi ≥ 0 for all i /∈ NR}. First, we view ({ x(j) } j∈NR , { x(i) } i∈NC ) , i.e. the set of vertices with zero catalysts together with these catalysts, near its initial point ({ x0j } j∈NR , { x0i } i∈NC ) as a perturbation of the diffusion on RNR × RNC+ , which is given by the unique 16 solution to the following system of SDEs: dx (j) t = b 0 jdt+ √√√√√2γ0j  ∑ i∈Cj x (i) t  dBjt , x(j)0 = x0j , for j ∈ NR and (2.4) dx (i) t = b 0 i dt+ √ 2γ0i x (i) t dB i t , x (i) 0 = x 0 i , for i ∈ NC , where for j ∈ NR, b0j = bj(x0) ∈ R and γ0j = γj(x0)x0j > 0 as x0j > 0 if its catalysts are all zero. Also, b0i = bi(x 0) > 0 as x0i = 0 for i ∈ NC and γ0i = γi(x 0) ∑ k∈Ci x 0 k > 0 if i ∈ NC ∩ R as i is a zero catalyst thus having at least one non-zero catalyst itself, or γ0i = γi(x 0) > 0 if i ∈ NC\R. Note that the non-negativity of b0i , i ∈ NC ensures that solutions starting in {x0i ≥ 0} remain there (also see definition of S0). Secondly, for j ∈ N2 we view this coordinate as a perturbation of the Feller branching process (with immigration) dx (j) t = b 0 jdt+ √ 2γ0jx (j) t dB j t , x (j) 0 = x 0 j , for j ∈ N2, (2.5) where b0j = (bj(x 0)∨ 0) (at the end of Section 2.3 the general case bj(x0) ∈ R is reduced to bj(x 0) ≥ 0 by a Girsanov transformation), γ0j = γj(x0) ∑ i∈Cj x 0 i > 0 if j ∈ R by definition of N2, i.e. at least one of the catalysts being positive, or γ0j = γj(x 0) > 0 if j /∈ R. As for i ∈ NC , the non-negativity of b0j , j ∈ N2 ensures that solutions starting in {x0j ≥ 0} remain there (see again definition of S0). Therefore we can view A as a perturbation of the generator A0 = ∑ j∈V b0j ∂ ∂xj + ∑ j∈NR γ0j  ∑ i∈Cj xi   ∂2 ∂x2j + ∑ i∈NC∪N2 γ0i xi ∂2 ∂x2i . (2.6) The coefficients b0i , γ 0 i found above for x 0 ∈ S now satisfy  γ0j > 0 for all j, b0j ≥ 0 if j /∈ NR, b0j > 0 if j ∈ (R ∪ C) ∩ Z, (2.7) where NR ∩ Z = ∅. (2.8) In the remainder of the paper we shall always assume the conditions (2.7) hold when dealing with A0 whether or not it arises from a particular x0 ∈ S as above. As we shall see in Subsection 2.2.1 the A0 martingale problem is then well-posed and the solution is a diffusion on S0 ≡ S ( x0 ) = {x ∈ Rd : xi ≥ 0 for all i ∈ V \NR = NC ∪N2}. (2.9) 17 Notation 2.1.8. In the following we shall use the notation NC2 ≡ NC ∪N2. Step 3: A key estimate. Set Bf := (A−A0)f = ∑ j∈V ( b̃j(x)− b0j ) ∂f ∂xj + ∑ j∈NR ( γ̃j(x)− γ0j )∑ i∈Cj xi   ∂2f ∂x2j + ∑ i∈NC2 ( γ̃i(x)− γ0i ) xi ∂2f ∂x2i , where for j ∈ V, b̃j(x) = bj(x), for j ∈ NR, γ̃j(x) = γj(x)xj , and for i ∈ NC2, γ̃i(x) = 1{i∈R}γi(x) ∑ k∈Ci xk + 1{i/∈R}γi(x). By using the continuity of the diffusion coefficients of A and the localization argument mentioned in Step 1 we may assume that the coefficients of the op- erator B are arbitrarily small, say less than η in absolute value. The key step (see Theorem 2.3.3) will be to find a Banach space of continuous functions with norm ‖·‖, depending on x0, so that for η small enough and λ0 > 0 large enough, ‖BRλf ‖≤ 1 2 ‖f ‖, ∀ λ > λ0. (2.10) Here Rλf = ∫ ∞ 0 e−λsPsfds (2.11) is the resolvent of the diffusion with generator A0 and Pt is its semigroup. The uniqueness of the resolvent of our strong Markov solution will then follow as in [13] and [4]. A sketch of the proof is given in Section 2.3. Remark 2.1.9. Under additional restrictions on the structure of the branching network our results carry over to the system of SDEs, where the additive form for the catalysts is replaced by a multiplicative form as follows. For j ∈ R we now consider dx (j) t = bj(xt)dt+ √√√√√2γj(xt)  ∏ i∈Cj x (i) t  x(j)t dBjt instead and for j /∈ R dx (j) t = bj(xt)dt+ √ 2γj(xt)x (j) t dB j t 18 as before. Indeed, if we impose that for all j ∈ R we have either |Cj | = 1 or |Cj | ≥ 2 and for all i1 6= i2, i1, i2 ∈ Cj : i1 ∈ Ci2 or i2 ∈ Ci1 , and if we assume that Hypothesis 2.1.2 holds, then we can show a result similar to Theorem 2.1.6. For instance, the following system of SDEs would be included. dx (1) t = b1(xt)dt+ √ 2γ1(xt)x (2) t x (3) t x (1) t dB 1 t , dx (2) t = b2(xt)dt+ √ 2γ2(xt)x (3) t x (4) t x (2) t dB 2 t , dx (3) t = b3(xt)dt+ √ 2γ3(xt)x (4) t x (1) t x (3) t dB 3 t , dx (4) t = b4(xt)dt+ √ 2γ4(xt)x (1) t x (2) t x (4) t dB 4 t . Note in particular, that the additional assumptions on the network ensure that at most one of either the catalysts in Cj or j itself can become zero, so that we obtain the same generator A0 as in the setting of additive catalysts if we set γ0j ≡ γj(x0) ∏ i∈{j}∪Cj :x0i >0 x 0 i (cf. the derivation of (2.4)). Remark 2.1.10. In [5] the Hölder condition on the coefficients was successfully removed but the restrictions on the network as stated in [7] were kept. As both [7] and [5] are based upon realizing the SDE in question as a perturbation of a well-understood SDE, one could start extending [5] to arbitrary networks by using the same generator and semigroup decomposition for the well-understood SDE as considered in this paper. 2.1.6 Weighted Hölder norms and semigroup norms In this section we describe the Banach space of functions which will be used in (2.10). In (2.10) we use the resolvent of the generator A0 with state space S0 = S ( x0 ) = {x ∈ Rd : xi ≥ 0 for all i ∈ NC2}. Note in particular that the state space and the realizations of the sets NR, R̄i etc. depend on x 0. Next we shall define the Banach space of weighted α-Hölder continuous func- tions on S0, Cαw(S0) ⊂ Cb(S0), in two steps. It will be the Banach space we look for and is a modification of the space of weighted Hölder norms used in [4]. Let f : S0 → R be bounded and measurable and α ∈ (0, 1). As a first step define the following seminorms for i ∈ NC : |f |α,i = sup { |f(x+ h)− f(x)| ( |h|−αxα/2i ∨ |h|−α/2 ) : |h| > 0, hk = 0 if k /∈ {i} ∪ R̄i, x, h ∈ S0 } . 19    Figure 2.3: Decomposition of the system of SDEs: unfilled circles, resp. filled circles, resp. squares are elements of NR, resp. NC , resp. N2. The definition of |f |α,i, i ∈ NC allows changes in i (filled circles) and the associated j ∈ R̄i (un- filled circles), the definition of |f |α,j , j ∈ N2 allows changes in j ∈ N2 (squares). Hence changes in all vertices are possible. For j ∈ N2 this corresponds to setting |f |α,j = sup { |f(x+ h)− f(x)| ( |h|−αxα/2j ∨ |h|−α/2 ) : hj > 0, hk = 0 if k 6= j, x ∈ S0 } . This definition is an extension of the definition in [7], p. 329. In our context the definition of |f |α,i, i ∈ NC had to be extended carefully by replacing the set Ri (in [7] equal to the set R̄i) by the set R̄i ⊂ Ri. Observe that the seminorms for i ∈ NC and j ∈ N2 taken together still allow changes in all coordinates (see Figure 2.3). The definition of |f |α,j , j ∈ N2 furthermore varies slightly from the one in [7]. We use our definition instead as it enables us to handle the coordinates i ∈ NC , j ∈ N2 without distinction. Secondly, set I = NC2. Then let |f |Cαw = maxj∈I |f |α,j , ‖f ‖Cαw= |f |Cαw+ ‖f ‖∞, where ‖f ‖∞ is the supremum norm of f . ‖f ‖Cαw is the norm we looked for and its corresponding Banach subspace of Cb(S0) is Cαw(S0) = {f ∈ Cb(S0) :‖f ‖Cαw<∞}, the Banach space of weighted α-Hölder continuous functions on S0. Note that the definition of the seminorms |f |α,j , j ∈ I depends on NC , R̄i etc. and hence on x0. Thus ‖f ‖Cαw depends on x0 as well. The seminorms |f |α,i are weaker norms near the spatial degeneracy at xi = 0 where we expect to have less smoothing by the resolvent. 20 Some more background on the choice of the above norms can be found in [4], Section 2. Bass and Perkins ([4]) consider |f |∗α,i ≡ sup { |f(x+ hei)− f(x)||h|−αxα/2i : h > 0, x ∈ Rd+ } , |f |∗α ≡ sup i≤d |f |∗α,i and ‖f ‖∗α≡ |f |∗α+ ‖f ‖∞ instead, where ei denotes the unit vector in the i-th direction in R d. They show that if f ∈ Cb(Rd+) is uniformly Hölder of index α ∈ (0, 1], and constant outside of a bounded set, then f ∈ Cα,∗w ≡ {f ∈ Cb(Rd+) :‖ f ‖∗α< ∞}. On the other hand, f ∈ Cα,∗w implies f is uniformly Hölder of order α/2. As it will turn out later (see Theorem 2.2.20) our norm ‖f ‖Cαw is equivalent to another norm, the so-called semigroup norm, defined via the semigroup Pt corresponding to the generatorA0 of our process. As we shall mainly investigate properties of the semigroup Pt on Cb(S0) in what follows, it is not surprising that this equivalence turns out to be useful in later calculations. In general one defines the semigroup norm (cf. [2]) for a Markov semigroup {Pt} on the bounded Borel functions on D where D ⊂ Rd and α ∈ (0, 1) via |f |α = sup t>0 ‖Ptf − f ‖∞ tα/2 , ‖f ‖α= |f |α+ ‖f ‖∞ . (2.12) The associated Banach space of functions is then Sα = {f : D → R : f Borel , ‖f ‖α<∞}. (2.13) Convention 2.1.11. Throughout this paper all constants appearing in state- ments of results and their proofs may depend on a fixed parameter α ∈ (0, 1) and {b0j , γ0j : j ∈ V } as well as on |V | = d. By (2.7) M0 = M0(γ0, b0) ≡ max i∈V { γ0i ∨ ( γ0i )−1 ∨ ∣∣b0i ∣∣} ∨ max i∈(R∪C)∩Z ( b0i )−1 <∞. (2.14) Given α ∈ (0, 1), d and 0 < M <∞, we can, and shall, choose the constants to hold uniformly for all coefficients satisfying M 0 ≤M . 2.1.7 Outline of the paper Proofs only requiring minor adaptations from those in [7] are usually omitted. A more extensive version of the proofs appearing in Sections 2.2 and 2.3 may be found on the arXiv at arXiv:0802.0035v2. The outline of the paper is as follows. In Section 2.2 the semigroup Pt corresponding to the generator A0 on the state space S0, as introduced in (2.6) and (2.9), will be investigated. We start with giving an explicit representation of the semigroup in Subsection 2.2.1. In Subsection 2.2.2 the canonical measure N0 is introduced which is used in Subsection 2.2.3 to prove existence and give a representation of derivatives of the semigroup. In Subsections 2.2.4 and 2.2.5 21 bounds are derived on the L∞ norms and on the weighted Hölder norms of those differentiation operators applied to Ptf , which appear in the definition of A0. Furthermore, at the end of Subsection 2.2.4 the equivalence of the weighted Hölder norm and semigroup norm is shown. Finally, in Section 2.3 bounds on the resolvent Rλ of Pt are deduced from the bounds on Pt found in Section 2.2. The bounds on the resolvent will then be used to obtain the key estimate (2.10). The remainder of Section 2.3 illustrates how to prove the uniqueness of solutions to the martingale problem MP(A,ν) from this, as in [7]. 2.2 Properties of the Semigroup 2.2.1 Representation of the semigroup In this subsection we shall find an explicit representation of the semigroup Pt corresponding to the generator A0 (cf. (2.6)) on the state space S0 and fur- ther preliminary results. We assume the coefficients satisfy (2.7) and Conven- tion 2.1.11 holds. Let us have a look at (2.4) and (2.5) again. For i ∈ NC or j ∈ N2 the processes x (i) t resp. x (j) t are Feller branching processes (with immigra- tion). If we condition on these processes, the processes x (j) t , j ∈ NR become independent time-inhomogeneous Brownian motions (with drift), whose dis- tributions are well understood. Thus if the associated process is denoted by xt = { x (j) t } j∈NR∪NC2 = { x (j) t } j∈V , the semigroup Ptf has the explicit repre- sentation Ptf(x) = (⊗i∈NC2 P ixi) [∫ R |NR| f ( {zj}j∈NR , { x (i) t } i∈NC2 ) (2.15) × ∏ j∈NR p γ0j 2I (j) t ( zj − xj − b0j t ) dzj   , where P ixi is the law of the Feller branching immigration process x (i) on C(R+,R+), started at xi with generator Ai0 = b0i ∂ ∂x + γ0i x ∂2 ∂x2 , I (j) t = ∫ t 0 ∑ i∈Cj x(i)s ds, and for y ∈ (0,∞) py(z) := e− z2 2y (2piy)1/2 . Remark 2.2.1. This also shows that the A0 martingale problem is well-posed. 22 For (y, z) = ( {yj}j∈NR , {zi}i∈NC2 ) and xNR ≡ {xj}j∈NR , let G(y, z) = Gt,xNR (y, z) = Gt,xNR ( {yj}j∈NR , {zi}i∈NC2 ) (2.16) = ∫ R |NR| f ( {uj}j∈NR , {zi}i∈NC2 ) ∏ j∈NR pγ0j 2yj ( uj − xj − b0j t ) duj . Notation 2.2.2. In the following we shall use the notations ENC2 = (⊗i∈NC2 P ixi) , INRt = {I(j)t } j∈NR , xNC2t = { x (i) t } i∈NC2 and we shall write E whenever we do not specify w.r.t. which measure we integrate. Now (2.15) can be rewritten as Ptf(x) = E NC2 [ Gt,xNR ( INRt , x NC2 t )] = ENC2 [ G ( INRt , x NC2 t )] . (2.17) Lemma 2.2.3. Let j ∈ NR, then (a) ENC2  ∑ i∈Cj x (i) t   = ∑ i∈Cj ( xi + b 0 i t ) , ENC2   (∑ i∈Cj x (i) t )2 = (∑ i∈Cj xi )2 + ∑ i∈Cj  2  ∑ k∈Cj b0k + γ 0 i  xi   t + ∑ i∈Cj    ∑ k∈Cj b0k + γ 0 i   b0i   t2, ENC2  (∑ i∈Cj ( x (i) t − xi ))2 = ∑ i∈Cj 2γ0i xit+ ∑ i∈Cj    ∑ k∈Cj b0k + γ 0 i   b0i   t2 and ENC2 [ I (j) t ] = ENC2  ∫ t 0 ∑ i∈Cj x(i)s ds   = ∑ i∈Cj ( xit+ b0i 2 t2 ) . (b) ENC2 [( I (j) t )−p] ≤ c(p)t−p min i∈Cj { (t+ xi) −p} ∀p > 0. Note. Observe that the requirement b0i > 0 if i ∈ (R ∪ C) ∩ Z as in (2.7) is crucial for Lemma 2.2.3(b). As i ∈ Cj , j ∈ NR implies i ∈ C ∩ Z, (2.7) guarantees b0i > 0. The bound (b) cannot be applied to i ∈ N2 in general, as (2.7) only gives b0i ≥ 0 in these cases. 23 Proof of (a). The first three results follow from Lemma 7(a) in [7] together with the independence of the Feller-diffusions under consideration. Proof of (b). Proceeding as in the proof of Lemma 7(b) in [7] we obtain ENC2 [( I (j) t )−p] ≤ cpe ∫ ∞ 0 ENC2 [ e−u −1I(j)t ] u−p−1du ≤ cpemin i∈Cj {∫ ∞ 0 P ixi [ e−u −1I(i)t ] u−p−1du } as I (j) t = ∑ i∈Cj ∫ t 0 x (i) s ds ≡ ∑i∈Cj I(i)t , where the Feller-diffusions under con- sideration are independent. Now we can proceed as in Lemma 7(b) of [7] to obtain the desired result. Lemma 2.2.4. Let Gt,xNR be as in (2.16). Then (a) for j ∈ NR∣∣∣∣∂Gt,xNR∂xj ( {yj}j∈NR , {zi}i∈NC2 )∣∣∣∣ = ∣∣∣∣∂Gt,xNR∂xj (y, z) ∣∣∣∣ ≤‖f ‖∞ (γ0j yj)−1/2, (2.18) and more generally for any k ∈ N, there is a constant ck such that∣∣∣∣∣∂ kGt,xNR ∂xkj (y, z) ∣∣∣∣∣ ≤ ck ‖f ‖∞ y−k/2j . (b) For j ∈ NR ∣∣∣∣∂Gt,xNR∂yj (y, z) ∣∣∣∣ ≤ c1 ‖f ‖∞ y−1j . (2.19) More generally there are constants ck, k ∈ N such that for l1, l2, j1, j2 ∈ NR,∣∣∣∣∣∂ m1+m2+k1+k2Gt,xNR ∂xm1l1 ∂x m2 l2 ∂yk1j1 ∂y k2 j2 (y, z) ∣∣∣∣∣ ≤ cm1+m2+k1+k2 ‖f ‖∞ y−m1/2l1 y−m2/2l2 y−k1j1 y−k2j2 for all m1,m2, k1, k2 ∈ N. (c) Let yNR = {yj}j∈NR and zNC2 = {zi}i∈NC2 , then for all zNC2 with zi ≥ 0, i ∈ NC2 we have that ( xNR , yNR ) → Gt,xNR(yNR , zNC2) is C3 on R |NR| × (0,∞)|NR|. Proof. This proceeds as in [7], Lemma 11, using the product form of the density. Lemma 2.2.5. If f is a bounded Borel function on S0 and t > 0, then Ptf ∈ Cb(S0) with |Ptf(x)− Ptf(x′)| ≤ c ‖f ‖∞ t−1 |x− x′| . 24 Proof. The outline of the proof is as in the proof of [7], Lemma 12. We shall nevertheless show the proof in detail as it illustrates some basic notational issues, which will appear again in later theorems. Note in particular the frequent use of the triangle inequality resulting in additional sums of the form ∑ j:j∈R̄i0 in the second part of the proof. Using (2.17), we have for x, x′ ∈ RNR ,∣∣Ptf(x, xNC2)− Ptf(x′, xNC2)∣∣ (2.20) = ∣∣∣ENC2[Gt,x(INRt , xNC2t )−Gt,x′(INRt , xNC2t )]∣∣∣ ≤ ‖f ‖∞ ∑ j∈NR |xj − x′j |√ γ0j ENC2 [( I (j) t )−1/2] (by (2.18)) ≤ c ‖f ‖∞ ∑ j∈NR |xj − x′j |√ γ0j t−1/2 min i∈Cj { (t+ xi) −1/2 } (by Lemma 2.2.3(b)) ≤ c ‖f ‖∞ t−1 ∑ j∈NR |xj − x′j |. Next we shall consider x, x′ = x+hei0 ∈ RNC2 where i0 ∈ NC2 is arbitrarily fixed. Assume h > 0 and let xh denote an independent copy of x(i0) starting at h but with b0i0 = 0. Then x (i0) + xh has law P i0xi0+h (additive property of Feller branching processes) and so if Ih(t) = ∫ t 0 xhs ds,∣∣Ptf(xNR , x′)− Ptf(xNR , x)∣∣ = ∣∣∣∣ENC2 [ Gt,xNR ({ I (j) t + 1{i0∈Cj}Ih(t) } j∈NR , { xit + 1{i=i0}x h t } i∈NC2 ) −Gt,xNR ({ I (j) t } j∈NR , xNC2t )]∣∣∣∣ . For what follows it is important to observe that {j ∈ NR : i0 ∈ Cj} = { j : j ∈ R̄i0 } , having made the definition of R̄i necessary. Next we shall use the triangle inequality to first sum up changes in the jth coordinates (where j ∈ NR such that i0 ∈ Cj) separately in increasing order, followed by the change in the coordinate i0. If Th = inf{t ≥ 0 : xht = 0} we thus obtain as a bound for the above (recall that ek denotes the unit vector in the kth direction): ∑ j:j∈R̄i0 c ‖f ‖∞ ENC2 [ Ih(t) ( I (j) t )−1] + 2 ‖f ‖∞ E[Th > t] = ∑ j:j∈R̄i0 c ‖f ‖∞ ENC2 [Ih(t)]ENC2 [( I (j) t )−1] + 2 ‖f ‖∞ E[Th > t] 25 by (2.19) and as ‖G‖∞≤‖f ‖∞ by the definition of G. Next we shall use that E[Th > t] ≤ htγ0i0 (for reference see equation (2.26) in Section 2.2.2). Together with Lemma 2.2.3(a), (b) we may bound the above by ∑ j:j∈R̄i0 c ‖f ‖∞ htt−1 min i∈Cj { (t+ xi) −1}+ 2 ‖f ‖∞ h tγ0i0 ≤ c ‖f ‖∞ ht−1. The case x′ = x + hei, i ∈ NC2 follows similarly. Note that for i ∈ N2 only the second term in the above bound is nonzero as the sum is taken over an empty set (R̄i = ∅ for i ∈ N2). Together with (2.20) (recall that the 1-norm and Euclidean norm are equivalent) we obtain the result via triangle inequality. Finally, we give elementary calculus inequalities that will be used below. Lemma 2.2.6. Let g : Rd+ → R be C2. Then for all ∆,∆′ > 0, y ∈ Rd+ and I1, I2 ⊂ {1, . . . , d}, |g(y + ∆∑i1∈I1 ei1 + ∆′∑i2∈I2 ei2)− g(y + ∆∑i1∈I1 ei1) (∆∆′) −g(y + ∆′∑i2∈I2 ei2) + g(y)| (∆∆′) ≤ sup {y′∈Qi∈{1,...,d}[yi,yi+∆+∆′]} ∑ i1∈I1 ∑ i2∈I2 ∣∣∣∣ ∂2∂yi1∂yi2 g(y′) ∣∣∣∣ . Also let f : Rd+ → R be C3. Then for all ∆1,∆2,∆3 > 0, y ∈ Rd+ and I1, I2, I3 ⊂ {1, . . . , d}, |f(y + ∆1 ∑ i1∈I1 ei1 + ∆2 ∑ i2∈I2 ei2 + ∆3 ∑ i3∈I3 ei3) (∆1∆2∆3) −f(y + ∆1 ∑ i1∈I1 ei1 + ∆3 ∑ i3∈I3 ei3) + f(y + ∆2 ∑ i2∈I2 ei2) (∆1∆2∆3) −f(y + ∆2 ∑ i2∈I2 ei2 + ∆3 ∑ i3∈I3 ei3) + f(y + ∆3 ∑ i3∈I3 ei3) (∆1∆2∆3) −f(y + ∆1 ∑ i1∈I1 ei1 + ∆2 ∑ i2∈I2 ei2) + f(y + ∆1 ∑ i1∈I1 ei1)− f(y)| (∆1∆2∆3) ≤ sup {y′∈Qi∈{1,...,d}[yi,yi+∆1+∆2+∆3]} ∑ i1∈I1 ∑ i2∈I2 ∑ i3∈I3 ∣∣∣∣ ∂3∂yi1∂yi2∂yi3 f(y′) ∣∣∣∣ . Proof. This is an extension of [7], Lemma 13, using the triangle inequality to split the terms under consideration into sums of differences in only one coordi- nate at a time. 26 2.2.2 Decomposition techniques In this subsection we cite relevant material from [7], namely Lemma 8, Propo- sition 9 and Lemma 10. Proofs and references can be found in [7]. Further background and motivation on the processes under consideration may be found in [11], Section II.7. Let {P 0x : x ≥ 0} denote the laws of the Feller branching process X with no immigration (equivalently, the 0-dimensional squared Bessel process) with generator L0f(x) = γxf ′′(x). Recall that the Feller branching process X can be constructed as the weak limit of a sequence of rescaled critical Galton-Watson branching processes. If ω ∈ C(R+,R+) let ζ(ω) = inf{t > 0 : ω(t) = 0}. There is a unique σ-finite measure N0 on Cex = {ω ∈ C(R+,R+) : ω(0) = 0, ζ(ω) > 0, ω(t) = 0 ∀t ≥ ζ(ω)} (2.21) such that for each h > 0, if Ξh is a Poisson point process on Cex with intensity hN0, then X = ∫ Cex νΞh(dν) has law P 0h . (2.22) Citing [11], N0 can be thought of being the time evolution of a cluster given that it survives for some positive length of time. The representation (2.22) decomposes X according to the ancestors at time 0. Moreover we also have N0[νδ > 0] = (γδ) −1 (2.23) and for t > 0 ∫ Cex νtdN0(ν) = 1. (2.24) For t > 0 let P ∗t denote the probability on Cex defined by P ∗t [A] = N0[A ∩ {νt > 0}] N0[νt > 0] . (2.25) Lemma 2.2.7. For all h > 0 P 0h [ζ > t] = P 0 h [Xt > 0] = 1− e−h/(tγ) ≤ h tγ . (2.26) Proposition 2.2.8. Let f : C(R+,R+) → R be bounded and continuous. Then for any δ > 0, lim h↓0 h−1E0h[f(X)1{Xδ>0}] = ∫ Cex f(ν)1{νδ>0}dN0(ν). The representation (2.22) leads to the following decompositions of the pro- cesses x (i) t , i ∈ NC2 that will be used below. Recall that x(i)t is the Feller branching immigration process with coefficients b0i ≥ 0, γ0i > 0 starting at xi and with law P ixi . In particular, we can make use of the additive property of Feller branching processes. 27 Lemma 2.2.9. Let 0 ≤ ρ ≤ 1. (a) We may assume x(i) = X ′0 +X1, where X ′0 is a diffusion with generator A′0f(x) = γ0i xf ′′(x) + b0i f ′(x) starting at ρxi, X1 is a diffusion with generator γ 0 i xf ′′(x) starting at (1− ρ)xi ≥ 0, and X ′0, X1 are independent. In addition, we may assume X1(t) = ∫ Cex νtΞ(dν) = Nt∑ j=1 ej(t), (2.27) where Ξ is a Poisson point process on Cex with intensity (1−ρ)xiN0, {ej , j ∈ N} is an iid sequence with common law P ∗t , and Nt is a Poisson random variable (independent of the {ej}) with mean (1−ρ)xitγ0i . (b) We also have ∫ t 0 X1(s)ds = ∫ Cex ∫ t 0 νsds1{νt 6=0}Ξ(dν) + ∫ Cex ∫ t 0 νsds1{νt=0}Ξ(dν) ≡ Nt∑ j=1 rj(t) + I1(t) and ∫ t 0 x(i)s ds = Nt∑ j=1 rj(t) + I2(t), (2.28) where rj(t) = ∫ t 0 ej(s)ds, I2(t) = I1(t) + ∫ t 0 X ′0(s)ds. (c) Let Ξh be a Poisson point process on Cex with intensity hiN0 (hi > 0), independent of the above processes. Set Ξx+h = Ξ + Ξh and Xht = ∫ νtΞ h(dν). Then Xx+ht ≡ x(i)t +Xh(t) = ∫ Cex νtΞ x+h(dν) +X ′0(t) (2.29) is a diffusion with generator A′0 starting at xi + hi. In addition ∫ Cex νtΞ x+h(dν) = N ′t∑ j=1 ej(t), (2.30) where N ′t is a Poisson random variable with mean ((1− ρ)xi +hi)(γ0i t)−1, such that {ej} and (Nt, N ′t) are independent. Also ∫ t 0 Xx+hs ds = N ′t∑ j=1 rj(t) + I2(t) + I h 3 (t), (2.31) where Ih3 (t) = ∫ Cex ∫ t 0 νsds1{νt=0}Ξ h(dν). 28 2.2.3 Existence and representation of derivatives of the semigroup Let A0 and Pt be as in Subsection 2.2.1. The first and second partial derivatives of Ptf w.r.t. xk, xl, k, l ∈ NC2 will be represented in terms of the canonical measure N0. Recall that by (2.17) Ptf(x) = E NC2 [ G ( INRt , x NC2 t )] , where INRt = { I (j) t } j∈NR with I (j) t = ∫ t 0 ∑ i∈Cj x (i) s ds. Notation 2.2.10. If X ∈ C ( R+,R NC2 + ) , η, η′, θ, θ′ ∈ Cex (for the definition of Cex see (2.21)) and k, l ∈ NC2, let G+k t,xNR ( X ; ∫ t 0 ηsds, θt ) ≡ Gt,xNR ({∫ t 0 ∑ i∈Cj X isds+ 1{k∈Cj} ∫ t 0 ηsds } j∈NR , { X it + 1{i=k}θt } i∈NC2 ) and G+k,+l t,xNR ( X ; ∫ t 0 ηsds, θt, ∫ t 0 η′sds, θ ′ t ) ≡ Gt,xNR ({∫ t 0 ∑ i∈Cj X is + 1{k∈Cj}ηs + 1{l∈Cj}η ′ sds } j∈NR , { X it + 1{i=k}θt + 1{i=l}θ ′ t } i∈NC2 ) . Note that if k ∈ N2 in the above we have 1{k∈Cj} = 0 for j ∈ NR, i.e. G+k t,xNR ( X ; ∫ t 0 ηsds, θt ) = G+k t,xNR ( X ; 0, θt ) , G+k,+l t,xNR ( X ; ∫ t 0 ηsds, θt, ∫ t 0 η′sds, θ ′ t ) = G+k,+l t,xNR ( X ; 0, θt, ∫ t 0 η′sds, θ ′ t ) (2.32) and for l ∈ N2 G+k,+l t,xNR ( X ; ∫ t 0 ηsds, θt, ∫ t 0 η′sds, θ ′ t ) = G+k,+l t,xNR ( X ; ∫ t 0 ηsds, θt, 0, θ ′ t ) . (2.33) If X ∈ C ( R+,R NC2 + ) , ν, ν′ ∈ Cex and k, l ∈ NC2, let ∆G+k t,xNR (X, ν) ≡ G+k t,xNR ( X ; ∫ t 0 νsds, νt ) −G+k t,xNR ( X ; 0, 0 ) 29 and ∆G+k,+l t,xNR (X, ν, ν′) (2.34) ≡ G+k,+l t,xNR ( X ; ∫ t 0 νsds, νt, ∫ t 0 ν′sds, ν ′ t ) −G+k,+l t,xNR ( X ; 0, 0, ∫ t 0 ν′sds, ν ′ t ) −G+k,+l t,xNR ( X ; ∫ t 0 νsds, νt, 0, 0 ) +G+k,+l t,xNR ( X ; 0, 0, 0, 0 ) . Proposition 2.2.11. If f is a bounded Borel function on S0 and t > 0 then Ptf ∈ C2b (S0) and for k, l ∈ V = {1, . . . , d} ‖(Ptf)kl ‖∞≤ c‖f ‖∞ t2 . Moreover if f is bounded and continuous on S0, then for all k, l ∈ NC2 (Ptf)k(x) = E NC2 [∫ ∆G+k t,xNR ( xNC2 , ν ) dN0(ν) ] , (2.35) (Ptf)kl(x) = E NC2 [∫ ∫ ∆G+k,+l t,xNR ( xNC2 , ν, ν′ ) dN0(ν)dN0(ν ′) ] . (2.36) Proof. The outline of this proof is similar to the one for [7], Proposition 14. We shall therefore only mention some changes due to the consideration of more than one catalyst at a time. With the help of Lemma 2.2.5 and using that Ptf = Pt/2(Pt/2f) one can eas- ily show that it suffices to consider bounded continuous f . In [7], Proposition 14 one only proves the existence of (Ptf)kl(x), k, l ∈ NC2 and its representation in terms of the canonical measure as in (2.36) based on (2.35). From the methods used it should then be clear how the easier formula (2.35) may have been found. Hence, let us also assume (Ptf)k exists and is given by (2.35) for k ∈ NC2. Let 0 < δ ≤ t. The role of δ will be explained at the end of this proof. In the first case where ν ′δ = νt = 0, use Lemmas 2.2.6 and 2.2.4(b) to see that for 30 k, l ∈ NC∣∣∣∆G+k,+l t,xNR ( xNC2 , ν, ν′ )∣∣∣ (2.37) = ∣∣∣∣∣G+k,+lt,xNR ( xNC2 ; ∫ t 0 νsds, 0, ∫ δ 0 ν′sds, 0 ) −G+k,+l t,xNR ( xNC2 ; 0, 0, ∫ δ 0 ν′sds, 0 ) −G+k,+l t,xNR ( xNC2 ; ∫ t 0 νsds, 0, 0, 0 ) +G+k,+l t,xNR ( xNC2 ; 0, 0, 0, 0 )∣∣∣∣ = ∣∣∣∣Gt,xNR ({∫ t 0 ∑ i∈Cj x(i)s ds+ 1{k∈Cj} ∫ t 0 νsds+ 1{l∈Cj} ∫ δ 0 ν′sds } j∈NR , xNC2t ) −Gt,xNR ({∫ t 0 ∑ i∈Cj x(i)s ds+ 1{l∈Cj} ∫ δ 0 ν′sds } j∈NR , xNC2t ) −Gt,xNR ({∫ t 0 ∑ i∈Cj x(i)s ds+ 1{k∈Cj} ∫ t 0 νsds } j∈NR , xNC2t ) +Gt,xNR ({∫ t 0 ∑ i∈Cj x(i)s ds } j∈NR , xNC2t )∣∣∣∣ ≤ ∑ j1:j1∈R̄k ∑ j2:j2∈R̄l c ‖f ‖∞ ( I (j1) t )−1 ( I (j2) t )−1 ∫ δ 0 ν′sds ∫ t 0 νsds (compare to (49) in [7]). For k or l ∈ N2 we obtain via (2.32) and (2.33)∣∣∣∆G+k,+l t,xNR ( xNC2 , ν, ν′ )∣∣∣ = 0. This is consistent with (2.37) if we consider the sum over an empty set to be zero (recall that R̄k = Rk ∩NR and thus R̄k = ∅ if k ∈ N2). Hence (2.37) is a bound for all k, l ∈ NC2. The other cases are proven as in [7] (for the last case use the trivial bound∣∣∣∆G+k,+l t,xNR (xNC2 , ν, ν′) ∣∣∣ ≤ 4 ‖f ‖∞) with the same modifications as just observed. 31 Combining all the cases we conclude that∣∣∣∆G+k,+lt,xNR (xNC2 , ν, ν′) ∣∣∣ ≤ { 1{ν′ δ =νt=0}   ∑ j1:j1∈R̄k ∑ j2:j2∈R̄l ( I (j1) t )−1 ( I (j2) t )−1 ∫ δ 0 ν′sds ∫ t 0 νsds   + 1{ν′δ=0,νt>0}   ∑ j:j∈R̄l ( I (j) t )−1 ∫ δ 0 ν′sds   + 1{ν′δ>0,νt=0}   ∑ j:j∈R̄k ( I (j) t )−1 ∫ t 0 νsds  + 1{ν′δ>0,νt>0} } c ‖f ‖∞ ≤ { 1{ν′δ=νt=0} (∫ t 0 x(k)s ds )−1(∫ t 0 x(l)s ds )−1 ∫ δ 0 ν′sds ∫ t 0 νsds + 1{ν′δ=0,νt>0} (∫ t 0 x(l)s ds )−1 ∫ δ 0 ν′sds + 1{ν′δ>0,νt=0} (∫ t 0 x(k)s ds )−1 ∫ t 0 νsds+ 1{ν′δ>0,νt>0} } c ‖f ‖∞ ≡ ḡt,δ ( xNC2 , ν, ν′ ) The remainder of the proof works similar to the proof in [7]. Some minor changes are necessary in the proof of continuity from below in x2 (now to be replaced by xNC2) following (59) in [7], by considering every coordinate on its own. Also, new mixed partial derivatives appear, which can be treated similarly to the ones already appearing in the proof of Proposition 14 in [7]. Other necessary technical changes will reappear in later proofs where they will be worked out in detail. They are thus omitted at this point. Remark 2.2.12. The necessity for introducing δ only becomes clear in the context of a complete proof. For instance, the derivation of (2.36) starts by defining Xh. , independent of x (l) and satisfying Xht = h+ ∫ t 0 √ 2γ0l X h s dB ′ s, (h > 0) (i.e. Xh has law P 0h) so that x (l) +Xh has law P lxl+h. Therefore (2.35) together with definition (2.34) implies 1 h [(Ptf)k(x+ hel)− (Ptf)k(x)] = 1 h ∫ ∫ ∫ ∆G+k,+l t,xNR ( xNC2 , ν,Xh ) ( 1{Xhδ =0} + 1{Xhδ >0} ) dN0(ν)dP NC2dP 0h . Now the first term can be made arbitrarily small for t fixed and δ ↓ 0+. The second term can be further rewritten with the help of Proposition 2.2.8 and will finally yield the representation (2.36) by first taking h ↓ 0+ and then δ ↓ 0+. 32 2.2.4 L∞ bounds of certain differentiation operators applied to Ptf and equivalence of norms We continue to work with the semigroup Pt on the state space S0 corresponding to the generator A0. Recall the definitions of the semigroup norm |f |α from (2.12) and of the associated Banach space of functions Sα from (2.13) in what follows. Proposition 2.2.13. If f is a bounded Borel function on S0 then for j ∈ NR∣∣∣∣ ∂∂xj Ptf(x) ∣∣∣∣ ≤ c ‖f ‖∞√tmax i∈Cj {√t+ xi} , (2.38) and ∣∣∣∣∣maxi∈Cj{xi} ∂ 2 ∂x2j Ptf(x) ∣∣∣∣∣ ≤ c ‖f ‖∞t . (2.39) If f ∈ Sα, then ∣∣∣∣ ∂∂xj Ptf(x) ∣∣∣∣ ≤ c|f |αt α 2− 12 max i∈Cj {√t+ xi} ≤ c|f |αt α 2−1, (2.40) and ∣∣∣∣∣maxi∈Cj{xi} ∂ 2 ∂x2j Ptf(x) ∣∣∣∣∣ ≤ c|f |αtα2−1. (2.41) Proof. The proof proceeds as in [7], Proposition 16 except for minor changes. The estimate in (2.38) can be obtained by mimicking the calculation in (2.20). (2.39) follows from a double application of (2.38), where we use that Pt and ∂∂xj commute. If f ∈ Sα, we proceed as in [2] and write∣∣∣∣ ∂∂xj P2tf(x)− ∂ ∂xj Ptf(x) ∣∣∣∣ = ∣∣∣∣ ∂∂xj Pt(Ptf − f)(x) ∣∣∣∣ . Applying the estimate (2.38) to g = Ptf − f and using the definition of |f |α we get ∣∣∣∣ ∂∂xj P2tf(x)− ∂ ∂xj Ptf(x) ∣∣∣∣ ≤ c ‖g‖∞√tmax i∈Cj {√t+ xi} ≤ c|f |αt α/2 √ tmax i∈Cj {√t+ xi} . This together with (2.38) ⇒ lim t→∞ ∣∣∣∣ ∂∂xj Ptf(x) ∣∣∣∣ = 0 33 implies that ∣∣∣∣ ∂∂xj Ptf(x) ∣∣∣∣ ≤ ∞∑ k=0 ∣∣∣∣ ∂∂xj (P2ktf − P2(k+1)tf) (x) ∣∣∣∣ ≤ |f |α ∞∑ k=0 ( 2kt )α 2− 12 c max i∈Cj { √ 2kt+ xi} ≤ |f |αtα2− 12 c max i∈Cj {√t+ xi} . This then immediately yields (2.40). Use (2.39) to derive (2.41) in the same way as (2.38) was used to prove (2.40). Notation 2.2.14. If w > 0, set pj(w) = wj j! e −w. For {rj(t)} and {ej(t)} as in Lemma 2.2.9, let Rk = Rk(t) = ∑k j=1 rj(t) and Sk = Sk(t) = ∑k j=1 ej(t). Notation 2.2.15. If X ∈ C ( R+,R NC2 + ) , Y, Y ′, Z, Z ′ ∈ C(R+,R+), η, η′, θ, θ′ ∈ Cex and m,n, k, l ∈ NC2, where m 6= n let Gm,n,+k,+l t,xNR ( X,Yt, Zt, Y ′ t , Z ′ t; ∫ t 0 ηsds, θt, ∫ t 0 η′sds, θ ′ t ) ≡ Gt,xNR ({∫ t 0 ∑ i∈Cj\{m,n} X isds+ 1{m∈Cj}Yt + 1{n∈Cj}Y ′ t + ∫ t 0 1{k∈Cj}ηs + 1{l∈Cj}η ′ sds } j∈NR , { 1{i/∈{m,n}}X it + 1{i=m}Zt + 1{i=n}Z ′ t + 1{i=k}θt + 1{i=l}θ′t } i∈NC2 ) . The notation indicates that the one-dimensional coordinate processes∫ t 0 Xms ds,X m t resp. ∫ t 0 Xns ds,X n t will be replaced by the processes Yt, Zt resp. Y ′t , Z ′t (note that for m ∈ N2 this only implies a change from Xmt into Zt). Additionally, we add ∫ t 0 νsds, θt, ∫ t 0 ν ′ sds and θ ′ t as before. The terms Gm,+k,+l t,xNR , Gm,+k t,xNR , Gm,n,+l t,xNR , Gm,n t,xNR , Gmt,xNR ,∆G m,+k,+l t,xNR etc. (2.42) will then be defined in a similar way, where for instance Gm t,xNR only refers to replacing the processes ∫ t 0 Xms ds,X m t via Yt, Zt but doesn’t involve adding processes. 34 Proposition 2.2.16. If f is a bounded Borel function on S0, then for i ∈ NC2∣∣∣∣ ∂∂xiPtf(x) ∣∣∣∣ ≤ c ‖f ‖∞√t√t+ xi , (2.43) and ∣∣∣∣xi ∂2∂x2i Ptf(x) ∣∣∣∣ ≤ cxi ‖f ‖∞t(t+ xi) ≤ c ‖f ‖∞ t . (2.44) If f ∈ Sα, then ∣∣∣∣ ∂∂xiPtf(x) ∣∣∣∣ ≤ c|f |αt α 2− 12√ t+ xi ≤ c|f |αtα2−1, and ∣∣∣∣xi ∂2∂x2i Ptf(x) ∣∣∣∣ ≤ c|f |αtα2−1. Proof. The outline of the proof is the same as for [7], Proposition 17. Part of the proof will be presented here with its notational modifications since some care is needed when working in a multi-dimensional setting and the formulas become more involved. As in the proof of Proposition 2.2.11 we assume w.l.o.g. that f is bounded and continuous. In what follows we shall illustrate the proof of (2.44) as (2.43) is easier. Consider second derivatives in k. The representation of (Ptf)kk in Proposition 2.2.11 and symmetry allow us to write for k ∈ NC2 (i.e. l = k) (Ptf)kk(x) = E NC2 [∫ ∫ ∆G+k,+k t,xNR ( xNC2 , ν, ν′ ) 1{νt=0,ν′t=0}dN0(ν)dN0(ν ′) ] + 2ENC2 [∫ ∫ ∆G+k,+k t,xNR ( xNC2 , ν, ν′ ) 1{νt=0,ν′t>0}dN0(ν)dN0(ν ′) ] +ENC2 [∫ ∫ ∆G+k,+k t,xNR ( xNC2 , ν, ν′ ) 1{νt>0,ν′t>0}dN0(ν)dN0(ν ′) ] ≡ E1 + 2E2 +E3. The idea for bounding |E1|, |E2| and |E3| is similar to the one in [7]. In what follows we shall illustrate the necessary changes to bound |E3|. Notation 2.2.17. We have N0[·∩{νt > 0}] = (γt)−1P ∗t [·] on {νt > 0}, where we used (2.25) and (2.23). Whenever we change integration w.r.t. N0 to integration w.r.t. P ∗t we shall denote this by (∗) =. 35 The decomposition of Lemma 2.2.9 (cf. (2.27) and (2.28)) with ρ = 0 gives |E3| (∗)= c t2 ∣∣∣∣E [∫ ∫ { Gk,+k,+k t,xNR ( xNC2 , RNt + I2(t), SNt +X ′ 0(t); (2.45)∫ t 0 νsds, νt, ∫ t 0 ν′sds, ν ′ t ) − Gk,+k,+k t,xNR ( xNC2 , RNt + I2(t), SNt +X ′ 0(t); 0, 0, ∫ t 0 ν′sds, ν ′ t ) − Gk,+k,+k t,xNR ( xNC2 , RNt + I2(t), SNt +X ′ 0(t); ∫ t 0 νsds, νt, 0, 0 ) + Gk,+k,+k t,xNR ( xNC2 , RNt + I2(t), SNt +X ′ 0(t); 0, 0, 0, 0 )} × dP ∗t (ν)dP ∗t (ν′) ]∣∣∣∣, where for instance Gk,+k,+k t,xNR ( xNC2 , RNt + I2(t), SNt +X ′ 0(t); ∫ t 0 νsds, νt, ∫ t 0 ν′sds, ν ′ t ) = Gt,xNR ({∫ t 0 ∑ i∈Cj\{k} X isds+ 1{k∈Cj} (RNt + I2(t)) + ∫ t 0 1{k∈Cj} (νs + ν ′ s) ds } j∈NR , { 1{i6=k}X it + 1{i=k} (SNt +X ′ 0(t)) + 1{i=k} (νt + ν′t) } i∈NC2 ) by Notation 2.2.15 and the comment following it. Recall that Rk = Rk(t) = ∑k j=1 rj(t) and Sk = Sk(t) = ∑k j=1 ej(t) with {rj(t)} and {ej(t)} as in Lemma 2.2.9. In particular, {ej , j ∈ N} is iid with common law P ∗t and rj(t) = ∫ t 0 ej(s)ds. We obtain (recall the definition of Gk t,xNR from (2.42)) |E3| = c t2 ∣∣∣∣E [ Gkt,xNR ( xNC2 , RNt+2 + I2(t), SNt+2 +X ′ 0(t) ) − 2Gkt,xNR ( xNC2 , RNt+1 + I2(t), SNt+1 +X ′ 0(t) ) + Gkt,xNR ( xNC2 , RNt + I2(t), SNt +X ′ 0(t) )]∣∣∣∣. Observe that in case k ∈ N2 the above notation Gkt,xNR (xNC2 , RNt + I2(t), SNt +X ′ 0(t)) only indicates that x (k) t gets changed into SNt +X ′ 0(t); for k ∈ N2 36 the indicated change of ∫ t 0 x (k) s ds into RNt + I2(t) has no impact on the term under consideration. Let w = xk/(γ 0 kt). The independence of Nt from ({ ∫ t 0 x (i) s ds, i ∈ Cj\{k}, j ∈ NR}, x(NC2)\{k}t , I2(t), X ′0(t), {el}, {rl}) yields |E3| = c t2 ∣∣∣ ∞∑ n=0 pn(w)E [ Gkt,xNR ( xNC2 , Rn+2 + I2(t), Sn+2 +X ′ 0(t) ) − 2Gkt,xNR ( xNC2 , Rn+1 + I2(t), Sn+1 +X ′ 0(t) ) +Gkt,xNR ( xNC2 , Rn + I2(t), Sn +X ′ 0(t) )]∣∣∣. Sum by parts twice and use |G| ≤‖f ‖∞ to bound the above by c ‖f ‖∞ 1 xkt ( w(3p0(w) + p1(w)) + ∞∑ n=2 w |pn−2(w) − 2pn−1(w) + pn(w)| ) ≤ c ‖f ‖∞ 1 xkt ( wp0(w) + wp1(w) + ∞∑ n=2 pn(w) |(w − n)2 − n| w ) ≤ c ‖f ‖∞ 1 xkt ( 2p1(w) + ∞∑ n=0 pn(w) (w − n)2 + n w ) ≤ c ‖f ‖∞ 1 xkt . We obtain another bound on |E3| if we use the trivial bound |G| ≤‖f ‖∞ in (2.45). This yields |E3| ≤ c ‖f ‖∞ t−2 and so |E3| ≤ c ‖f ‖∞ t(t+ xk) . Combine the bounds on |E1|, |E2| and |E3| to obtain (2.44). The bounds for f ∈ Sα are obtained from the above just as in the proof of Proposition 2.2.13. Recall Convention 2.1.11, as stated in (2.14), for the definition of M 0 in what follows. Notation 2.2.18. Set J (j) t = γ 0 j 2I (j) t , j ∈ NR. Lemma 2.2.19. For each M ≥ 1, α ∈ (0, 1) and d ∈ N there is a c = c(M,α, d) > 0 such that if M 0 ≤M , then |fg|α ≤ c|f |Cαw ‖g‖∞ + ‖f ‖∞ |g|α (2.46) and ‖fg‖α≤ c (‖f ‖Cαw‖g‖∞ + ‖f ‖∞ |g|α) . (2.47) 37 Proof. Compared to the proof of [7], Lemma 18, the derivation of a bound for the second error term E2 below becomes more involved. Again the triangle- inequality has to be used to express multi-dimensional coordinate changes via one-dimensional ones. Let ( xNR , xNC2 ) ∈ R|NR| × R|NC2|+ and define f̃(y) = f(y) − f(x). Then (2.15) gives |Pt(fg)(x)− (fg)(x)| (2.48) ≤ |Pt(f̃g)(x)| + |f(x)||Ptg(x)− g(x)| ≤ ‖g‖∞ ENC2  ∫ R|NR| ∣∣∣f̃(zNR , xNC2t )∣∣∣ ∏ j∈NR p J (j) t ( zj − xj − b0j t ) dzj   + ‖f ‖∞ |g|αtα/2. The above expectation can be bounded by three terms as follows: ENC2  ∫ ∣∣∣f̃(zNR , xNC2t )∣∣∣ ∏ j∈NR p J (j) t ( zj − xj − b0j t ) dzj   (2.49) ≤ ENC2 [∫ {∣∣∣f̃(zNR , xNC2t )− f̃(zNR , xNC2)∣∣∣ + ∣∣f(zNR , xNC2)− f(xNR + b0NRt, xNC2)∣∣ + ∣∣f(xNR + b0NRt, xNC2)− f(xNR , xNC2)∣∣} × ∏ j∈NR p J (j) t ( zj − xj − b0j t ) dzj ] ≡ E1 +E2 +E3. For all three terms we shall use the triangle inequality to sum up changes in different coordinates separately. The definition of |f |α,i gives E1 ≤ ∑ i∈NC2 |f |α,iENC2 [(∣∣∣x(i)t − xi∣∣∣α x−α/2i ) ∧ ∣∣∣x(i)t − xi∣∣∣α/2 ] ≤ ∑ i∈NC2 |f |α,i (( ENC2 [∣∣∣x(i)t − xi∣∣∣2 ]α/2 x −α/2 i ) ∧ENC2 [∣∣∣x(i)t − xi∣∣∣2 ]α/4) . We now proceed as in the derivation of a bound on E1 in the proof of Lemma 18 in [7], using Lemma 2.2.3(a) (alternatively compare with estimation of E2 be- low). We finally obtain E1 ≤ c ∑ i∈NC2 |f |α,itα/22α/2 ≤ c|f |Cαw tα/22α/2. 38 Similarly we have E2 ≤ ∑ k∈NR min i:k∈R̄i { |f |α,iENC2 [∫ ((∣∣zk − (xk + b0kt)∣∣α x−α/2i )∧ ∣∣zk − (xk + b0kt)∣∣α/2) ∏ j∈NR p J (j) t ( zj − xj − b0j t ) dzj ]} ≤ c ∑ k∈NR min i:k∈R̄i { |f |α,iENC2 [(∣∣∣J (k)t ∣∣∣α/2 x−α/2i ) ∧ ∣∣∣J (k)t ∣∣∣α/4 ]} ≤ c ∑ k∈NR min i:k∈R̄i { |f |α,i (( ENC2 [∣∣∣J (k)t ∣∣∣]α/2 x−α/2i ) ∧ ENC2 [∣∣∣J (k)t ∣∣∣]α/4 )} as ∫ |z|βpJ(z)dz ≤ cJβ/2 for β ∈ (0, 1). Next use Lemma 2.2.3(a) which shows that ENC2 [ J (k) t ] = γ0k2E NC2 [ I (k) t ] ≤ ∑l∈Ck cM2(t2 + xlt). Put this in the above bound on E2 to see that E2 can be bounded by c ∑ k∈NR min i:k∈R̄i  |f |α,i     (∑ l∈Ck (t2 + xlt) )α/2 x −α/2 i   ∧ (∑ l∈Ck (t2 + xlt) )α/4   k∈NR≤ c|f |Cαw ∑ k∈NR    ∑ l∈Ck t2 + xlt max i:k∈R̄i xi   α/2 ∧ (∑ l∈Ck ( t2 + t max i:k∈R̄i xi ))α/4 k∈NR≤ c|f |Cαw tα/2 ∑ k∈NR     t max i:k∈R̄i xi + 1   α/2 ∧ ( 1 + max i:k∈R̄i xi t )α/4 ≤ c|f |Cαw tα/22α/2. For the third term E3 we finally have E3 ≤ ∑ k∈NR min i:k∈R̄i { |f |α,i ((∣∣b0kt∣∣α x−α/2i ) ∧ (∣∣b0kt∣∣α/2))} ≤ c|f |Cαw ∑ k∈NR ∣∣b0kt∣∣α/2 ≤ c|f |Cαw tα/2. Put the above bounds on E1, E2 and E3 into (2.49) and then in (2.48) to conclude that |Pt(fg)(x)− (fg)(x)| ≤ (‖g‖∞ c|f |Cαw+ ‖f ‖∞ |g|α) tα/2 and so by definition of the semigroup norm |fg|α ≤ c|f |Cαw ‖g‖∞ + ‖f ‖∞ |g|α. This gives (2.46) and (2.47) is then immediate. 39 Theorem 2.2.20. There exist 0 < c1 ≤ c2 such that c1|f |Cαw ≤ |f |α ≤ c2|f |Cαw . (2.50) This implies that Cαw = Sα and so Sα contains C1 functions with compact support in S0. Proof. The idea of the proof was taken from the proof of Theorem 19 in [7]. The second inequality in (2.50) follows immediately by setting g = 1 in Lemma 2.2.19. For the first inequality let x, h ∈ S0, t > 0 and use Propositions 2.2.13 and 2.2.16 to see that |f(x+ h)− f(x)| (2.51) ≤ |Ptf(x+ h)− f(x+ h)|+ |Ptf(x)− f(x)|+ |Ptf(x+ h)− Ptf(x)| ≤ 2|f |αtα/2 + |Ptf(x+ h)− Ptf(x)| ≤ 2|f |αtα/2 + c|f |αtα2− 12  ∑ j∈NR |hj | max l∈Cj {√t+ xl} + ∑ i∈NC2 hi√ t+ xi   , where we used the triangle inequality together with hl ≥ 0, l ∈ Cj ⊂ NC2 for all j ∈ NR. By setting t = |h| and bounding (maxl∈Cj{√t+ xl})−1 and (√t+ xi)−1 by(√ t )−1 we obtain as a first bound on (2.51) c|f |α|h|α/2. (2.52) Next only consider h ∈ S0 such that there exists i ∈ NC2 and j ∈ {i} ∪ R̄i such that hj 6= 0 and hk = 0 if k /∈ {i} ∪ R̄i. (2.51) becomes |f(x+ h)− f(x)| ≤ 2|f |αtα/2 + c|f |αtα2− 12   ∑ j:j∈R̄i |hj | max l∈Cj {√t+ xl} + hi√ t+ xi   ≤ 2|f |αtα/2 + c|f |αtα2− 12 1√ t+ xi |h|. In case xi > 0 set t = |h|2 xi and bound (√ t+ xi )−1 by (√ xi )−1 to get as a second upper bound c|f |αx−α/2i |h|α. (2.53) The first inequality in (2.50) is now immediate from (2.52) and (2.53) and the proof is complete. Note. Special care was needed when choosing h ∈ S0 in the last part of the proof as it only works for those h which are to be considered in the definition of | · |Cαw . Note that this was the main reason to define the weighted Hölder norms for R̄i instead of Ri. 40 Remark 2.2.21. The equivalence of the two norms will prove to be crucial later in Section 2.3, where we show the uniqueness of solutions to the martingale problem MP(A,ν) as stated in Theorem 2.1.6. All the estimates of Section 2.2 are obtained in terms of the semigroup norm. In Section 2.3 we shall further need estimates on the norm of products of certain functions. At this point we shall have to rely on the result of Lemma 2.2.19 for weighted Hölder norms. The equivalence of norms now yields a similar result in terms of the semigroup norm. 2.2.5 Weighted Hölder bounds of certain differentiation operators applied to Ptf The xj , j ∈ NR derivatives are much easier. Notation 2.2.22. We shall need the following slight extension of our notation for ENC2 : ENC2 = ENC2 xNC2 = (⊗i∈NC2 P ixi) . Notation 2.2.23. To ease notation let T − 12 k ( t, xNC2 ) ≡ { min l∈Ck { (t+ xl) −1/2} , k ∈ NR, (t+ xk) −1/2, k ∈ NC2. Proposition 2.2.24. If f is a bounded Borel function on S0, then for all x, h ∈ S0, j ∈ NR, i ∈ Cj and arbitrary k ∈ V ,∣∣∣∣ ∂∂xj Ptf(x+ hkek)− ∂ ∂xj Ptf(x) ∣∣∣∣ ≤ c ‖f ‖∞t3/2 |hk|T− 1 2 k ( t, xNC2 ) (2.54) and∣∣∣∣∣(x+ hkek)i ∂ 2Ptf ∂x2j (x+ hkek)− xi ∂ 2Ptf ∂x2j (x) ∣∣∣∣∣ ≤ c ‖f ‖∞t3/2 |hk|T− 1 2 k ( t, xNC2 ) . (2.55) If f ∈ Sα, then∣∣∣∣ ∂∂xj Ptf(x+ hkek)− ∂ ∂xj Ptf(x) ∣∣∣∣ ≤ c|f |αtα2− 32 |hk|T− 12k (t, xNC2) (2.56) and∣∣∣∣∣(x + hkek)i ∂ 2Ptf ∂x2j (x+ hkek)− xi ∂ 2Ptf ∂x2j (x) ∣∣∣∣∣ ≤ c|f |αtα2− 32 |hk|T− 1 2 k ( t, xNC2 ) . (2.57) Proof. The focus will be on proving (2.55) as (2.54) is simpler. Again, it suffices to consider f bounded and continuous. For increments in xk , k ∈ NR the statement follows as in the proof of [7], Proposition 22. 41 Consider increments in xk, k ∈ NC2. We start with observing that for hk ≥ 0 (xi + δkihi) ∂2Ptf ∂x2j (x+ hkek)− xi ∂ 2Ptf ∂x2j (x) = δkihiE NC2 xNC2+hkek [ ∂2 ∂x2j Gt,xNR ( INRt , x NC2 t )] + xi ( ENC2 xNC2+hkek −ENC2 xNC2 )[ ∂2 ∂x2j Gt,xNR ( INRt , x NC2 t )] ≡ E1 +E2, by arguing as in the proof of [7], Proposition 22. The bound on E1 is derived as in that proof, using Lemmas 2.2.4(a) and 2.2.3(b). For E2 we use the decompositions (2.29), (2.30), (2.31) and notation from Lemma 2.2.9 with ρ = 12 . Recall the notation G k t,xNR from (2.42) and the definition of Rk and Sk as in Notation 2.2.14. Then |E2| = xi ∣∣∣∣∣E [ ∂2 ∂x2j Gkt,xNR ( xNC2 , RN ′t + I2(t) + I h 3 (t), SN ′t +X ′ 0(t) ) − ∂ 2 ∂x2j Gkt,xNR ( xNC2 , RNt + I2(t), SNt +X ′ 0(t) )]∣∣∣∣∣ ≤ xi ∣∣∣∣∣E [ ∂2 ∂x2j Gkt,xNR ( xNC2 , RN ′t + I2(t) + I h 3 (t), SN ′t +X ′ 0(t) ) − ∂ 2 ∂x2j Gkt,xNR ( xNC2 , RN ′t + I2(t), SN ′t +X ′ 0(t) )]∣∣∣∣∣ + xi ∣∣∣∣∣E [ ∂2 ∂x2j Gkt,xNR ( xNC2 , RN ′t + I2(t), SN ′t +X ′ 0(t) ) − ∂ 2 ∂x2j Gkt,xNR ( xNC2 , RNt + I2(t), SNt +X ′ 0(t) )]∣∣∣∣∣ ≡ E2a +E2b. E2a can be bounded as in [7], using Lemmas 2.2.4(b) and 2.2.3(b), and the independence of xNC2 and Ih3 (t). Next turn to E2b. Recall that Sn = Sn(t) =∑n l=1 el(t), Rn = Rn(t) = ∑n l=1 rl(t) and pk(w) = e −wwk/k!. In the first term of E2b we may condition on N ′ t as it is independent from the other random variables and in the second term we do the same for Nt. Thus, if w ′ = w + hk γ0kt 42 and w = xk 2γ0kt , then by Lemma 2.2.4(a) and Lemma 2.2.3(b), E2b = xi ∣∣∣∣ ∞∑ n=0 (pn(w ′)− pn(w))E [ ∂2 ∂x2j Gkt,xNR ( xNC2 , Rn + I2(t), Sn +X ′ 0(t) )]∣∣∣∣ ≤ cxi ∞∑ n=0 ∣∣∣∣∣ ∫ w′ w p′n(u)du ∣∣∣∣∣ ‖f ‖∞ ×   ENC2 [( I (j) t )−1] , k /∈ Cj , min i∈Cj\{k} { ENC2 [(∫ t 0 x (i) s ds )−1]} ∧ E [(∫ t 0 X ′0(s)ds )−1] , k ∈ Cj   ≤ c ‖f ‖∞ xi ∞∑ n=0 ∣∣∣∣∣ ∫ w′ w p′n(u)du ∣∣∣∣∣ t−1 minl∈Cj { (t+ xl) −1} , where we used that X ′0 starts at xk 2 and thus by Lemma 2.2.3(b) E [(∫ t 0 X ′0(s)ds )−1] ≤ ct−1 ( t+ xk 2 )−1 ≤ ct−1(t+ xk)−1. We therefore obtain with i ∈ Cj E2b ≤ c ‖f ‖∞ xi ∫ w′ w ∞∑ n=0 pn(u) |n− u| u dut−1(t+ xi)−1 ≤ c ‖f ‖∞ ((∫ w′ w 1√ u du ) ∧ (∫ w′ w 2du )) t−1, where we used ∑∞ n=0 pn(u) |n−u| u = 1 uE|N − u| ≤ 1u √ E|N − u|2 = 1√ u and∑∞ n=0 pn(u) |n−u| u ≤ ∑∞ n=0 pn(u) ( n u + 1 ) = E|N |u + 1 = 2 with N being Poisson distributed with parameter u. Hence E2b ≤ c ‖f ‖∞ (w′ − w) ( 1√ w ∧ 2 ) t−1 = c ‖f ‖∞ hk t ( √ t√ xk ∧ 2 ) t−1. As ( 1√ xk ∧ 2√ t ) ≤ c 1√ t+xk we finally get E2b ≤ c ‖f ‖∞ t−3/2hk(t+ xk)−1/2. The bounds (2.56) and (2.57) can be derived from the first two by an argu- ment similar to the one used in the proof of Proposition 2.2.13 (alternatively refer to the end of the proof of Proposition 22 in [7]). In what follows recall Notation 2.2.23. 43 Proposition 2.2.25. If f is a bounded Borel function on S0, then for all x, h ∈ S0, i ∈ NC2 and arbitrary k ∈ V ,∣∣∣∣ ∂∂xiPtf(x+ hkek)− ∂ ∂xi Ptf(x) ∣∣∣∣ ≤ c ‖f ‖∞t3/2 |hk|T− 1 2 k ( t, xNC2 ) (2.58) and∣∣∣∣(x+ hkek)i ∂2Ptf∂x2i (x+ hkek)− xi ∂2Ptf ∂x2i (x) ∣∣∣∣ ≤ c ‖f ‖∞t3/2 |hk|T− 1 2 k ( t, xNC2 ) . (2.59) If f ∈ Sα, then∣∣∣∣ ∂∂xiPtf(x+ hkek)− ∂ ∂xi Ptf(x) ∣∣∣∣ ≤ c|f |αtα2− 32 |hk|T− 12k (t, xNC2) and∣∣∣∣(x + hkek)i ∂2Ptf∂x2i (x+ hkek)− xi ∂2Ptf ∂x2i (x) ∣∣∣∣ ≤ c|f |αtα2− 32 |hk|T− 12k (t, xNC2) . Proof. Proposition 2.2.25 is an extension of Proposition 23 in [7]. The last two inequalities follow from the first two by an argument similar to the one used in the proof of Proposition 2.2.13 (alternatively refer to the end of the proof of Proposition 22 in [7]). As the proof of (2.58) is similar to, but much easier than, that of (2.59), we only prove the latter. As usual we may assume f is bounded and continuous. Recall the notation ∆G+i,+i t,xNR (X, ν, ν′) from (2.34). Proposition 2.2.11 gives (Ptf)ii(x) = 4∑ n=1 ENC2 [ ∆nGt,xNR ( xNC2 )] , (2.60) where ∆1Gt,xNR (X) ≡ ∫ ∫ ∆G+i,+i t,xNR (X, ν, ν′)1{νt=ν′t=0}dN0(ν)dN0(ν ′), ∆2Gt,xNR (X) ≡ ∫ ∫ ∆G+i,+i t,xNR (X, ν, ν′)1{νt>0,ν′t=0}dN0(ν)dN0(ν ′), ∆3Gt,xNR (X) ≡ ∫ ∫ ∆G+i,+i t,xNR (X, ν, ν′)1{νt=0,ν′t>0}dN0(ν)dN0(ν ′) and ∆4Gt,xNR (X) ≡ ∫ ∫ ∆G+i,+i t,xNR (X, ν, ν′)1{νt>0,ν′t>0}dN0(ν)dN0(ν ′) (∗) = c t2 ∫ ∫ ∆G+i,+i t,xNR (X, ν, ν′)1{νt>0,ν′t>0}dP ∗ t (ν)dP ∗ t (ν ′). 44 Let us consider first the increments in xk , k ∈ NC2. Increments in xk , k ∈ NR will follow at the end of this section in Lemma 2.2.30. Let hk ≥ 0 and use (2.60) to obtain |(x+ hkek)i(Ptf)ii(x+ hkek)− xi(Ptf)ii(x)| (2.61) ≤ 4∑ n=1 ∣∣∣xi (ENC2xNC2+hkek −ENC2xNC2 )[ ∆nGt,xNR ( xNC2 )]∣∣∣ + hk |(Ptf)kk(x + hkek)| . The last term on the right hand side can be bounded via (2.44) as follows: hk |(Ptf)kk(x+ hkek)| ≤ hk c ‖f ‖∞ t(t+ xk) ≤ c ‖f ‖∞ hkt−3/2(t+ xk)−1/2, where we used hk ≥ 0. In the following Lemmas 2.2.26, 2.2.27 and 2.2.29 we again use the decom- positions from Lemma 2.2.9 with ρ = 12 to bound the first four terms in (2.61). Lemma 2.2.26. For k ∈ NC2 (and i ∈ NC2) we have∣∣∣xi (ENC2xNC2+hkek −ENC2xNC2 )[ ∆1Gt,xNR ( xNC2 )]∣∣∣ ≤ c ‖f ‖∞ t3/2(t+ xk)1/2 hk. Proof. This Lemma corresponds to Lemma 24 in [7]. In [7] one considered ∆G+i,+i(·) as a second order difference, thus obtaining terms involving (t+ xi)−2. In our setting this method will not work for i 6= k as we do in fact need terms of the form (t+ xi) −1(t+ xk)−1. Instead, we shall bound the left hand side by reasoning as for the E2-term in Proposition 22 of [7] (part of the proof can be found in this paper in the proof of Proposition 2.2.24), but with ∂ 2 ∂x2j G(·), j ∈ NR replaced by ∆G+i,+i(·), i ∈ NC2. Lemma 2.2.27. For k ∈ NC2 (and i ∈ NC2) and n = 2, 3 we have∣∣∣xi (ENC2xNC2+hkek −ENC2xNC2 )[ ∆nGt,xNR ( xNC2 )]∣∣∣ ≤ c ‖f ‖∞ t3/2(t+ xk)1/2 hk. (2.62) Proof. By symmetry we only need to consider n = 2. As before let w = xk 2γ0 k t , w′ = w+ hk γ0kt , Sn = ∑n l=1 el(t) and Rn = ∑n l=1 rl(t). Let Qh be the law of I h 3 (t) as defined after (2.31). As this random variable is independent of the others appearing below we may condition on it and use (2.29), (2.30) and (2.31) to 45 conclude xiE NC2 xNC2+hkek [ ∆2Gt,xNR ( xNC2 )] = xiE [∫ ∫ ∫ { Gk,+i,+i t,xNR ( xNC2 , I2(t) + z +RN ′t , X ′ 0(t) + SN ′t ;∫ t 0 νsds, νt, ∫ t 0 ν′sds, 0 ) −Gk,+i,+i t,xNR ( xNC2 , I2(t) + z +RN ′t , X ′ 0(t) + SN ′t ; 0, 0, ∫ t 0 ν′sds, 0 ) −Gk,+i,+i t,xNR ( xNC2 , I2(t) + z +RN ′t , X ′ 0(t) + SN ′t ; ∫ t 0 νsds, νt, 0, 0 ) +Gk,+i,+i t,xNR ( xNC2 , I2(t) + z +RN ′t , X ′ 0(t) + SN ′t ; 0, 0, 0, 0 )} × 1{νt>0}1{ν′t=0}dN0(ν)dN0(ν′)dQh(z) ] . When working under ENC2 xNC2 there is no Ih3 (t) term. Hence we obtain the same formula with z replaced by 0 and N ′t replaced by Nt. The difference of these terms can be bounded by a difference dealing with the change from z to 0 and the change from N ′t to Nt separately. For the second term we recall that pn(u) = e −uun/n! and observe that N ′t is independent of the other random variables. Hence we may condition on its value to see that the l.h.s. of (2.62) is at most xi ∣∣∣∣E [∫ ∫ ∫ { ∆Gk,+i,+i t,xNR ( xNC2 , I2(t) + z +RN ′t , X ′ 0(t) + SN ′t ;∫ t 0 νsds, νt, ∫ t 0 ν′sds, 0 ) −∆Gk,+i,+i t,xNR ( xNC2 , I2(t) +RN ′t , X ′ 0(t) + SN ′t ; ∫ t 0 νsds, νt, ∫ t 0 ν′sds, 0 )} × 1{νt>0}1{ν′t=0}dN0(ν)dN0(ν′)dQh(z) ]∣∣∣∣ + xi ∣∣∣∣ ∞∑ n=0 (pn(w ′)− pn(w))E [∫ ∫ ∆Gk,+i,+i t,xNR ( xNC2 , I2(t) +Rn, X ′ 0(t) + Sn; ∫ t 0 νsds, νt, ∫ t 0 ν′sds, 0 ) × 1{νt>0}1{ν′t=0}dN0(ν)dN0(ν′) ]∣∣∣∣ ≡ Ea +Eb. The first term can be rewritten as the sum of two second order differences (one in z, one in ∫ t 0 ν′sds). Together with Lemma 2.2.6, Lemma 2.2.4(b) and 46 Lemma 2.2.3(b) we therefore obtain (terms including empty sums are again understood as being zero) Ea ≤ 2xic ‖f ‖∞ ∑ j1:j1∈R̄i ∑ j2:j2∈R̄k E     ( I (j1) t )−1 , k /∈ Cj1 , min m∈Cj1\{k} {(∫ t 0 x (m) s ds )−1} ∧ (∫ t 0 X ′ 0(s)ds )−1 , k ∈ Cj1   × (∫ t 0 X ′0(s)ds )−1]∫ ∫ t 0 ν′sdsdN0(ν ′)N0[νt > 0] ∫ zdQh(z) ≤ xic ‖f ‖∞ t−2(t+ xi)−1(t+ xk)−1tt−1hkt ≤ c ‖f ‖∞ hkt−3/2(t+ xk)−1/2. Turning to Eb observe that we have the sum of two first order differences (both in ∫ t 0 ν ′ sds). Together with the triangle inequality, Lemma 2.2.4(b) and Lemma 2.2.3(b) we therefore obtain Eb ≤ cxi ∞∑ n=0 ∣∣∣∣∣ ∫ w′ w p′n(u)du ∣∣∣∣∣ ‖f ‖∞ ∑ j1:j1∈R̄i E     ( I (j1) t )−1 , k /∈ Cj1 , min m∈Cj1\{k} {(∫ t 0 x (m) s ds )−1} ∧ (∫ t 0 X ′0(s)ds )−1 , k ∈ Cj1     × ∫ ∫ t 0 ν′sdsdN0(ν ′)N0[νt > 0] ≤ cxi ∞∑ n=0 ∣∣∣∣∣ ∫ w′ w p′n(u)du ∣∣∣∣∣ ‖f ‖∞ t−1(t+ xi)−1tt−1. Now proceed again as in the estimation of E2b in the proof of Proposi- tion 2.2.24 to get Eb ≤ cxit−1/2hk(t+ xk)−1/2 ‖f ‖∞ t−1(t+ xi)−1tt−1 ≤ c ‖f ‖∞ hkt−3/2(t+ xk)−1/2. The above bounds on Ea and Eb give the required result. Notation 2.2.28. Let Gm,n6=m t,xNR (X,Yt, Zt, Y ′ t , Z ′ t) ≡ { Gm,n t,xNR (X,Yt, Zt, Y ′ t , Z ′ t) if n 6= m Gm t,xNR (X,Yt, Zt) if n = m. Expressions such as Gm,n6=m,+k,+l t,xNR ( X,Yt, Zt, Y ′ t , Z ′ t; ∫ t 0 ηsds, θt, ∫ t 0 η ′ sds, θ ′ t ) will be defined similarly. 47 Lemma 2.2.29. For k ∈ NC2 (and i ∈ NC2) we have∣∣∣xi (ENC2xNC2+hkek −ENC2xNC2 )[ ∆4Gt,xNR ( xNC2 )]∣∣∣ ≤ c ‖f ‖∞ t3/2(t+ xk)1/2 hk. Proof. Let E ≡ xi ∣∣∣(ENC2xNC2+hkek −ENC2xNC2 )[ ∆4Gt,xNR ( xNC2 )]∣∣∣ . (2.63) We use the same setting and notation as in Lemma 2.2.27. Proceeding as in the estimation of the l.h.s. in (2.62), thereby not only decomposing x(k) but also x(i) (the respective parts of the decomposition of x(k) and x(i) are designated via upper indices k resp. i and are independent for k 6= i), we have xiE NC2 xNC2+hkek [ ∆4Gt,xNR ( xNC2 )] = xiE [∫ ∫ ∫ ∆Gk,i6=k,+i,+i t,xNR ( xNC2 , I (k) 2 (t) + z +R (k) N ′(k) t , X ′(k) 0 (t) +S (k) N ′(k) t , I (i) 2 (t) +R (i) N (i) t , X ′(i) 0 (t) + S (i) N (i) t ; ∫ t 0 νsds, νt, ∫ t 0 ν′sds, ν ′ t ) × 1{νt>0}1{ν′t>0}dN0(ν)dN0(ν′)dQh(z) ] . Now let for k = i Ĝn(z) ≡ E [ Gkt,xNR ( xNC2 , I (k) 2 (t) + z +R (k) n , X ′(k) 0 (t) + S (k) n )] , respectively for k 6= i, Ĝn ( z,N ′(k) t ) ≡ E [ Gk,i t,xNR ( xNC2 , I (k) 2 (t) + z +R (k) N ′(k) t , X ′(k) 0 (t) + S (k) N ′(k) t , I (i) 2 (t) +R (i) n , X ′(i) 0 (t) + S (i) n )] . Note that the expectation in the definition of Ĝn ( z,N ′(k) t ) excludes the random variable N ′(k) t . Use w ′(k) = xk 2γ0 k t + hk γ0 k t (i.e. ρ = 1/2) to obtain for k = i xkE NC2 xNC2+hkek [ ∆4Gt,xNR ( xNC2 )] (2.64) (∗) = c xk t2 ∞∑ n=0 pn ( w ′(k) )∫ ( Ĝn+2 − 2Ĝn+1 + Ĝn ) (z)dQh(z), and use w(i) = xi 2γ0i t to obtain for k 6= i xiE NC2 xNC2+hkek [ ∆4Gt,xNR ( xNC2 )] (2.65) (∗) = c xi t2 ∞∑ n=0 pn ( w(i) ) E [∫ ( Ĝn+2 − 2Ĝn+1 + Ĝn )( z,N ′(k) t ) dQh(z) ] . 48 A similar argument holds for xiE NC2 xNC2 [ ∆4Gt,xNR ( xNC2 )] . Indeed, if k = i replace z by 0 and replace w ′(k) by w(k) = xk 2γ0kt in (2.64). If k 6= i replace z by 0 and replace N ′(k) t by N (k) t in (2.65). Let us first investigate the case k = i. Define Ĥn(z) = Ĝn(z)− Ĝn(0) to get for E as in (2.63), E ≤ cxk t2 ∣∣∣∣∣ ∞∑ n=0 pn ( w ′(k) )∫ ( Ĥn+2 − 2Ĥn+1 + Ĥn ) (z)dQh(z) ∣∣∣∣∣ + c xk t2 ∣∣∣∣∣ ∞∑ n=0 ( pn ( w ′(k) ) − pn ( w(k) ))( Ĝn+2 − 2Ĝn+1 + Ĝn ) (0) ∣∣∣∣∣ ≡ E1 +E2. We can bound E1 by c xk t2 ∞∑ n=0 ∣∣∣(pn−2 − 2pn−1 + pn)(w′(k))∣∣∣ sup n≥0 ∣∣∣∣ ∫ Ĥn(z)dQh(z) ∣∣∣∣ , where pn(w) ≡ 0 if n < 0. By using qn(w) = wpn(w) and ∑∞ n=0 |(qn−2−2qn−1+ qn)(w)| ≤ 2 (see [7], (109)) we obtain E1 ≤ cxk t2 1 w′(k) sup n≥0 ∣∣∣∣ ∫ Ĥn(z)dQh(z) ∣∣∣∣ . Next observe that Ĥn(z) is zero for k ∈ N2 (recall that for k ∈ N2 the indicated change from ∫ t 0 x (k) s ds into I (k) 2 (t) + z +R (k) n resp. I (k) 2 (t) +R (k) n has no impact on the terms under consideration) and is a first order difference for k ∈ NC for which we obtain as usual∣∣∣∣ ∫ Ĥn(z)dQh(z) ∣∣∣∣ ≤ c ‖f ‖∞ t−1(t+ xk)−1 ∫ zdQh(z) ≤ c ‖f ‖∞ t−1(t+ xk)−1hkt ≤ c ‖f ‖∞ hkt−1/2(t+ xk)−1/2. Together with w(k) = xk 2γ0kt and w ′(k) = xk 2γ0kt + hk γ0kt this gives E1 ≤ c ‖f ‖∞ hkt−3/2(t+ xk)−1/2. 49 For E2 we obtain with ‖G‖∞≤‖f ‖∞ and Fubini’s theorem E2 ≤ c ‖f ‖∞ xk t2 ∞∑ n=0 ∣∣∣(pn−2 − 2pn−1 + pn)(w′(k)) (2.66) − (pn−2 − 2pn−1 + pn) ( w(k) )∣∣∣ ≤ c ‖f ‖∞ xk t2 ∫ w′(k) w(k) ∞∑ n=0 ∣∣(p′n−2 − 2p′n−1 + p′n)(u)∣∣ du. As pn(u) = e −u un n! we have p ′ n(u) = −pn(u) + pn−1(u) and thus we obtain in case 0 < u < 1 for the integrand ∞∑ n=0 ∣∣(p′n−2 − 2p′n−1 + p′n)(u)∣∣ ≤ 8. For u ≥ 1 we obtain for the integrand as an upper bound p0(u) + p1(u) ∣∣∣∣3 1u − 1 ∣∣∣∣+ ∞∑ n=2 pn(u) ∣∣∣∣n(n− 1)(n− 2)u3 − 3n(n− 1)u2 + 3nu − 1 ∣∣∣∣ ≤ e−u(1 + 3 + u) + 1 u3 ∞∑ n=2 pn(u) ∣∣(n− u)3 − 3n(n− u) + 2n∣∣ ≤ e−u(4 + u) + 1 u3 ( E|Nu − u|3 + 3 √ EN2uE(Nu − u)2 + 2ENu ) , where Nu is Poisson with mean u. Note that E|Nu − u|m ≤ cmum/2 for m ∈ N and u ≥ 1. We also have ENu = u and EN2u = u2 + u. This yields as an upper bound for the integrand in (2.66) for u ≥ 1 cu− 3 2 + 1 u3 ( c3u 3/2 + 3 √ (u2 + u) c2u1 + 2u ) ≤ cu− 32 . We thus get for E2 E2 ≤ c ‖f ‖∞ xk t2 ∫ w′(k) w(k) ( u+ 1 2γ0k )−3/2 du ≤ c ‖f ‖∞ xk t2 ∣∣∣w′(k) − w(k)∣∣∣ (w(k) + 1 2γ0k )−3/2 ≤ c ‖f ‖∞ xk t2 hk t ( xk + t 2γ0kt )−3/2 ≤ c ‖f ‖∞ hkt−3/2 (t+ xk)−1/2 . Together with the bound on E1 the assertion now follows for k = i. 50 Next investigate the case k 6= i. Define Ĥ1n ( z,N ′(k) t ) = Ĝn ( z,N ′(k) t ) − Ĝn ( 0, N ′(k) t ) , Ĥ2n ( N ′(k) t , N (k) t ) = Ĝn ( 0, N ′(k) t ) − Ĝn ( 0, N (k) t ) to get E ≤ cxi t2 ∣∣∣∣ ∞∑ n=0 pn ( w(i) ) E [∫ ( Ĥ1n+2 − 2Ĥ1n+1 + Ĥ1n )( z,N ′(k) t ) dQh(z) ]∣∣∣∣ + c xi t2 ∣∣∣∣ ∞∑ n=0 pn ( w(i) ) E [∫ ( Ĥ2n+2 − 2Ĥ2n+1 + Ĥ2n )( N ′(k) t , N (k) t ) dQh(z) ]∣∣∣∣. Recall that the expectation in the definition of Ĝn ( z,N ′(k) t ) and thus of Ĥ1n ( z,N ′(k) t ) excludes the random variable N ′(k) t . To bound E we thus take expectation w.r.t. N ′(k) t , too. Rewriting this yields E ≤ cxi t2 ∞∑ n=0 ∣∣∣(pn−2 − 2pn−1 + pn)(w(i))∣∣∣ × sup n≥0 { E [∣∣∣∣ ∫ Ĥ1n ( z,N ′(k) t ) dQh(z) ∣∣∣∣ ] +E [∣∣∣∣ ∫ Ĥ2n ( N ′(k) t , N (k) t ) dQh(z) ∣∣∣∣ ]} and by using qn(w) = wpn(w) and ∑∞ n=0 |(qn−2− 2qn−1 + qn)(w)| ≤ 2 again we obtain E ≤ cxi t2 1 w(i) sup n≥0 { E [∣∣∣∣ ∫ Ĥ1n ( z,N ′(k) t ) dQh(z) ∣∣∣∣ ] +E [∣∣∣∣ ∫ Ĥ2n ( N ′(k) t , N (k) t ) dQh(z) ∣∣∣∣ ]} . Next observe that Ĥ1n ( z,N ′(k) t ) is zero for k ∈ N2 and is a first order difference for k ∈ NC for which we obtain∣∣∣∣ ∫ Ĥ1n ( z,N ′(k) t ) dQh(z) ∣∣∣∣ ≤ c ‖f ‖∞ t−1(t+ xk)−1 ∫ zdQh(z) ≤ c ‖f ‖∞ hkt−1/2(t+ xk)−1/2. 51 The other term can be bounded as follows: ∣∣∣Ĥ2n(N ′(k)t , N (k)t )∣∣∣ ≤ ∞∑ N=0 ∣∣∣pN(w′(k))− pN(w(k))∣∣∣ × ∣∣∣E[Gk,it,xNR ( xNC2 , I (k) 2 (t) +R (k) N , X ′(k) 0 (t) + S (k) N , I (i) 2 (t) +R (i) n , X ′(i) 0 (t) + S (i) n )]∣∣∣ ≤ ∞∑ N=0 ∣∣∣pN(w′(k))− pN(w(k))∣∣∣ ‖G‖∞, where w(k) = xk 2γ0kt and w ′(k) = xk 2γ0kt + hk γ0kt . As done before in the proof of Proposition 2.2.24 we use ∞∑ N=0 ∣∣∣∣∣ ∫ w′(k) w(k) p′N(u)du ∣∣∣∣∣ ≤ ct−1/2hk(t+ xk)−1/2 to finally get with ‖G‖∞≤‖f ‖∞∣∣∣∣ ∫ Ĥ2n ( N ′(k) t , N (k) t ) dQh(z) ∣∣∣∣ ≤ ct−1/2hk(t+ xk)−1/2 ‖f ‖∞ . Plugging our results into our estimate for E we get E ≤ cxi t2 t xi c ‖f ‖∞ hkt−1/2(t+ xk)−1/2 ≤ c ‖f ‖∞ hkt−3/2(t+ xk)−1/2, which proves our assertion. Finally we consider the increments in xk, k ∈ NR. Lemma 2.2.30. If f is a bounded Borel function on S0, then for all x, h ∈ S0, i ∈ NC2 and k ∈ NR∣∣∣∣xi ∂2Ptf∂x2i (x+ hkek)− xi ∂2Ptf ∂x2i (x) ∣∣∣∣ ≤ c ‖f ‖∞t3/2 |hk|minl∈Ck { (t+ xl) −1/2 } . Proof. Except for the necessary adaptations, already used in the proofs of the preceding assertions, the proof proceeds analogously to Lemma 27 in [7]. Continuation of the proof of Proposition 2.2.25. Use Lemmas 2.2.26, 2.2.27 and 2.2.29 in (2.61) together with the calculation following (2.61) to obtain the bound for increments in xk, k ∈ NC2. Lemma 2.2.30 gives the corresponding bound for increments in xk, k ∈ NR which completes the proof of (2.59). 52 2.3 Proof of Uniqueness As in Section 3, [7], it is relatively straightforward to use the results from the previous sections on the semigroup Pt to prove bounds on the resolvent Rλ of Pt. We shall then use these bounds to complete the proof of uniqueness of solu- tions to the martingale problem MP(A,ν) satisfying Hypothesis 2.1.1 and 2.1.2, where ν is a probability on S =  x ∈ Rd+ : ∏ j∈R  ∑ i∈Cj xi + xj   > 0   (recall (2.3) and Lemma 2.1.5) and Af(x) = ∑ j∈R γj(x)  ∑ i∈Cj xi  xjfjj(x) +∑ j /∈R γj(x)xjfjj(x) + ∑ j∈V bj(x)fj(x). (2.67) The proof of uniqueness is identical to the one in [7] except for minor changes such as the replacement of xcj by ∑ i∈Cj xi at the appropriate places. Note in particular the change in the definition of the state space S. In what follows we shall give a sketch of the proofs and indicate where statements have to be modified. For explicit calculations the reader is referred to [7], Sections 3 and 4. Notation 2.3.1. For i ∈ NC2 let ȳi = ( {yj}j∈R̄i , yi ) , ȳiēi = ∑ j∈R̄i yjej + yiei and R̄i = R |R̄i| × R+, (2.68) where we understand this to be ȳi = (yi) in case i ∈ N2, i.e. R̄i = ∅. For f ∈ C2b (S0) let ∂f ∂x̄i = ({ ∂ ∂xj f } j∈R̄i , ∂ ∂xi f ) , ∣∣∣∣ ∂f∂x̄i ∣∣∣∣ = ∑ j∈R̄i ∣∣∣∣ ∂∂xj f ∣∣∣∣+ ∣∣∣∣ ∂∂xi f ∣∣∣∣ (2.69) and ∣∣∣∣ ∣∣∣∣ ∂f∂x̄i ∣∣∣∣ ∣∣∣∣ ∞ = sup {∣∣∣∣ ∂f∂x̄i (x) ∣∣∣∣ : x ∈ S0 } , (2.70) where S0 = {x ∈ Rd : xi ≥ 0 for all i ∈ NC2} as defined in (2.9). Also introduce ∆if =   { xi ∂2 ∂x2j f } j∈R̄i , xi ∂2 ∂x2i f   . Define |∆if | and ‖∆if ‖∞ similarly to (2.69) and (2.70). 53 With the help of these notations A0 (see (2.6)) can be rewritten to A0f(x) = ∑ j∈V b0jfj(x) + ∑ j∈NR γ0j  ∑ i∈Cj xi   fjj(x) + ∑ i∈NC2 γ0i xifii(x) (2.71) = ∑ i∈NC2 〈 b0 i, ∂f ∂x̄i (x) 〉 + ∑ i∈NC2 〈 γ0 i ,∆if(x) 〉 , where 〈·, ·〉 denotes the standard scalar product in Rk, k ∈ N. To prevent over- counting in case R̄i1 ∩ R̄i2 6= ∅ for i1 6= i2, i1, i2 ∈ NC (see also definition (2.68)) the vector b0i was replaced by b 0 i in the above formula, where b 0 i has certain coordinates set to zero so that the above equality holds. The same applies to the vector γ0 i . The details are left to the interested reader. Theorem 2.3.2. There is a constant c such that for all f ∈ Cαw(S0), λ ≥ 1 and k ∈ NC2, (a) ∣∣∣∣ ∣∣∣∣∂Rλf∂x̄k ∣∣∣∣ ∣∣∣∣ ∞ + ||∆kRλf ||∞ ≤ cλ−α/2|f |Cαw . (b) ∣∣∣∣∂Rλf∂x̄k ∣∣∣∣ Cαw + |∆kRλf |Cαw ≤ c|f |Cαw . Note. This result is slightly weaker than the corresponding Theorem 34 in [7] as |f |α,k is replaced by |f |Cαw in (a). Proof. Firstly we obtain a result similar to Proposition 30 in [7]. This is an easy consequence of Proposition 2.2.13 and Proposition 2.2.16, using the equivalence of norms shown in Theorem 2.2.20 and states that there is a constant c such that (a) For all f ∈ Cαw(S0), t > 0, x ∈ S0, and i ∈ NC2,∣∣∣∣∂Ptf∂x̄i (x) ∣∣∣∣ ≤ c|f |Cαw tα/2−1/2(t+ xi)−1/2 ≤ c|f |Cαw tα/2−1, (2.72) and ‖∆iPtf ‖∞≤ c|f |Cαw tα/2−1. (2.73) (b) For all f bounded and Borel on S0 and all i ∈ NC2,∣∣∣∣ ∣∣∣∣∂Ptf∂x̄i ∣∣∣∣ ∣∣∣∣ ∞ ≤ c ‖f ‖∞ t−1. Note in particular that Theorem 2.2.20 gave Cαw = Sα and that every function in Cαw(S0) is by definition bounded. Secondly, an easy consequence of Propositions 2.2.24, 2.2.25 and the triangle inequality, using the equivalence of norms shown in Theorem 2.2.20 and the equivalence of the maximum norm and Euclidean norm of finite dimensional 54 vectors, is a result similar to Proposition 32, [7]: There is a constant c such that for all f ∈ Cαw(S0), i, k ∈ NC2 and h̄i ∈ R̄i, (a)∣∣∣∣∂Ptf∂x̄k ( x+ h̄iēi )− ∂Ptf ∂x̄k (x) ∣∣∣∣ ≤ c|f |Cαw t−3/2+α/2(t+ xi)−1/2 ∣∣h̄i∣∣ , (2.74) (b)∣∣∆k(Ptf)(x+ h̄iēi)−∆k(Ptf)(x)∣∣ ≤ c|f |Cαw t−3/2+α/2(t+ xi)−1/2 ∣∣h̄i∣∣ . (2.75) Finally recall that Rλf(x) = ∫∞ 0 e −λtPtf(x)dt is the resolvent associated with Pt. Now the remainder of the proof works as in the proof of Theorem 34 in [7]: Part (a) of Theorem 2.3.2 is obtained by integrating (2.72) resp. (2.73) over time. Part (b) follows by integrating (2.72) resp. (2.73) over the time- interval from zero to some fixed value t̃ > 0 and (2.74) resp. (2.75) over the time interval from t̃ to infinity. Appropriate choices for t̃ now yield the required bounds. Here the choices of t̃ are in fact easier due to the replacement of | · |α,i in [7] by | · |Cαw . Proof of Theorem 2.1.6. The existence of a solution to the martingale problem for MP(A, ν) follows by standard methods (a result of Skorokhod yields ex- istence of approximating solutions, then use a tightness-argument), e.g. see the proof of Theorem 1.1 in [1]. Note in particular that Lemma 2.1.5 ensures that solutions remain in S ⊂ Rd+. The uniform boundedness in M of the term E [∑ i ∣∣∣XM,iT ∣∣∣] that appears in the proof of Theorem 1.1 in [1] can easily be re- placed by the uniform boundedness in M of E [∑ i∈V (X M,i T ) 2 ] via a Gronwall- type argument. At the end of this section we shall reduce the proof of uniqueness to the following theorem. The theorem investigates uniqueness of a perturbation of the operator A0 as defined in (2.6) (also refer to (2.71)) with coefficients satisfying (2.7) and (2.8). A0 is the generator of a unique diffusion on S(x0) given by (2.9) with semigroup Pt and resolvent Rλ given by (2.11). For the definition of M0 refer to (2.14). In what follows x0 ∈ S will be arbitrarily fixed. Theorem 2.3.3. Assume that Ãf(x) = ∑ j∈NR γ̃j(x)  ∑ i∈Cj xi   fjj(x) (2.76) + ∑ j∈NC2 γ̃j(x)xjfjj(x) + ∑ j∈V b̃j(x)fj(x), x ∈ S ( x0 ) , where b̃k : S ( x0 )→ R and γ̃k : S(x0)→ (0,∞), Γ̃ = d∑ k=1 ||γ̃k||Cαw + ∣∣∣∣∣∣b̃k∣∣∣∣∣∣Cαw <∞. 55 Let ̃0 = d∑ k=1 ∣∣∣∣γ̃k − γ0k∣∣∣∣∞ + ∣∣∣∣∣∣b̃k − b0k∣∣∣∣∣∣∞ , where b0k, γ 0 k, k ∈ V satisfy (2.7). Let Bf = (à − A0)f . (a) There exists 1 = 1(M 0) > 0 and λ1 = λ1(M 0, Γ̃) ≥ 0 such that if ̃0 ≤ 1 and λ ≥ λ1 then BRλ : Cαw → Cαw is a bounded operator with ‖BRλ‖≤ 1/2. (b) If we assume additionally that γ̃k and b̃k are Hölder continuous of index α ∈ (0, 1), constant outside a compact set and b̃k|{xk=0} ≥ 0 for all k ∈ V \NR, then the martingale problem MP(Ã, ν) has a unique solution for each probability ν on S(x0). Proof. Let R̃λ be the associated resolvent operator of the perturbation operator Ã. Using the definition B = à −A0 and recalling (2.71) we get for f ∈ Cαw that ‖BRλf ‖Cαw≤ ∑ i∈NC2 ∣∣∣∣ ∣∣∣∣ 〈( b̃(x) − b0 ) i , ∂Rλf ∂x̄i (x) 〉∣∣∣∣ ∣∣∣∣ Cαw + ∑ i∈NC2 ∣∣∣∣ ∣∣∣∣ 〈( γ̃(x) − γ0) i ,∆iRλf(x) 〉∣∣∣∣ ∣∣∣∣ Cαw . Using (2.46) (recall in particular the discussion on the reasons for using two different norms from Remark 2.2.21) we obtain for instance for arbitrary i ∈ NC and j ∈ R̄i∣∣∣∣(b̃j(x)− b0j) ∂Rλf∂xj (x) ∣∣∣∣ Cαw ≤ c [∣∣∣∣∣∣b̃j(x) − b0j ∣∣∣∣∣∣Cαw ∣∣∣∣ ∣∣∣∣∂Rλf∂xj (x) ∣∣∣∣ ∣∣∣∣ ∞ + ∣∣∣∣∣∣b̃j(x) − b0j ∣∣∣∣∣∣∞ ∣∣∣∣∂Rλf∂xj (x) ∣∣∣∣ α ] ≤ c [( Γ̃ +M0 ) λ−α/2|f |Cαw + ̃0|f |Cαw ] by Theorem 2.3.2, (2.50) and the assumptions of this theorem. By arguing similarly for the other terms we get indeed ‖ BRλf ‖Cαw≤ 12 ‖ f ‖Cαw for λ big enough thus finishing the proof of part (a). For part (b) we proceed as in the proof of [7], Theorem 37. The proof of Theorem 37 in [7] involves the proof of Lemma 38 in [7], where one shows that for f ∈ Cαw R̃λf = Rλf + R̃λBRλf. (2.77) Note that the proof of Lemma 38 relies amongst others on an estimate, derived in Corollary 33 of [7], which we now obtain for free in Proposition 2.2.11 as we treated all vertices in one step only. The proof of Theorem 37 now concludes as follows. Iteration of (2.77) yields R̃λf(x) = ∞∑ n=0 Rλ((BRλ)nf)(x). 56 Using ‖BRλ ‖Cαw≤ 1/2 from part (a) and ‖f ‖∞≤‖f ‖Cαw we get λ ‖Rλ((BRλ)nf)‖∞≤‖(BRλ)nf ‖∞≤‖(BRλ)nf ‖Cαw≤ 2−n ‖f ‖Cαw . Thus the series converges uniformly and the error term approaches zero. The uniqueness of MP(Ã, ν) now follows from the uniqueness of its resolvents R̃λ. Continuation of the proof of Theorem 2.1.6. Recall “Step 1: Reduction of the problem”, in Subsection 2.1.5. The remainder of the proof of uniqueness of MP (A, δx0) works analogously to [7] (compare the proof of Theorem 4 on pp. 380-382 in [7]) except for minor changes, making again use of Lemma 2.1.5. The main step consists in using a localization argument of [13] (see e.g. the argument in the proof of Theorem 1.2 of [4]), which basically states that it is enough if for each x0 ∈ S the martingale problem MP (Ã, δx0) has a unique solution, where bi = b̃i and γi = γ̃i agree on some neighbourhood of x 0. By comparing the definition of A (see (2.67)) and à (see (2.76)) one chooses b̃k(x) = bk(x) for all k ∈ V, γ̃j(x) = xjγj(x) for j ∈ NR, γ̃j(x) =  ∑ i∈Cj xi   γj(x) for j ∈ R\NR γ̃j(x) = γj(x) for j /∈ R. 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MR532498 58 Chapter 3 Long-term Behaviour of a Cyclic Catalytic Branching System1 3.1 Introduction 3.1.1 Basics In this paper we investigate the long-term behaviour of the following system of stochastic differential equations (SDEs) for d ≥ 2: dX it = √ 2γiX itX i+1 t dB i t + d∑ j=1 Xjt qjidt, i ∈ {1, . . . , d}, (3.1) where Xd+1t ≡ X1t . We shall assume the γi and qji, i 6= j to be given positive constants and the X i0 ≥ 0, i ∈ {1, . . . , d} to be given initial conditions. (qji) is a Q-matrix modelling mutations from type j to type i. This system involves both cyclic catalytic branching and mutation between types. The extension of the cyclic setup to arbitrary networks (see Subsec- tion 3.2.6 at the end of this paper) is straightforward. Existence of solutions shall be shown by standard methods. To show weak uniqueness we shall employ the results of Dawson and Perkins [3] once we show that a solution does not hit 0 ∈ Rd in finite time. The given system of SDEs can be understood as a stochastic analogue to a system of ODEs for the concentrations yj , j = 1, . . . , d of a type Tj . Then yj/ẏj corresponds to the rate of growth of type Tj and one obtains the following ODEs (see Hofbauer and Sigmund [6]): for independent replication ẏj = bjyj , auto- catalytic replication ẏj = γjy 2 j and catalytic replication ẏj = γj (∑ i∈Cj yi ) yj . In the cyclic catalytic case type Tj+1 catalyzes the replication of type j, i.e. the growth of type j is proportional to the mass of type j + 1 present at time t. The cyclic catalytic case represents the simplest form of mutual help between different types. It was firstly introduced by Eigen and Schuster (see Eigen and Schuster [4]). 1A version of this chapter will be submitted for publication. Kliem, S.M. (2009) Long-term Behaviour of a Cyclic Catalytic Branching System. 59 The system of SDEs can be obtained as a limit of branching particle systems. The growth rate of types in the ODE setting now corresponds to the branching rate in the stochastic setting, i.e. type j branches at a rate proportional to the mass of type j + 1 at time t. Results on weak uniqueness for catalytic branching networks can be found for instance in [3] and Kliem [9]. The former proved weak uniqueness for cat- alytic replication under the restriction to networks with at most one catalyst per reactant, which includes the hypercyclic case. The latter removed this re- striction. Both papers allow more general diffusion- and drift- coefficients under some Hölder-continuity conditions. These conditions were weakened in Bass and Perkins [1] to continuity only. Our main interest shall be the long-time behaviour of the above system. In particular, we shall investigate survival and coexistence of types. Such questions naturally arise in biological competition models. For instance, Fleischmann and Xiong [5] investigated a cyclically catalytic super-Brownian motion. They showed global segregation (noncoexistence) of neighbouring types in the limit and other results on the finite time survival-extinction but they were not able to determine, if the overall sum dies out in the limit or not. In this paper we shall show that in our SDE-setup the overall sum converges to zero but does not hit zero in finite time. To further analyze the relative behaviour of types while they approach zero, we turned our attention to the normalized processes Y it ≡ X it/ ∑ j X j t - note that X i t/X j t = Y i t /Y j t - and showed weak convergence to a unique stationary distribution that does not charge the set where at least one of the coordinates is zero. 3.1.2 Main results and outline of the paper As a first step we shall show existence and nonnegativity of solutions X it , i ∈ {1, . . . , d} to the above SDE by standard methods in Subsection 3.2.1. As a next step we shall prove in Subsection 3.2.2 that the sum of all coordinates, i.e. St ≡ ∑d i=1X i t , converges to zero but does not hit zero in finite time a.s. We then establish the weak uniqueness of the system by Theorem 4 of [3] or Theorem 1.6 of [9]. Secondly, from Subsection 3.2.3 on we shall change our focus to the normal- ized processes, i.e. to Y it = X i t/St to get some insight on the relative behaviour of types. Existence of solutions follows again by standard methods and the weak uniqueness of solutions in [0, 1]d follows by establishing a connection between the system at hand and the original system of SDEs. In Subsection 3.2.4 we show that any stationary distribution for Yt does not charge the set where at least one of the coordinate processes becomes extinct. We shall use this result in Subsection 3.2.5 to prove weak convergence to a unique stationary distribution by adapting the proof of Theorem 2.3 of Dawson, Greven, den Hollander, Sun and Swart [2] to our setup. Finally, in Subsection 3.2.7 we shall give a complete analysis of the case d = 2 by using methods of speed and scale. 60 3.2 Main Results 3.2.1 Existence and nonnegativity Let (Ω,F , (F)t,P) be a filtered probability space that satisfies the usual condi- tions (cf. Rogers and Williams [10], Introduction to Chapter IV). Consider the following system of SDEs for d ≥ 2: dX it = √ 2γiX itX i+1 t dB i t + d∑ j=1 Xjt qjidt, i ∈ {1, . . . , d}, (3.2) where Xd+1t ≡ X1t . We shall assume the γi and qji, i 6= j to be given strictly positive constants and the X i0 ≥ 0, i ∈ {1, . . . , d} to be given initial conditions. As the qji model mutations from type j to type i we impose qii = − ∑ j:j 6=i qij ⇐⇒ ∑ j qij = 0. (3.3) Let qmax = max 1≤i,j≤d |qij |, qmin = min 1≤i,j≤d |qij | > 0 and γmax = max 1≤i≤d γi > 0. First we shall investigate the existence of solutions to (3.2). Lemma 3.2.1. There exists a solution to the given system of SDEs (3.2). Reference for the proof. Existence follows by standard methods, see for in- stance Theorem V.3.10 in Ethier and Kurtz [7]. Next, we shall show that all solutions to (3.2) stay in the first quadrant (here we replaced the terms under the square root with their absolute values to be able to consider solutions on all of Rd). For this purpose we shall first show that the local time of the coordinate processes at zero is zero. Corollary 3.2.2. Let i ∈ {1, . . . , d} be arbitrarily fixed. Then the local time l0t at zero of the process X it is zero. Proof. The proof proceeds along the lines of standard techniques for local times. Let i ∈ {1, . . . , d} be arbitrarily fixed. By [10], IV.(45.3) (“occupation density formula”) we have for ϕ(x) = 1[0,](x),  > 0,∫ t 0 ϕ(X is)d < X i >s= ∫ t 0 1{0≤Xis≤}2γ i ∣∣X isX i+1s ∣∣ ds = ∫  0 lat da = ∫ R lat ϕ(a)da. Next recall from Theorem IV.(44.2) that without loss of generality lat is right- continuous in a. We also know that the processes under consideration are con- tinuous. Hence, 0 ≤ l0t = lim ↓0+ 1  ∫  0 lat da ≤ lim ↓0+ 1  ∫ t 0 1{0<Xis≤}2γ i ∣∣X i+1s ∣∣ ds = 0, the last by dominated convergence, proving the assumption. 61 Notation 3.2.3. In what follows we shall denote the martingale part of X it by Mit ≡ ∫ t 0 √ 2γiX isX i+1 s dB i s, i ∈ {1, . . . , d}. Lemma 3.2.4. The processes X it , i ∈ {1, . . . , d} are nonnegative if we start at X i0 ≥ 0, ∀i ∈ {1, . . . , d}. We also obtain that the Mit, i ∈ {1, . . . , d} are martingales. Proof. To the purpose of proving this Lemma we shall use that ( X it )− = ∫ t 0 −1{Xis≤0}dX is + 1 2 l0t = ∫ t 0 −1{Xis≤0}dX is, (3.4) the last by Corollary 3.2.2 Before we continue, observe that Mit ≡ ∫ t 0 √ 2γiX isX i+1 s dBis is a martingale. Indeed, Mi is a continuous local martingale and so it suffices to show that E [ <Mi >t ] <∞ for all t > 0. To show this defines a sequence of stopping times Tn ≡ inf{t ≥ 0 : maxdi=1 |X it | ≥ n}. As Cauchy Schwarz’ inequality yields E [ <Mi >t∧Tn ] ≤ C ∫ t 0 √ E [ (X is∧Tn) 2 ] E [ (X i+1s∧Tn) 2 ] ds <∞, (3.5) Mi·∧Tn is a continuous martingale. In particular, E [(Mit∧Tn)2] = E[<Mi·∧Tn >t] and we obtain that E [ (X it∧Tn) 2 ] ≤ C  (X i0)2 + E [(Mit∧Tn)2]+ E    ∫ t∧Tn 0 d∑ j=1 Xjsqjids   2     (3.5) ≤ C ( 1 + ∫ t 0 d max i=1 E [ (X is∧Tn) 2 ] ds ) . Hence Gronwall’s lemma gives maxdi=1 E [ (X it∧Tn) 2 ] ≤ CeCt. As Tn → ∞ for n → ∞ we can now apply the monotone convergence theorem in (3.5) to get E [ <Mi >t ] <∞ for all t > 0. Thus Mi is indeed a continuous martingale. Taking expectations in (3.4) this implies E [( X it )−] ≤ E  ∫ t 0 d∑ j=1 1Xjs≤0(−Xjs )qjids   . Sum both sides over i and use (3.3) to obtain 0 ≤∑i E[(X it)−] ≤ 0. Thus X it ≥ 0 a.s. for all i ∈ {1, . . . , d} and t ≥ 0. 62 3.2.2 The overall sum and uniqueness In what follows we shall investigate the behaviour of our system for t→∞. We shall show that the sum of all coordinates converges to zero but does not hit zero at a finite time. At the end of this Subsection we shall use this result to establish the weak uniqueness of solutions to (3.2). Notation 3.2.5. Let St ≡ ∑d i=1 X i t . Corollary 3.2.6. St converges a.s. for t→∞. Proof. First note that by Lemma 3.2.4, St is a nonnegative process. Using (3.2) we obtain for St dSt = d∑ i=1 √ 2γiX itX i+1 t dB i t + d∑ i=1 d∑ j=1 Xjt qjidt (3.3) = d∑ i=1 √ 2γiX itX i+1 t dB i t . Using that the Mit are martingales as shown in Lemma 3.2.4, we obtain that St is a nonnegative martingale and thus a.s. convergent. Lemma 3.2.7. St > 0 for all t ≥ 0 a.s., given that S0 > 0. Proof. First observe that < S >t= ∫ t 0 d∑ i=1 2γiX isX i+1 s ds ≤ 2γmax ∫ t 0 S2sds. Next we shall use a time-change to be able to take advantage of this inequality. Let Is ≡ d∑ i=1 2γiX isX i+1 s and Ct ≡ ∫ t 0 Is 2γmaxS2s ds for t ≤ τ, where τ ≡ inf{t ≥ 0 : ∃ > 0 such that Is = 0 ∀s ∈ [t, t+ ]}. Note that if St0 = 0 for some t0 > 0, then St = 0 for all t ≥ t0 by the optional sampling theorem as St is a continuous nonnegative martingale. As Is ≤ 2γmaxS2s we therefore have Ss > 0 for all s < τ . Also note that Ct < ∞ for all t < τ as 0 ≤ Is and Is ≤ 2γmaxS2s which yields 0 ≤ Is 2γmaxS2s ≤ 1 for all 0 ≤ s < τ. (3.6) In particular the definition of τ implies that Ct is a continuous strictly increas- ing function defining a homeomorphism between [0, τ ] and [0, ξ] for ξ < ∞ respectively [0,∞) if ξ = ∞ (let us also denote this by [0, ξ]), where ξ ≡ Cτ . Let D : [0, ξ] → [0, τ ] be the continuous strictly increasing inverse to Ct. Under this time-change we now obtain for St, Zt ≡ SDt , t ≤ ξ, 63 where < Z >t=< S >Dt= ∫ Dt 0 Isds = ∫ t 0 IDs 2γmaxS 2 Ds IDs ds = 2γmax ∫ t 0 Z2sds, i.e. d < Z >t= 2γmaxZ 2 t dt, ∀t ≤ ξ. Thus Z is a geometric Brownian motion (see for instance Karatzas and Shreve [8], Exercise 5.5.31). In particular Zt > 0 ∀t ≤ ξ if ξ < ∞ respectively Zt > 0 ∀t < ξ if ξ = ∞ follows, given that Z0 = S0 > 0. We therefore obtain  SDt > 0 ∀t ≤ ξ ⇒ St > 0 ∀t ≤ τ, if ξ <∞ respectively SDt > 0 ∀t < ξ ⇒ St > 0 ∀t < τ, if ξ = ∞ ( (3.6)⇒ τ = ∞ ) . (3.7) In what follows let η = inf{t ≥ 0 : St = 0}. To finish our proof it remains to show that η = ∞ a.s. By (3.7) it can easily be seen that we have τ ≤ η and by continuity of St the infimum in the definition of η is attained given η < ∞. Finally note that ST = 0 implies St = 0 ∀t ≥ T as St is a nonnegative martingale. Indeed, T is a stopping time and so this follows from the optional sampling theorem (see for instance [7], II.2.13). Let us suppose by contradiction that η <∞. We are left with two cases. 1. case. τ(ω) = η(ω) <∞. This yields a contradiction as Sτ = Sη = 0 by definition of η but Sτ > 0 by (3.7). Before investigating the second case we shall need a Corollary on the be- haviour of the martingales Mit. Corollary 3.2.8. We have for Mit = ∫ t 0 √ 2γiX isX i+1 s dBis that ∣∣Mit̄ −Mit∣∣→ 0 for t̄ ≥ t→∞ a.s. Proof. Mit is a continuous martingale with d < Mi >t= 2γiX itX i+1t dt and Mi0 = 0. By [10], Theorem IV.(34.11) we can time-change Mit such that Mit = B<Mi>t , where Bt is a Brownian motion on a suitably extended probability space. Now 0 ≤<Mi >t̄ − <Mi >t= ∫ t̄ t 2γiX isX i+1 s ds ≤ ∫ t̄ t d∑ i=1 2γiX isX i+1 s ds =< S >t̄ − < S >t→ 0 for t̄ ≥ t→∞ as St converges a.s. and thus lim t→∞ < S >t<∞ a.s. Hence we obtain∣∣Mit̄ −Mit∣∣ = ∣∣B<Mi>t̄ −B<Mi>t ∣∣→ 0 for t̄ ≥ t→∞ a.s. (3.8) as required. 64 Continuation of the proof of Lemma 3.2.7. 2. case. τ(ω) < η(ω) <∞. Suppose τ < η <∞. This implies that there exists  > 0 such that I |[τ,τ+) = 0 and S|[τ,τ+) > 0. By definition of It, St and the continuity of the processes under consideration this requires that there exists δ = δ(ω) > 0 and i = i(ω) ∈ {1, . . . , d} such that X it |[τ,τ+δ) = 0 and X i+1t |[τ,τ+δ) > δ. (3.9) Next consider increments in the ith coordinate to see that for all 0 ≤ t < δ 0 = X iτ+t −X iτ = (Miτ+t −Miτ)+ ∫ τ+t τ d∑ j=1,j 6=i ( Xjsqji −X isqij ) ds ≥ (Miτ+t −Miτ)− (d− 1) max i6=j qij ∫ τ+t τ X isds+ ∫ τ+t τ X i+1s qi+1,ids ≥ (Miτ+t −Miτ)+ tqi+1,iδ. As I |[τ,τ+) = 0 and d < S >s= Isds we get d<S>sds ∣∣∣ [τ,τ+) = 0. Similarly to the proof of Corollary 3.2.8 this implies d<M i>s ds ∣∣∣ [τ,τ+) = 0. Rewriting Mit = B<Mi>t as done in the Corollary we get Mi|[τ,τ+) ≡ const. and thus Miτ+t−Miτ = 0. Plugging this in the above inequality we obtain 0 ≥ tqi+1,iδ > 0, a contradiction. Taking both cases together we have shown that η = ∞ a.s., i.e. inf{t ≥ 0 : St = 0} = ∞ a.s. Remark 3.2.9. We have actually shown more. As η <∞ was not used in the proof of case 2 of Lemma 3.2.7, we have moreover shown that τ = η = ∞. Also observe that the proof of Lemma 3.2.7 only uses qij ≥ 0, j 6= i and qi+1,i > 0 for all i ∈ {1, . . . , d} and so in particular holds for nearest neighbour random walk on the circle, even though uniqueness for this case remains open. Lemma 3.2.10. The overall sum St of our processes converges to 0 a.s., i.e. St → 0 for t → ∞. As the processes X it ≤ St, i ∈ {1, . . . , d} are nonnegative, they converge to 0 a.s. as well. Proof. We shall prove the assertion by constructing a contradiction. The a.s.- existence of a limit S∞ was shown in Corollary 3.2.6. Suppose by contradiction that S∞ = S∞(ω) > 0 for ω element of a set of positive measure. Choose 0 <  < S∞ arbitrarily small. By (3.8) there exists T = T (, ω) ≥ 0 such that for all t̄ > t ≥ T∣∣Mit̄ −Mit∣∣ ≤ , i ∈ {1, . . . , d}, and |St − S∞| ≤ . (3.10) 65 Now observe that for t̄ > t ∣∣X it̄ −X it ∣∣ ≥ − ∣∣Mit̄ −Mit∣∣+ ∫ t̄ t d∑ j=1,j 6=i ( Xjsqji −X isqij ) ds ≥ − ∣∣Mit̄ −Mit∣∣+ qmin ∫ t̄ t d∑ j=1,j 6=i Xjsds− (d− 1)qmax ∫ t̄ t X isds ≥ − ∣∣Mit̄ −Mit∣∣+ qmin ∫ t̄ t (Ss −X is)ds− (d− 1)qmax ∫ t̄ t X isds = − ∣∣Mit̄ −Mit∣∣+ qmin ∫ t̄ t Ssds− [(d− 1)qmax + qmin] ∫ t̄ t X isds. Hence we have for t̄ > t ≥ T∣∣X it̄ −X it ∣∣ ≥ −+ qmin(t̄− t) (S∞ − )− [(d− 1)qmax + qmin] (t̄− t) sup s∈[t,t̄] X is. This is equivalent to sup s∈[t,t̄] X is ≥ −+ qmin(t̄− t) (S∞ − )− ∣∣X it̄ −X it ∣∣ [(d− 1)qmax + qmin] (t̄− t) . (3.11) In what follows fix i ∈ {1, . . . , d} arbitrary and use the following notation 0 ≤ I ≡ lim inf t→∞ X i t ≤ lim sup t→∞ X it ≡ S ≤ S∞ <∞. We shall prove that S∞ > 0 implies I > 0, which will provide us with the desired contradiction in the end. 1. case: S∞ > 0 and I < S. As X it is a continuous process, there exists an increasing sequence {tn}n∈N, tn →∞ for n→∞, independent of the choice of , such that X itn = I + δ(S − I), n ∈ N, where 0 < δ < 1 fixed but arbitrarily small (see Figure 3.1). Without loss of generality let sup s∈[t2n−1,t2n] X is = X i t2n−1 = X i t2n = I + δ(S − I), n ∈ N. Applying (3.11) with t = t2n−1, t̄ = t2n (choose n ∈ N such that t ≥ T ) gives I + δ(S − I) ≥ −+ qmin(t2n − t2n−1) (S∞ − ) [(d− 1)qmax + qmin] (t2n − t2n−1) . (3.12) 66 PSfrag replacements I t1 t2 t3 t4 S I + δ(S − I) Figure 3.1: The definition of t2n−1 and t2n. As the choice of n may depend on  we have to find an estimate for the term∣∣∣ t2n−t2n−1 ∣∣∣ before considering  → 0+. First observe that for 0 ≤ u < v to be specified later ∣∣X iv −X iu∣∣ ≤ ∣∣Miv −Miu∣∣+ ∫ v u d∑ j=1 ∣∣Xjsqji∣∣ ds ≤ ∣∣Miv −Miu∣∣+ (v − u)qmax sup 0≤s<∞ Ss and thus as St converges a.s. v − u ≥ ∣∣X iv −X iu∣∣− ∣∣Miv −Miu∣∣ qmax sup 0≤s<∞ Ss . Now (3.8) yields that for all θ > 0 small there exists T ′ = T ′(θ, ω) (note that T ′ is independent of the choice of ) such that for all v > u ≥ T ′ v − u ≥ ∣∣X iv −X iu∣∣− θ qmax sup 0≤s<∞ Ss . (3.13) Moreover, choose T ′ such that for all t2n−1 ≥ T ′ we have inf s∈[t2n−1,t2n] X is < I + δ 2 (S − I), which is possible by definition of I, S and δ. We get ∃u, v ∈ [t2n−1, t2n] s.t. |X iv −X iu| > δ 2 (S − I) (3.13)⇒ v − u ≥ δ 2 (S − I)− θ qmax sup 0≤s<∞ Ss . By choosing θ sufficiently small we obtain that there exists T ′ = T ′(θ, ω, δ) such that |t2n − t2n−1| ≥ v − u > const(ω, δ) > 0 for all t2n−1 ≥ T ′. Let us return to (3.12). Letting → 0+ yields I + δ(S − I) ≥ qminS∞ [(d− 1)qmax + qmin] . 67 Now let δ → 0+ to finally obtain I > 0. 2. case: S∞ > 0 and I = S. As I = S is equivalent to X it being convergent, we can choose T as defined in (3.10) to additionally satisfy |X is − I | ≤ , ∀s ≥ T. Now (3.11) gives with t̄ > t = T , t̄ arbitrary I +  ≥ sup s∈[T,t̄] X is ≥ −+ qmin(t̄− T ) (S∞ − )− ∣∣X it̄ −X iT ∣∣ [(d− 1)qmax + qmin] (t̄− T ) ≥ −+ qmin(t̄− T ) (S∞ − )− 2 [(d− 1)qmax + qmin] (t̄− T ) . Taking → 0+ yields I > 0 once more. Taking both cases together we have shown lim inf t→∞ X i t > 0 for all i ∈ {1, . . . , d}, given S∞ > 0, as i was chosen arbitrary in the calculations. As the X it are con- tinuous processes this gives a contradiction to lim t→∞ < S >t= ∫∞ 0 ∑d i=1 2γ iX isX i+1 s ds < ∞ a.s., as this requires that the limit inferior of the nonnegative integrand becomes zero for t → ∞. S∞ > 0 thus gives a contradiction, i.e. we have shown S∞ = 0 a.s. Lemma 3.2.11. The solution to the given SDE (3.2) is unique in law for all X0 ∈ Rd s.t. X i0 ≥ 0, i ∈ {1, . . . , d} and ∏d i=1 ( X i0 +X i+1 0 ) > 0. Proof. We shall apply Theorem 4 in [3] (or alternatively Theorem 1.6 of [9]). The Hypotheses under which this Theorem is stated hold, except for condition (3) in Hypothesis 2. Here we have that bi(x) > 0 if xi = 0, except for the case where x = 0 ∈ Rd (here bi(0) = 0). Thus we have to consider this case separately. By modifying the drift coefficients in a small open neighbourhood of 0, say in B(0) with  > 0 arbitrarily small, we can achieve that the drift coefficients satisfy all conditions such as Hölder continuity on compact subsets of Rd+ and |bi(x)| ≤ c(1 + |x|) and additionally are strictly positive on B(0). For instance, let b̃i(x) ≡ bi(x) + (− |x|) ∨ 0. By solving the system with these modified coefficients we obtain existence and uniqueness of the modified solution. Finally take  > 0 so small that the starting point x0 /∈ B(0). As the diffusion and drift coefficients of the modified SDE are identical with the ones of the original SDE on (B(0)) c , every solution to (3.2) is unique until it hits B(0). By taking  ↓ 0+ and recalling that we showed that every solution to (3.2) does not hit 0 in finite time (see Lemma 3.2.7) we obtain the assertion. 68 3.2.3 The normalized processes Corollary 3.2.12. The corresponding SDEs for the normalized processes Y it ≡ X it St (3.14) with 0 ≤ Y it ≤ 1 are as follows. dY it =− Y it ∑ j 6=i √ 2γjY jt Y j+1 t dB j t + (1− Y it ) √ 2γiY it Y i+1 t dB i t (3.15) + Y it ∑ j 6=i 2γjY jt Y j+1 t dt+ (Y i t − 1)2γiY it Y i+1t dt+ d∑ j=1 Y jt qjidt. Idea of the proof. The proof is an easy application of Itô’s formula. In what follows we shall consider the above system of SDEs for the Y it with- out referring to their derivation via the X it and ask for existence and uniqueness of solutions. As we have not shown nonnegativity of the Y it yet, we replace the terms under the square root with their absolute values. Proposition 3.2.13. The SDE (3.15) started at Y0 ∈ [0, 1]d\∂[0, 1]d with∑ i Y i 0 = 1 has a unique in law solution. Moreover the solution satisfies Yt ∈ [0, 1]d and ∑ i Y i t = 1 for all t ≥ 0 a.s. Proof. Existence follows immediately from the existence of solutionsXt to (3.2). Indeed, let X0 ≡ Y0, then Y it ≡ X it/ ∑ j X j t solves (3.15) by Corollary 3.2.12 with Y i0 = X i 0/ ∑ j X j 0 = Y i 0 / ∑ j Y j 0 = Y i 0 . As in Corollary 3.2.2 one can show that the local times at zero of the pro- cesses Y it are zero. The nonnegativity of the processes Y i t can be shown as follows. Lemma 3.2.14. The processes Y it , i ∈ {1, . . . , d} are nonnegative. Proof. We have for all 1 ≤ i ≤ d and t ≤ TM = TM (ω) ≡ inf{t ≥ 0 : maxi |Y it | ≥ M}, M ∈ N fixed, ( Y it )− = ∫ t 0 −1{Y is≤0}dY is = mart. + ∫ t 0 −1{Y is≤0}  Y is ∑ j 6=i 2γjY js Y j+1 s + (Y i s − 1)2γiY is Y i+1s + d∑ j=1 Y js qji   ds. 69 We can bound the above by mart. + ∫ t 0 C(M) ( Y is )− − 1{Y is≤0}∑ j 6=i Y js qjids ≤ mart. + ∫ t 0 C(M) ( Y is )− + C ∑ j 6=i ( Y js )− ds. Taking expectations on both sides and summing over all coordinates we get for all t ≥ 0, ∑ i E [( Y it∧TM )−] ≤ ∫ t 0 C(M) ∑ i E [( Y is∧TM )−] + C ∑ i ∑ j 6=i E [( Y js∧TM )−] ds ≤ C(M) ∫ t 0 ∑ i E [( Y is∧TM )−] ds. An application of Gronwall’s lemma yields E [( Y it∧TM )−] = 0. Take M → ∞ and use Fatou’s lemma to obtain the claim. As we shall show in the following Corollary ∑ i Y i 0 = 1 implies ∑ i Y i t = 1 for all t ≥ 0. Corollary 3.2.15. Every solution (Y it )1≤i≤d to (3.15) with ∑ i Y i 0 = 1 has to satisfy ∑ i Y i t = 1 for all t ≥ 0 a.s. Proof. We have from (3.15) and (3.3) that d ( d∑ i=1 Y it ) = ( 1− d∑ i=1 Y it ) d∑ j=1 [√ 2γjY jt Y j+1 t dB j t − 2γjY jt Y j+1t dt ] = ( 1− d∑ i=1 Y it ) (dNt − d < N >t) , where Nt ≡ ∫ t 0 d∑ j=1 √ 2γjY js Y j+1 s dBjs and Nt∧TM is a martingale starting at 0. Setting Dt ≡ d∑ i=1 Y it and applying Itô’s formula we obtain (D0 − 1 = 0) (Dt∧TM − 1)2 = mart. + ∫ t∧TM 0 { 2(Ds − 1)(−1 +Ds) + 1 2 2(1−Ds)2 } d < N >s giving E(Dt∧TM − 1)2 ≤ 3C(M) ∫ t 0 E(Ds∧TM − 1)2ds. Here we used that d < N >s≤ C(M)ds for s ≤ t∧ TM . Now Gronwall’s lemma yields Dt − 1 ≡ 0 for all t ≤ TM a.s. Take M →∞ to obtain the claim. 70 Continuation of the proof of Proposition 3.2.13. It remains to prove the uniqueness of solutions to (3.15). Observe that Corol- lary 3.2.15 implies 0 ≤ Y it ≤ 1 by the nonnegativity of the Y it . Now suppose that Yt is a solution to (3.15) with Y0 ∈ [0, 1]d\∂[0, 1]d and such that ∑ i Y i 0 = 1. The following Lemma gives the existence of processes X i t such that Y it = X i t/ ∑ i X i t . This will enable us later to derive uniqueness of solutions to (3.15) from uniqueness of solutions to (3.2). Lemma 3.2.16. Given a process (Y it )1≤i≤d that satisfies (3.15) with Y0 ∈ [0, 1]d\∂[0, 1]d and with ∑i Y i0 = 1, we can find a system of processes (X it )1≤i≤d that satisfies (3.2) and (3.14) with St ≡ ∑ iX i t . Proof. We start with a motivation for the proof. Let (Y it )1≤i≤d be given by (3.14). Definition (3.14) and St = ∑ iX i t implied in the former setting that dSt = d∑ i=1 dX it = d∑ i=1 √ 2γiX itX i+1 t dB i t = St d∑ i=1 √ 2γiY it Y i+1 t dB i t , i.e. dSt ≡ StdMt (3.16) being solved by St = S0 exp { Mt − 1 2 <M >t } . In the given setting the above calculation may be taken as a motivation to define Rt ≡ R0 exp { Mt − 1 2 <M >t } , where Mt ≡ ∫ t 0 d∑ i=1 √ 2γiY is Y i+1 s dB i s, such that (3.16) holds with St replaced by Rt and R0 > 0 to be chosen arbi- trarily. Let TM ≡ inf{t ≥ 0 : maxi Y it ≥M}. For t ≤ TM we have <M >t<∞ a.s. This finally leads to the definition of (X it)1≤i≤d, for t ≤ TM , by X it ≡ Y it Rt = Y it R0 exp { Mt − 1 2 <M >t } , (3.17) where X i0 ≡ Y i0R0. Let us check that (X it)1≤i≤d as defined in (3.17) satisfies (3.2). Indeed, we have dX it (3.17) = RtdY i t + Y i t dRt+ < Y i, R >t (3.15),(3.16) = Rt √ 2γiY it Y i+1 t dB i t +Rt d∑ j=1 Y jt qjidt (3.17) = √ 2γiX itX i+1 t dB i t + d∑ j=1 Xjt qjidt, 71 which proves our claim after taking M → ∞ and observing that by Corol- lary 3.2.15, Rt = ∑ i Y i t Rt = ∑ i X i t . Conclusion of the proof of Proposition 3.2.13. The uniqueness of the solution to (3.15) follows from the uniqueness of the solution (X it)1≤i≤d to (3.2). Indeed, observe that the uniqueness of Xt yields the uniqueness of Rt = ∑ i Y i t Rt = ∑ iX i t and thus of Y i t = X i t/Rt. 3.2.4 Properties of a stationary distribution to the system (3.15) of normalized processes Recall that we are given the system of SDEs (3.15) and look for solutions sat- isfying ∑ i Y i t = 1, where 0 ≤ Y it ≤ 1, 1 ≤ i ≤ d. In what follows we shall look for stationary distributions to this system. By Proposition IV.9.2 of [7], a measure pi is a stationary distribution for our process, that is, it is a stationary distribution for A, where A denotes the generator of our system of SDEs, if and only if ∫ Agdpi = 0 for all g ∈ C2. Hence we shall investigate necessary properties of a measure pi satisfying ∫ [0,1]d Agdpi = 0 for all g ∈ C2. In particular we want to show the following Proposition. Proposition 3.2.17. If pi is stationary then it does not put mass on the set N ≡ {y ∈ [0, 1]d : ∃i ∈ {1, . . . , d} : yi = 0}, i.e. on the set where at least one of the coordinate processes becomes extinct. Proof. The generator A of our diffusion Y can be determined to be Ag(x) = d∑ i=1 ∂ig(x)bi(x) + 1 2 d∑ i,j=1 ∂ijg(x)aij(x), (3.18) where bi(x) = xi ∑ j 6=i 2γjxjxj+1 + (xi − 1)2γixixi+1 + d∑ j=1 xjqji, (3.19) aii(x) = (xi) 2 ∑ j 6=i 2γjxjxj+1 + (1− xi)2 2γixixi+1 and for i 6= j aij(x) = xixj ∑ k/∈{i,j} 2γkxkxk+1 − (1− xi)xj2γixixi+1 − xi(1− xj)2γjxjxj+1 (see [10], V.(1.7) for the definition of A). In what follows we shall try to find a function g ∈ C2 which leads to a contradiction to ∫ Agdpi = 0 in case pi puts mass on the set N . Thereby we shall take advantage of the observation that for xi = 0 for i ∈ {1, . . . , d} arbitrarily fixed we have aii(x) = 0 and bi(x) ≥ ∑ j 6=i xjqmin = (1− xi)qmin = qmin > 0, (3.20) 72 where we set qmin ≡ min i6=j qij . To make things easier we shall fix i ∈ {1, . . . , d} arbitrarily and only look for functions g(x) = g(xi), g ∈ C2. Thus we obtain for (3.18) with f(xi) = f(x) ≡ ∂ig(x) = ∂ig(xi), f ∈ C1 that ∫ [0,1]d Agdpi = 0 is equivalent to ∫ [0,1]d f(xi)  xi∑ j 6=i 2γjxjxj+1 + (xi − 1)2γixixi+1 + d∑ j=1 xjqji   dpi(x) = − ∫ [0,1]d 1 2 ∂if(xi)  (xi)2∑ j 6=i 2γjxjxj+1 + (1− xi)2 2γixixi+1   dpi(x). Using that x ∈ [0, 1]d we get ∫ [0,1]d  − |f(xi)|C1xi + f(xi) ∑ j 6=i xjqji   dpi(x) ≤ C2 ∫ [0,1]d |∂if(xi)|xidpi(x), where all constants under consideration are nonnegative. Assuming that f is nonnegative and as ∑ j 6=i xjqji ≥ (1− xi)qmin (see (3.20)) we finally get ∫ [0,1]d f(xi) [−C1xi + (1− xi)qmin] dpi(x) ≤ C2 ∫ [0,1]d |∂if(xi)|xidpi(x). (3.21) In what follows we shall try to find a nonnegative function f ∈ C1 which gives a contradiction to the assumption that pi puts mass on the set N . As we want to investigate the behaviour of pi on the set N , we shall define a function f ∈ C1 with support in [0, 1] and then “squeeze” the function, i.e. rescale it in such a way that the support of the new function f lies in [0, ]. This way we localize equation (3.21) at xi = 0. Let us make this more precise. Suppose we are given f ∈ C1+ with support in [0, 1] or (−∞, 1] and let f(x) ≡ f ( x  ) then f ∈ C1+ with support in [0, ] or (−∞, ] and f ′(x) = f ′ (x  ) 1  . Choose for instance f(x) = exp ( − 11−x ) 1(x ≤ 1). Plugging this into (3.21) and abbreviating A ≡ [0, ]× [0, 1]d−1 we obtain for arbitrary 0 <  < 1∫ A f (x  ) [−C1+ (1− )qmin] dpi(x, z) ≤ ∫ A ∣∣∣f ′(x  )∣∣∣C2x  dpi(x, z), 73 where we assumed without loss of generality that i = 1. This yields ∫ A exp ( − 1 1− x ){ [−C1+ (1− )qmin]− 1( 1− x )2 x C2 } dpi(x, z) (3.22) ≡ I1() + I2() ≤ 0 for all  > 0. For the first part I1 of the integral observe that the absolute value of the integrand is bounded for 0 <  ≤ 0 via C10 + qmin. Hence we can apply the dominated convergence theorem to the first integral to obtain lim ↓0+ I1() = e −1qminpi ({0} × [0, 1]d−1) . (3.23) For the second part of the integral I2 note that for x ∈ [0, ] the absolute value of the integrand is bounded by 4e−2 xC2 ≤ 4e−2C2. As this result is uniform in  we can apply the dominated convergence theorem again to obtain lim ↓0+ I2() = 0. Plugging this and (3.23) into (3.22) we get with qmin > 0 e−1qminpi ({0} × [0, 1]d−1) ≤ 0 ⇒ pi({0} × [0, 1]d−1) = 0, i.e. pi does not put mass on N as stated above. 3.2.5 Stationary distribution Recall that we are given the system of SDEs (3.15) which we rewrite as dYt = σ(Yt)dBt + b(Yt)dt and look for solutions satisfying ∑ i Y i t = 1, where 0 ≤ Y it ≤ 1, 1 ≤ i ≤ d. Proposition 3.2.18. The above system of SDEs has a unique stationary dis- tribution pi supported by S ≡ ([0, 1]d\∂[0, 1]d) ∩ {y : ∑i yi = 1}. Moreover, L(Yt|Y0 = y) ⇒ pi holds for all y ∈ S. Proof. Let Yt be the unique strong Markov solution to (3.15). Recall that we already showed that every equilibrium distribution for A doesn’t put mass on N = {y : ∃i : yi = 0} in Proposition 3.2.17 and that ∑ i Y i t = 1 for all t ≥ 0 in Proposition 3.2.13. Hence, if Yt has any equilibrium distributions, they are concentrated on S ≡ ([0, 1]d\∂[0, 1]d) ∩ { y : ∑ i yi = 1 } . In what follows we shall consider the process Ỹt ≡ (Y 1t , . . . , Y d−1t ) ∈ [0, 1]d−1 instead. The martingale problem for the resulting SDE for Ỹ is consequently 74 well-posed as the corresponding martingale problem for Y is well-posed. For x̃ ∈ S̃ ≡ ( [0, 1]d−1 ∩ { d−1∑ i=1 x̃i ≤ 1 }) \ { x̃ : ∃i : x̃i = 0 or d−1∑ i=1 x̃i = 1 } = ( [0, 1]d−1\∂[0, 1]d−1) ∩ { x̃ : 0 < ∑ i x̃i < 1 } , ãij(x̃) is non-singular by Corollary A.1.1 of the Appendix. Also observe that S̃ is an open subset of [0, 1]d−1 compact. Now the reasoning of [2], Section 3.1 can be applied to show that the system of SDEs for Ỹ has a unique stationary distribution p̃i supported by S̃ and that L(Ỹt|Ỹ0 = ỹ) ⇒ p̃i holds for all ỹ ∈ S̃. Note that, as in [2], the non-singularity of ãij(x̃) on S̃ is crucial. The claim now follows from Ỹt ≡ (Y 1t , . . . , Y d−1t ) and Y dt = 1− ∑d−1 i=1 Y i t . A complete proof is given in the Appendix, Subsection A.2. 3.2.6 Extension to arbitrary networks Instead of (3.1) we can consider dX it = √ 2γiX it ∑ j∈Ci Xjt dB i t + d∑ j=1 Xjt qjidt, i ∈ {1, . . . , d}, (3.24) where Ci ⊂ {1, . . . , d} with i /∈ Ci and |Ci| ≥ 1. We can think of Ci as the set of catalysts of i. The cyclic case corresponds to Ci = {i + 1}. We shall assume as above that γi and qji, i 6= j are given positive constants and the X i0 ≥ 0, i ∈ {1, . . . , d} are given initial conditions. (qji) is again a Q-matrix modelling mutations from type j to type i. For this setup the above proofs directly carry over. Observe in particular that (3.9) changes to the requirement that there exists δ = δ(ω) > 0 and i = i(ω), j = j(ω) ∈ {1, . . . , d} such that X it |[τ,τ+δ) = 0 and Xjt |[τ,τ+δ) > δ. Also note that the restriction on the state space for the initial condition in Lemma 3.2.11 changes to ∏d i=1 ( X i0 + ∑ j∈Ci X j 0 ) > 0 as we now use Theo- rem 1.6 of [9]. 3.2.7 Complete analysis of the case d = 2 Remark 3.2.19. We shall denote γi by γi in what follows, as for instance γ 2 might easily be misunderstood. Recall that the normalized processes Y it = Xit St for our given SDE satisfy (3.15). 75 Corollary 3.2.20. For d = 2 we obtain the following SDE for Yt ≡ Y 1t (note that Y 2t = 1− Y 1t ) dYt = √ 2(1− Yt)Yt ((Yt)2γ2 + (Yt − 1)2γ1)dBt (3.25) + 2(1− Yt)Yt (Ytγ2 + (Yt − 1)γ1) dt+ Yt (q11 − q21) dt+ q21dt. Idea of the proof. We can calculate the SDE for Y 1t from (3.15), using our cyclic definition Y 2+1t = Y 1 t . We get in particular that d < Y 1 >t= 2(1− Y 1t )Y 1t ( (Y 1t ) 2γ2 + (Y 1 t − 1)2γ1 ) dt. Hence we can rewrite Y 1t on a possibly enlarged probability space in terms of a Brownian motion Bt as above (cf. [10], Theorem IV.(34.11)). In what follows we shall prove existence and pathwise uniqueness of solutions to the SDE (3.25) under the constraint Yt ∈ [0, 1]. First observe that (3.25) implies that dYt = σ(Yt)dBt + b(Yt)dt, (3.26) where Yt ∈ I ≡ [0, 1] with σ(x) = √ 2(1− x)x (x2γ2 + (x− 1)2γ1) (3.27) and b(x) = 2(1− x)x (xγ2 + (x− 1)γ1) + x (q11 − q21) + q21. Lemma 3.2.21. If Y0 ∈ [0, 1], the SDE (3.26) has a pathwise unique solution, taking values in [0, 1]. Proof. Replace σ(x) and b(x) in (3.26) by continuous functions σ̃(x), b̃(x) with compact support such that they coincide with σ(x), b(x) on [0, 1]. Then existence of solutions to the SDE with modified coefficients follows by Theorem V.3.10 in [7]. By reasoning as in Lemma 3.2.14 (first consider (Yt) −, then (1 − Yt)−) we can further show that any solution to the modified SDE satisfies 0 ≤ Yt ≤ 1 a.s. and is therefore a solution to the given SDE as well. Pathwise uniqueness of a solution follows from the Yamada-Watanabe path- wise-uniqueness theorem for 1-dim. diffusions (see V.(40.1) in [10] and replace the term under the square root in (3.27) by its absolute value). As every one-dimensional diffusion can be uniquely characterized by its scale function and speed measure, we shall calculate the scale function as a first step towards investigating the long-time behaviour of the given one-dimensional SDE for Yt = Y 1 t . Lemma 3.2.22. The scale function of Y is given by (up to increasing affine transformations) s′(x) = ∣∣x2γ2 + (x − 1)2γ1∣∣− “ 1+ q11 2γ2 − q212γ1 ” |1− x| q11 γ2 |x|− q21 γ1 × exp { − ( q11 + q21√ γ1γ2 ) arctan [ (γ1 + γ2)x− γ1√ γ1γ2 ]} . 76 This yields in particular that s′(0) = s′(1) = ∞. We obtain moreover s(0) = { const. > −∞, q21γ1 < 1, −∞, o.w. and s(1) = { const. <∞, −q11γ2 < 1, ∞, o.w. Idea of the proof. Calculate the scale function as in [10], chapter V.28. Proposition 3.2.23. We shall show the following result on speed and scale. (i) 0 is   recurrent never hit recurrent never hit   and 1 is   recurrent recurrent never hit never hit   for   −q11 < γ2, q21 < γ1, −q11 < γ2, q21 ≥ γ1, −q11 ≥ γ2, q21 < γ1, −q11 ≥ γ2, q21 ≥ γ1,   , where “0 is recurrent” should mean P (∃n such that Yt 6= 0 for all t > n) = 0 and “1 never hit” that P (∃t ≥ 0 : Yt = 1) = 0. (ii) In all cases ∫∞ 0 1{0,1}(Yt)dt = 0. (iii) The scale function of Y is given (up to increasing affine transformations) as in Lemma 3.2.22. (iv) The speed measure of the diffusion Z ≡ s(Y ) in natural scale on [s(0), s(1)] (read this as [s(0), s(1)] = (−∞, s(1)] for −∞ = s(0) < s(1) <∞ etc.) is m(dz) = e 2 R s−1(z) 1 2 2b(u)σ(u)−2du σ(s−1(z))−21{z∈(s(0),s(1))}dz = 1 (s′σ)2 ◦ s−1(z)1{z∈(s(0),s(1))}dz. In particular, m puts no mass on the endpoints s(0) and s(1). Idea of the proof. We shall mimic the calculations of [10], V.48, p. 287f. Corollary 3.2.24. We obtain as a result on the limiting distribution the fol- lowing. Let {Pt} be the transition function of our diffusion in natural scale on R. Then for each x: lim t→∞ ‖pi − Pt(x, ·)‖= 0, where pi(dz) ≡ m(dz) m(R) . Here ‖·‖ denotes the total variation norm of a measure. Idea of the proof. This follows easily from [10], Theorem V.54.5. 77 Bibliography [1] Bass, R.F. and Perkins, E.A. Degenerate stochastic differential equa- tions arising from catalytic branching networks. Electron. J. Probab. (2008) 13, 1808–1885. 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MR0838085 [8] Karatzas, I. and Shreve, S.E. Brownian Motion and Stochastic Calcu- lus, second edition. Springer, New York, 1991. MR1121940 [9] Kliem, S. Degenerate Stochastic Differential Equations for Catalytic Branching Networks. To appear in Ann. Inst. H. Poincaré Probab. Statist. [10] Rogers, L.C.G. and Williams, D. Diffusions, Markov Processes, and Martingales, vol. 2, Reprint of the second (1994) edition. Cambridge Math- ematical Univ. Press, Cambridge, 2000. MR1780932 78 Chapter 4 Convergence of Rescaled Competing Species Processes to a Class of SPDEs1 4.1 Introduction We investigate convergence of certain rescaled models that have their applica- tions in biology. Such convergence results can for instance be used to relate the limits to questions of coexistence and survival of types in the original models. We start by introducing the underlying models and concepts for our later definitions in Subsections 4.1.1, 4.1.2 and 4.1.3. In Subsection 4.1.4 an overview of the results of this paper follows. Finally, in Subsection 4.1.5 we outline the remaining parts of the paper. 4.1.1 The voter model and the Lotka-Volterra model An extensive introduction to the voter model can be found in Liggett [7], Chap- ter V. In short, the 1-dimensional voter model is a process ξt : Z → {0, 1} with the following interpretation. x ∈ Z is seen as an individual with political opin- ion 0 or 1. This is the common interpretation which gives the model its name. Alternatively we can think of Z as space occupied by two populations 0 and 1. If ξt(x) = 0, at time t, the coordinate x is occupied by an individual of population 0. As we shall consider approximate densities later on, this interpretation will suit our purpose better in what follows. The evolution of the process in time is given via infinitesimal rates. Following the notation in [7], let c(x, ξ) denote the rate at which the coordinate ξ(x) flips from 0 to 1 or from 1 to 0 when the system is in state ξ. Then the process ξt will satisfy P(ξt(x) 6= ξ0(x)) = c(x, ξ0)t+ o(t) for t ↓ 0+. For the voter model, the rates can for instance be given by a random walk kernel 1A version of this chapter will be submitted for publication. Kliem, S.M. (2009) Conver- gence of Rescaled Competing Species Processes to a Class of SPDEs. 79 on Z, i.e. 0 ≤ p(x) ≤ 1 and ∑x∈Z p(x) = 1 such that 0 → 1 at rate c(x, ξ) = ∑ y p(x− y)ξ(y), 1 → 0 at rate c(x, ξ) = ∑ y p(x− y)(1− ξ(y)). Under certain conditions on the rate or kernel, it can now be shown that the given rates determine indeed a unique, {0, 1}Z-valued Markov process ξt. A possible interpretation of the kernel p(·) is that at exponential times with rate 1, the individual x ∈ Z selects a site at random according to the kernel p(x − ·) and in case this site has opposite opinion, changes its opinion to the opinion of the selected site. The exponential times and choices according to the random kernel are independent for all x ∈ Z. Finally observe that a special case of this model is the case where we fix a finite set N ⊂ Z of neighbours of 0. If we choose the random walk kernel p(x) = 1|N |1(x ∈ N ), then a neighbour gets chosen with equal probability. Moreover, fi(x, ξ) = 1 |N | ∑ y∈N 1(ξ(y) = i) = ∑ y∈Z p(x− y)1(ξ(y) = i), i = 0, 1 (4.1) can be understood as the frequency of type i in the neighbourhood x+N of x in configuration ξ. In general, we can set fi(x, ξ) = ∑ y∈Z p(x−y)1(ξ(y) = i), i = 0, 1 to rewrite the rates from above to 0 → 1 at rate c(x, ξ) = f1(x, ξ), (4.2) 1 → 0 at rate c(x, ξ) = f0(x, ξ). For the Lotka-Volterra model we consider rate-changes 0 → 1 at rate (4.3) c(x, ξ) = f1(x, ξ) (f0(x, ξ) + α01f1(x, ξ)) = f1(x, ξ) + (α01 − 1)f21 (x, ξ), 1 → 0 at rate c(x, ξ) = f0(x, ξ) (f1(x, ξ) + α10f0(x, ξ)) = f0(x, ξ) + (α10 − 1)f20 (x, ξ) instead, where we used that f0(x, ξ) + f1(x, ξ) = 1 by definition. The definition will become clear in the Subsection to follow (choose λ = 1). Observe in partic- ular that if we choose α01, α10 close to 1, the Lotka-Volterra model can be seen as a small perturbation of the voter model. Finally, we can consider biased voter models by multiplying the rate c(x, ξ) of the change 0 → 1 by a factor of (1 + τ), i.e. (4.2) becomes 0 → 1 at rate c(x, ξ) = (1 + τ)f1(x, ξ), (4.4) 1 → 0 at rate c(x, ξ) = f0(x, ξ) 80 instead. For τ > 0 small we thus have a slight favour for type 1 and for τ < 0 small we have a slight favour for type 0. The biased Lotka-Volterra model is constructed analogously. 4.1.2 Spatial versions of the Lotka-Volterra model As a further example consider spatial versions of the Lotka-Volterra model with finite range as introduced in [10] (they considered ξ(x) ∈ {1, 2} instead of {0, 1}). They use rates 0 → 1 at rate c(x, ξ) = λf1(x, ξ) λf1(x, ξ) + f0(x, ξ) (f0(x, ξ) + α01f1(x, ξ)) , (4.5) 1 → 0 at rate c(x, ξ) = f0(x, ξ) λf1(x, ξ) + f0(x, ξ) (f1(x, ξ) + α10f0(x, ξ)) , where α01, α10 ≥ 0, λ > 0. Here fi is as in (4.1) and N = {y : 0 < |y| ≤ R} with R ≥ 1. We can think of R as the finite interaction range of the model. [10] use this model to obtain results on the parameter regions for coexistence, founder control and spatial segregation of types 0 and 1 in the context of a model that incorporates short-range interactions and dispersal. As a conclusion they obtain that the short-range interactions alter the predictions of the mean-field model. Following [10] we can interpret the rates as follows. The second multiplica- tive factor of the rate governs the density-dependent mortality of a particle, the first factor represents the strength of the instantaneous replacement by a parti- cle of opposite type. The mortality of type 0 consists of two parts, f0 describes the effect of intraspecific competition, α01f1 the effect of interspecific competi- tion. [10] assume that the intraspecific competition is the same for both species. The replacement of a particle of opposite type is regulated by the fecundity pa- rameter λ. The first factors of both rate-changes added together yield 1. Thus they can be seen as weighted densities of the two species. If λ > 1, species 1 has a higher fecundity than species 0. 4.1.3 Long-range limits In [9], Mueller and Tribe show that the approximate densities of type 1 of rescaled biased voter processes, defined as in (4.1) and (4.4) with τ = θN , con- verge to continuous space time densities which solve the heat equation with drift, driven by Fisher-Wright noise, namely ∂u ∂t = ∆u 6 + 2θ(1− u)u+ √ 4u(1− u)Ẇ . (4.6) Observe that [9] scale space by 1/N and consider N = {0 < |y| ≤ N−1/2}. Hence, the number of neighbours of x ∈ Z/N is increasing, namely |N | = 2c(N)N1/2 with c(N) N→∞→ 1, and we thus obtain long-range interactions. Fi- nally, they also rescale time by speeding up the rates of change c(x, ξ) as follows. 81 Let p(N)(x) = 1|N |1(x ∈ N ), then in the Nth-model we have 0 → 1 at rate cN(x, ξN) = (N1/2 + θN−1/2) |N | ∑ y∈Z/N p(N)(x− y)ξN (y) = 2c(N)(N + θ)f1 ( x, ξN ) , 1 → 0 at rate cN(x, ξN) = N1/2|N | ∑ y∈Z/N p(N)(x− y)(1− ξN (y)) = 2c(N)Nf0 ( x, ξN ) . They fix θ ≥ 0, i.e. consider the case where the opinion of type 1 is slightly dominant. See the introduction and Theorem 2 of [9] for more details. In [3] it was shown that stochastic spatial Lotka-Volterra models, suitably rescaled in space and time, converge weakly to super-Brownian motion with linear drift. As they choose the parameters α01, α10 (recall (4.3)) such that N ( α (N) i(1−i) − 1 ) → θi ∈ R, i = 0, 1 (4.7) (see (H3) in [3]) their models can also be interpreted as small perturbations of the voter model. [3] extended the main results of Cox, Durrett and Perkins [2], which proved similar results for long-range voter models. Both papers treat the low density regime, i.e. where only a finite number of individuals of type 1 is present. Instead of investigating limits for approximate densities, both papers define measure-valued processes XNt by XNt = 1 N ′ ∑ x∈Z/(MN √ N) ξNt (x)δx, i.e. they assign mass 1/N ′, N ′ = N ′(N) to each individual of type 1 and consider weak limits in the space of finite Borel measures on R. In particular, they establish the tightness of the sequence of measures and the uniqueness of the martingale problem, solved by any limit point. Note that both papers use a different scaling in comparison to [9]. Using the notation in [2], for d = 1 they take N ′ = N and the space is scaled by MN √ N with MN/ √ N → ∞ (see for instance Theorem 1.1 of [2] for d = 1) in the long-range setup. According to this notation, [9] used MN = √ N , which is at the threshold of the results in [2], but not included. By letting MN = √ N in our setup several non-linear terms will arise in our limiting SPDE below. Also note the brief discussion of the case where MN/ √ N → 0 in d = 1 before (H3) in [2]. Additionally, [2] and [3] consider fixed kernel models in dimensions d ≥ 2 respectively d ≥ 3 with MN = 1 and a fixed random walk kernel q satisfying some additional conditions such that p(x) = q( √ Nx) on x ∈ Z/(MN √ N). Finally, in Cox and Perkins [4], the results of [3] for d ≥ 3 are used to relate the limiting super-Brownian motions to questions of coexistence and survival of a rare type in the original Lotka-Volterra model. 82 4.1.4 Overview of results In the present paper we first prove tightness of the local densities for scaling limits of more general particle systems. The generalization includes two features. Firstly, we shall extend the model in [9] to limits of small perturbations of the long-range voter model, including the setup from [10]. As the rates in [10] (see (4.5)) include taking ratios, we extend our perturbations to a set of power series (for extensions to polynomials of degree 2 recall (4.3)), thereby including certain analytic functions. Recall in particular from (4.7) that we shall allow the coefficients of the power series to depend on N . Secondly, we shall combine both long-range interaction and fixed kernel in- teraction for the perturbations. As we shall see, the tightness results will carry over. As a special case we shall be able to consider rescaled Lotka-Volterra models with long-range dispersal and short-range competition, i.e. where (4.3) gets generalized to 0 → 1 at rate c(x, ξ) = f1(x, ξ) (g0(x, ξ) + α01g1(x, ξ)) , 1 → 0 at rate c(x, ξ) = f0(x, ξ) (g1(x, ξ) + α10g0(x, ξ)) . Here fi(x, ξ), i = 0, 1 is the density corresponding to a long-range kernel p L and gi(x, ξ), i = 0, 1 is the density corresponding to a fixed kernel p F (also recall the interpretation of both multiplicative factors in Subsection 4.1.2). Finally, in the case of long-range interactions only we show the limit points are solutions of a SPDE similar to (4.6) but with a drift depending on the choice of our perturbation and small changes in constants due to simple differences in scale factors. Hence, we obtain a class of SPDEs that can be characterized as the limit of perturbations of the long-range voter model. If the limiting initial condition u0 satisfies ∫ u0(x)dx < ∞, we can show the weak uniqueness of solutions to the limiting SPDE and therefore show weak convergence of the rescaled particle densities to this unique law. When there exists a fixed kernel, the question of uniqueness of all limit points and of identifying the limit remains an open problem. Also, when we consider long-range interactions only with ∫ u0(x)dx = ∞ the proof of weak uniqueness of solutions to the limiting SPDE remains open. The proof of our results generalizes the work done in [9]. In [9], limits are considered for both the long-range contact process and the long-range voter process. Full details are given for the contact process. For the voter process, once the approximate martingale problem is derived, almost all of the remaining steps are left to the reader. Many arguments of our proof are similar to [9] but as additions and adaptations are needed due to our broader setup and as they did not provide details for the long-range voter model we shall sometimes be more detailed. 83 4.1.5 Outline of the paper In Section 4.2 to follow we shall first set up our model and give the main results. Then we shall reformulate the model so that it can be approached by the methods used in [9]. A statement of the main results in the reformulated setting follows. In Section 4.3 we shall introduce a graphical construction for each approxi- mating model ξN· . This allows us to write out the time-evolution of our models. By integrating it against a test function and summing over x ∈ Z/N we finally obtain an approximate martingale problem for the N th-process. We define the approximate density A(ξNt )(x) as the average density of particles of type 1 on Z/N in an interval centered at x of length 2/ √ N (see (4.10) below). By choos- ing a specific test function, the properties of which are under investigation at the beginning of Section 4.4, an approximate Green’s function representation for the approximate densities A(ξNt )(·) is derived towards the end of Section 4.4 and bounds on error-terms appearing in it are given. Making use of the Green’s function representation, tightness of A(ξNt )(·) is proven in Section 4.5. Here the main part of the proof consists in finding estimates on pth-moment differences. In Section 4.6 the tightness of the approximate densities is used to show tightness of the measure corresponding to the sequence of configurations ξNt . Finally, in the special case with no fixed kernel, every limit is shown to solve a certain SPDE. In Section 4.7 we prove that this SPDE has a unique weak solution if < u0, 1 >< ∞. In this case, weak uniqueness of the limits of the sequence of approximate densities follows. 4.2 Main Results of the Paper 4.2.1 The model We define a sequence of rescaled competing species models ξNt in dimension d = 1, which can be described as perturbations of voter models. In the N th- model the sites are indexed by x ∈ N−1Z. We label the state of site x at time t by ξNt (x) where ξ N t (x) = 0 if the site is occupied at time t by type 0 and ξNt (x) = 1 if it is occupied by type 1. In what follows we shall write x ∼ y if and only if 0 < |x− y| ≤ N−1/2, i.e. if and only if x is a neighbour of y. Observe that each x has 2c(N)N1/2, c(N) N→∞→ 1 neighbours. The rates of change incorporate both long-range models and fixed kernel models with finite range. The long-range interaction takes into account the densities of the neighbours of x at long-range, i.e. f (N) i (x, ξ) ≡ 1 2c(N) √ N ∑ 0<|y−x|≤1/√N, y∈Z/N 1(ξN (y) = i), i = 0, 1 84 and the fixed kernel interaction considers g (N) i (x, ξ) ≡ ∑ y∈Z/N p(N(x− y))1(ξN (y) = i), i = 0, 1, where p(x) is a random walk kernel on Z of finite range, i.e. 0 ≤ p(x) ≤ 1,∑ x∈Z p(x) = 1 and p(x) = 0 for all |x| ≥ Cp. In what follows we shall often abbreviate f (N) i (x, ξ) by f (N) i and g (N) i (x, ξ) by g (N) i if the context is clear. Now define the rates of change of our configurations. At site x in configura- tion ξN ∈ {0, 1}Z/N the coordinate ξN (x) makes transitions 0 → 1 at rate Nf (N)1 + f (N)1 { g (N) 0 G (N) 0 ( f (N) 1 ) + g (N) 1 H (N) 0 ( f (N) 1 )} , (4.8) 1 → 0 at rate Nf (N)0 + f (N)0 { g (N) 0 G (N) 1 ( f (N) 0 ) + g (N) 1 H (N) 1 ( f (N) 0 )} , where G (N) i , H (N) i , i = 0, 1 are power series as follows. Hypothesis 4.2.1. We assume that G (N) i (x) ≡ ∞∑ m=0 α (m+1,N) i x m and H (N) i (x) ≡ ∞∑ m=0 β (m+1,N) i x m, x ∈ [0, 1] with i = 0, 1, α (m+1,N) i , β (m+1,N) i ∈ R, m ≥ 0 and that there exists N0 ∈ N such that sup N≥N0 ∑ i=0,1 ∞∑ m=0 { α (m+1,N) i ∨ 0 + β(m+1,N)i ∨ 0 +m ∣∣∣α(m+1,N)i ∧ 0∣∣∣ +m ∣∣∣β(m+1,N)i ∧ 0∣∣∣} <∞. Remark 4.2.2. The above rates determine indeed a unique, {0, 1}Z/N -valued Markov process ξNt for N ≥ N0 with N0 as in Hypothesis 4.2.1 as we now show. See for instance Theorem B3, p.3 in Liggett [8] and note the uniform boundedness assumption on the rates from p.1 of [8]. Following the notation in [8], let c(x, ξN ) denote the rate at which the coordinate ξN (x) flips from 0 to 1 or from 1 to 0 when the system is in state ξN . Then using (4.8), 0 ≤ f (N) i , g (N) i ≤ 1, i = 0, 1 and Hypothesis 4.2.1 yields sup x∈Z/N sup ξN∈{0,1}Z/N c ( x, ξN ) ≤ N + ( ∞∑ m=0 ∣∣∣α(m+1,N)0 ∣∣∣+ ∣∣∣β(m+1,N)0 ∣∣∣ ) ∨ ( ∞∑ m=0 ∣∣∣α(m+1,N)1 ∣∣∣+ ∣∣∣β(m+1,N)1 ∣∣∣ ) ≡ N + C0(N) <∞ 85 and sup x∈Z/N ∑ u∈Z/N sup ξN∈{0,1}Z/N ∣∣c(x, ξN)− c(x, ξNu )∣∣ ≤ sup x∈Z/N ∑ u∼x sup ξN∈{0,1}Z/N ∣∣c(x, ξN)− c(x, ξNu )∣∣ + sup x∈Z/N ∑ u∈Z/N sup ξN∈{0,1}Z/N ∑ i=0,1 ∣∣∣g(N)i (x, ξN)− g(N)i (x, ξNu )∣∣∣C0(N) ≤ 2c(N)N1/22(N + C0(N)) + sup x∈Z/N ∑ u∈Z/N 2p(N(x− u))C0(N) ≤ 2c(N)N1/22(N + C0(N)) + 2C0(N) <∞, where ξNu (v) = { ξN (v), v 6= u, 1− ξN (v), v = u. Following [8], the two conditions are sufficient to ensure the above. Additionally, the closure in the space of continuous functions on {0, 1}Z/N of the operator Ωf ( ξN ) = ∑ x c ( x, ξN ) ( f(ξNx )− f(ξN ) ) , which is defined on the space of finite cylinder functions on {0, 1}Z/N , is the Markov generator of the process ξNt . Remark 4.2.3. Observe in particular that f (N) 0 +f (N) 1 = 1 and g (N) 0 +g (N) 1 = 1. Hence the special case of no fixed kernel can be obtained by choosing G (N) i ≡ H (N) i , i = 0, 1 and we get 0 → 1 at rate Nf (N)1 + f (N)1 G(N)0 ( f (N) 1 ) , (4.9) 1 → 0 at rate Nf (N)0 + f (N)0 G(N)1 ( f (N) 0 ) . For the configurations ξNt ∈ {0, 1}Z/N we define approximate densities A(ξNt ) via A(ξNt )(x) = 1 2c(N)N1/2 ∑ y∼x ξNt (y), x ∈ N−1Z (4.10) and note that A(ξNt )(x) = f (N) 1 ( x, ξNt ) . By linearly interpolating between sites we obtain approximate densities A(ξNt )(x) for all x ∈ R. Notation 4.2.4. Set C1 ≡ {f : R → [0, 1] cont.} and let C1 be equipped with the topology of uniform convergence on compact sets. We obtain that t 7→ A(ξNt ) is cadlag C1-valued, where we used that 0 ≤ A(ξNt )(x) ≤ 1 for all x ∈ N−1Z. Hence, we can consider the law of A(ξN ) on the space of cadlag C1-valued paths with the Skorokhod topology. 86 4.2.2 Main results Before stating our main results we need some more notation. Notation 4.2.5. For f, g : N−1Z → R, we set < f, g >= 1 N ∑ x f(x)g(x). Let ν be a measure on N−1Z. Then we set < ν, f >= ∫ fdν. Remark 4.2.6. We can rewrite every configuration ξNt in terms of its corre- sponding measure. Let νNt ≡ 1 N ∑ x δx1(ξ N t (x) = 1), then < ξNt , φ >=< ν N t , φ > . Definition 4.2.7. Let S be a Polish space and let D(S) denote the space of cadlag paths from R+ to S with the Skorokhod topology. Following the first definition on p.148 of Perkins [11], we shall say that a collection of processes with paths in D(S) is C-tight if and only if it is tight in D(S) and all weak limit points are a.s. continuous. Recall that for Polish spaces, tightness and weak relative compactness are equivalent. Remark 4.2.8. In what follows we shall investigate tightness of {A(ξN· ) : N ≥ N0} in D(C1) and tightness of {νNt : N ≥ N0} in D(M(R)), where M(R) is the space of Radon measures equipped with the vague topology (M(R) is indeed Polish, see Dawson [5], Section 3.1.3). Theorem 4.2.9. Suppose that A(ξN0 ) → u0 in C1. Let the transition rates of ξN (x) be as in (4.8) with G (N) i , H (N) i , i = 0, 1 satisfying Hypothesis 4.2.1. Then( A(ξNt ) : t ≥ 0 ) are C-tight as cadlag C1-valued processes and the ( νNt : t ≥ 0 ) are C-tight as cadlag Radon measure valued processes with the vague topology. If ( A ( ξNkt ) , νNkt ) t≥0 converges to (ut, νt)t≥0, then νt(dx) = ut(x)dx for all t ≥ 0. Remark 4.2.10. The above applies in particular to models where G (N) i , H (N) i are finite sums (see Hypothesis 4.2.1). Hypothesis 4.2.11. Let us consider the special case with no fixed kernel (see Remark 4.2.3). Additionally to Hypothesis 4.2.1 we assume that α (m+1,N) i N→∞→ α(m+1)i for all i = 0, 1 and m ≥ 0 87 with (take sgn(0) = 0) sgn ( α (m+1,N) i ) ≥ 0 for all N ≥ N0 or sgn ( α (m+1,N) i ) ≤ 0 for all N ≥ N0 for all i = 0, 1,m ≥ 0 and that lim N→∞ ∑ i=0,1 ∞∑ m=0 { α (m+1,N) i ∨ 0 +m ∣∣∣α(m+1,N)i ∧ 0∣∣∣} (4.11) = ∑ i=0,1 ∞∑ m=0 { α (m+1) i ∨ 0 +m ∣∣∣α(m+1)i ∧ 0∣∣∣} . Remark 4.2.12. The additional conditions of Hypothesis 4.2.11 are necessary to transform the given rates into rates with positive coefficients in a uniform way in Subsection 4.2.3 and to later characterize limit points of the approximate densities by taking limits in N →∞ inside infinte sums in (4.55). Definition 4.2.13. Under the assumptions of Theorem 4.2.9 and Hypothe- sis 4.2.11 we let for x ∈ [0, 1], Gi(x) ≡ lim N→∞ G (N) i (x) = lim N→∞ ∞∑ m=0 α (m+1,N) i x m = ∞∑ m=0 α (m+1) i x m, i = 0, 1. This is well-defined by (4.11) and Royden [12], Proposition 11.18. Theorem 4.2.14. We obtain under the assumptions of Theorem 4.2.9 and Hypothesis 4.2.11 for the special case with no fixed kernel that the limit points of A(ξNt ) are continuous C1-valued processes ut which solve ∂u ∂t = ∆u 6 + (1− u)u {G0(u)−G1(1− u)}+ √ 2u(1− u)Ẇ (4.12) with initial condition u0. If we assume additionally < u0, 1 >< ∞, then ut is the unique (in law) [0, 1]-valued solution to the above SPDE. Remark 4.2.15. As an example consider spatial versions of the Lotka-Volterra model as introduced in Subsection 4.1.2. In what follows we shall choose the competition and fecundity parameters near one and we shall consider the long- range case. Namely, the model exhibits the following rates: 0 → 1 at rate N [ λ(N)f (N) 1 λ(N)f (N) 1 + f (N) 0 ( f (N) 0 + α (N) 01 f (N) 1 )] , 1 → 0 at rate N [ f (N) 0 λ(N)f (N) 1 + f (N) 0 ( f (N) 1 + α (N) 10 f (N) 0 )] . We suppose that λ(N) ≡ 1 + λ ′ N , α (N) 01 ≡ 1 + α01 N , α (N) 10 ≡ 1 + α10 N . 88 Using f (N) 0 + f (N) 1 = 1 we can therefore rewrite the rates as 0 → 1 at rate (N + λ′) f (N)1 ( 1 + α01 N f (N) 1 )∑ n≥0 ( −λ ′ N f (N) 1 )n , (4.13) 1 → 0 at rate Nf (N)0 ( 1 + α10 N f (N) 0 )∑ n≥0 ( −λ ′ N f (N) 1 )n = Nf (N) 0 ( 1 + α10 N f (N) 0 )∑ k≥0 ( f (N) 0 )k (λ′ N )k ∑ n≥k ( n k )( −λ ′ N )n−k . Here we used that ∣∣∣f (N)i ∣∣∣ ≤ 1, i = 0, 1 and that ∣∣∣λ′N ∣∣∣ → 0 for N → ∞. We can use the explicit calculations for a geometric series, in particular that we have∑ n≥0 n|q|n < ∞ and ∑ n≥k |q|n−k ( n k ) = 1 (1−|q|)k+1 for |q| < 1 to check that Hypothesis 4.2.1 and Hypothesis 4.2.11 are satisfied. Using Theorem 4.2.14 we further obtain that the limit points of A(ξNt ) are continuous C1-valued processes ut which solve ∂u ∂t = ∆u 6 + (1− u)u {(λ′ + u (α01 − λ′))− (−λ′ + (1− u) (α10 + λ′))} + √ 2u(1− u)Ẇ = ∆u 6 + (1− u)u {λ′ − α10 + u (α01 + α10)}+ √ 2u(1− u)Ẇ by rewriting the above rates (4.13) in the form (4.9) and taking the limit for N → ∞. For < u0, 1 ><∞, ut is the unique weak [0, 1]-valued solution to the above SPDE. 4.2.3 Reformulation We proceed as in [9]. Recall from Hypothesis 4.2.1 and (4.8) that the rates of change are given by 0 → 1 at rate Nf (N)1 + g(N)0 ∞∑ m=1 α (m,N) 0 ( f (N) 1 )m + g (N) 1 ∞∑ m=1 β (m,N) 0 ( f (N) 1 )m , (4.14) 1 → 0 at rate Nf (N)0 + g(N)0 ∞∑ m=1 α (m,N) 1 ( f (N) 0 )m + g (N) 1 ∞∑ m=1 β (m,N) 1 ( f (N) 0 )m , where α (m,N) j , β (m,N) j ∈ R for all j = 0, 1,m ∈ N. Following [9] we shall model each term in the rate-changes via independent families of i.i.d. Poisson processes. For instance, if α (m,N) 0 is non-negative, the term g (N) 0 α (m,N) 0 ( f (N) 1 )m of the rate-change 0 → 1 in (4.14) can be modeled via i.i.d. Poisson processes( Qt(x; y1, . . . , ym; z) : x, y1, . . . , ym, z ∈ N−1Z ) 89 of rate α (m,N) 0 (2c(N))mNm/2 p(N(x− z)). At a jump of Qt(x; y1, . . . , ym; z) the voter at x adopts the opinion 1 provided that all of y1, . . . , ym have opinion 1 and z has opinion 0. As we want to allow the α (m,N) i , β (m,N) i to be negative, too, we first rewrite (4.14) with the help of f (N) 0 + f (N) 1 = 1 and g (N) 0 + g (N) 1 = 1 in a form where all resulting coefficients are non-negative. Corollary 4.2.16. We can rewrite our transitions as follows. 0 → 1 at rate (4.15) ( N − θ(N) ) f (N) 1 + f (N) 1   ∑ i=0,1 a (N) i g (N) i + ∑ m≥2,i,j=0,1 q (0,m,N) ij g (N) i f (N) j ( f (N) 1 )m−2  , 1 → 0 at rate ( N − θ(N) ) f (N) 0 + f (N) 0   ∑ i=0,1 b (N) i g (N) i + ∑ m≥2,i,j=0,1 q (1,m,N) ij g (N) i f (N) j ( f (N) 0 )m−2  , with corresponding θ(N), a (N) i , b (N) i , q (k,m,N) ij ∈ R+, i, j, k = 0, 1,m ≥ 2. Proof. We shall drop the superscripts of f (N) i , g (N) i , i = 0, 1 in what follows to simplify notation. Suppose for instance α (m,N) 0 < 0 for some m ≥ 1 in (4.14). Using that −xm = (1− x) m−1∑ k=1 xk − x and recalling that 1− f1 = f0 we obtain g0α (2k+1,N) 0 f 2k+1 1 = g0 {( −α(2k+1,N)0 ) f0 2k∑ l=1 f l1 + α (2k+1,N) 0 f1 } . Finally, we can use g0 = 1− g1 to obtain g0α (2k+1,N) 0 f 2k+1 1 = g0 ( −α(2k+1,N)0 ) f0 2k∑ l=1 f l1+g1 ( −α(2k+1,N)0 ) f1+α (2k+1,N) 0 f1. All terms on the r.h.s. but the last can be accommodated into an existing representation (4.15) as follows: q (0,m,N) 00 → q(0,m,N)00 + ( −α(2k+1,N)0 ) for 2 ≤ m ≤ 2k + 1, a (N) 1 → a(N)1 + ( −α(2k+1,N)0 ) . 90 Finally, we can assimilate the last term into the first part of the rate 0 → 1, i.e. we replace θ(N) → θ(N) − α(2k+1,N)0 . As we use the representation (4.15), a change in the first part of the rate 0 → 1 also impacts the rate 1 → 0 in its first term. Therefore we have to fix the rate 1 → 0 by adding a term of (−α(2k+1,N)0 )f0 = g0f0(−α(2k+1,N)0 ) + g1f0(−α(2k+1,N)0 ) to the second and third term of the rate, i.e. by replacing b (N) 0 → b(N)0 + ( −α(2k+1,N)0 ) , b (N) 1 → b(N)1 + ( −α(2k+1,N)0 ) . The case m = 2k, k ≥ 1 follows similarly and the general case with multiple negative α′s and/or β′s now follows inductively. Remark 4.2.17. The above construction yields the following non-negative co- efficients: θ(N) ≡ ∑ j=0,1 ∞∑ n=1 ( −α(n,N)j ) 1 ( α (n,N) j < 0 ) + ( −β(n,N)j ) 1 ( β (n,N) j < 0 ) , q (0,m,N) 00 ≡ ∞∑ n=m ( −α(n,N)0 ) 1 ( α (n,N) 0 < 0 ) , a (N) 0 ≡α(1,N)0 1 ( α (1,N) 0 ≥ 0 ) + { ∞∑ n=1 ( −β(n,N)0 ) 1 ( β (n,N) 0 < 0 ) + ∞∑ n=1 ( −α(n,N)1 ) 1 ( α (n,N) 1 < 0 ) + ∞∑ n=1 ( −β(n,N)1 ) 1 ( β (n,N) 1 < 0 )} , q (0,m,N) 01 ≡α(m,N)0 1 ( α (m,N) 0 ≥ 0 ) , q (0,m,N) 10 ≡ ∞∑ n=m ( −β(n,N)0 ) 1 ( β (n,N) 0 < 0 ) , a (N) 1 ≡β(1,N)0 1 ( β (1,N) 0 ≥ 0 ) + { ∞∑ n=1 ( −α(n,N)0 ) 1 ( α (n,N) 0 < 0 ) + ∞∑ n=1 ( −α(n,N)1 ) 1 ( α (n,N) 1 < 0 ) + ∞∑ n=1 ( −β(n,N)1 ) 1 ( β (n,N) 1 < 0 )} , q (0,m,N) 11 ≡β(m,N)0 1 ( β (m,N) 0 ≥ 0 ) , q (1,m,N) 01 ≡ ∞∑ n=m ( −α(n,N)1 ) 1 ( α (n,N) 1 < 0 ) , b (N) 0 ≡α(1,N)1 1 ( α (1,N) 1 ≥ 0 ) + { ∞∑ n=1 ( −β(n,N)1 ) 1 ( β (n,N) 1 < 0 ) + ∞∑ n=1 ( −α(n,N)0 ) 1 ( α (n,N) 0 < 0 ) + ∞∑ n=1 ( −β(n,N)0 ) 1 ( β (n,N) 0 < 0 )} , 91 q (1,m,N) 00 ≡α(m,N)1 1 ( α (m,N) 1 ≥ 0 ) , q (1,m,N) 11 ≡ ∞∑ n=m ( −β(n,N)1 ) 1 ( β (n,N) 1 < 0 ) , b (N) 1 ≡β(1,N)1 1 ( β (1,N) 1 ≥ 0 ) + { ∞∑ n=1 ( −α(n,N)1 ) 1 ( α (n,N) 1 < 0 ) + ∞∑ n=1 ( −α(n,N)0 ) 1 ( α (n,N) 0 < 0 ) + ∞∑ n=1 ( −β(n,N)0 ) 1 ( β (n,N) 0 < 0 )} , q (1,m,N) 10 ≡β(m,N)1 1 ( β (m,N) 1 ≥ 0 ) . By Hypothesis 4.2.1 this implies in particular that there exists N0 ∈ N such that sup N≥N0 ∑ i,j,k=0,1 ∑ m≥2 q (k,m,N) ij <∞. Remark 4.2.18. Observe that we can rewrite the transition rates in (4.15) such that a (N) i = b (N) i = 0, i = 0, 1, i.e. 0 → 1 at rate (4.16)( N − θ(N) ) f (N) 1 + f (N) 1 ∑ m≥2,i,j=0,1 q (0,m,N) ij g (N) i f (N) j ( f (N) 1 )m−2 , 1 → 0 at rate( N − θ(N) ) f (N) 0 + f (N) 0 ∑ m≥2,i,j=0,1 q (1,m,N) ij g (N) i f (N) j ( f (N) 0 )m−2 . Indeed, using that f (N) 0 + f (N) 1 = 1 we can change for instance a (N) 0 g (N) 0 + q (0,2,N) 00 g (N) 0 f (N) 0 ( f (N) 1 )0 + q (0,2,N) 01 g (N) 0 f (N) 1 ( f (N) 1 )0 with a (N) 0 , q (0,2,N) 00 , q (0,2,N) 01 nonnegative into( a (N) 0 + q (0,2,N) 00 ) g (N) 0 f (N) 0 ( f (N) 1 )0 + ( a (N) 0 + q (0,2,N) 01 ) g (N) 0 f (N) 1 ( f (N) 1 )0 , where the new coefficients are nonnegative again. Recall Remark 4.2.17 together with Hypothesis 4.2.1. We now introduce hy- potheses directly on the q (k,m,N) ij as the primary variables. Observe in particular that they will be assumed to be non-negative. Hypothesis 4.2.19. Assume that there exists N0 ∈ N such that sup N≥N0 ∑ i,j,k=0,1 ∑ m≥2 q (k,m,N) ij <∞ for non-negative q (k,m,N) ij , i, j, k = 0, 1 and m ≥ 2. We can use this condition as in Remark 4.2.2 to show that the rewritten rates can be used to determine a {0, 1}Z/N-valued Markov process ξNt for N ≥ N0. 92 Hypothesis 4.2.20. In the special case with no fixed kernel, i.e. where q (k,m,N) 00 = q (k,m,N) 10 and q (k,m,N) 01 = q (k,m,N) 11 ⇐⇒ q(k,m,N)0j = q(k,m,N)1j , j = 0, 1 (see Remark 4.2.3 and Remark 4.2.17) we assume that θ(N) N→∞→ θ, q (k,m,N) 0j N→∞→ q(k,m)0j for all j, k = 0, 1 and m ≥ 2 and lim N→∞ ∑ j,k=0,1 ∑ m≥2 q (k,m,N) 0j = ∑ j,k=0,1 ∑ m≥2 q (k,m) 0j . (4.17) Remark 4.2.21. Observe that if we assume that the q (k,m,N) 0j , j, k = 0, 1,m ≥ 2 were obtained from α (m,N) j , j = 0, 1,m ≥ 1 as described earlier in Remark 4.2.17 and Remark 4.2.18, then (4.11) implies (4.17). Indeed, use for instance [12], Proposition 11.18 together with Remark 4.2.17. Notation 4.2.22. For k = 0, 1 and a ∈ R we let Fk(a) = { 1− a, k = 0, a, k = 1. By the above it remains to prove the following theorem. The claim of The- orem 4.2.9 will then follow immediately and Theorem 4.2.14 will follow using Corollary 4.2.24 below. Theorem 4.2.23. Suppose that A(ξN0 ) → u0 in C1. Let the transition rates of ξN (x) be as in (4.16) and q (k,m,N) ij satisfying Hypothesis 4.2.19. Then the( A(ξNt ) : t ≥ 0 ) are C-tight as cadlag C1-valued processes and the ( νNt : t ≥ 0 ) are C-tight as cadlag Radon measure valued processes with the vague topology. If ( A ( ξNkt ) , νNkt ) t≥0 converges to (ut, νt)t≥0, then νt(dx) = ut(x)dx for all t ≥ 0. For the special case with no fixed kernel we further obtain that if Hypothe- sis 4.2.20 holds, then the limit points of A(ξNt ) are continuous C1-valued pro- cesses ut which solve ∂u ∂t = ∆u 6 + ∑ k=0,1 (1−2k) ∑ m≥2,j=0,1 q (k,m) 0j Fj(u) (F1−k(u)) m−1 Fk(u)+ √ 2u(1− u)Ẇ (4.18) with initial condition u0. If we assume additionally < u0, 1 >< ∞, then ut is the unique (in law) [0, 1]-valued solution to the above SPDE. In the next Corollary we assume there is no fixed kernel and the q (k,m) 0j , j, k = 0, 1,m ≥ 2 are defined from the α(m)j , j = 0, 1,m ≥ 1 as in Remark 4.2.17 and Remark 4.2.18 without the N ’s. 93 Corollary 4.2.24. Under the assumption above, the SPDE (4.18) may be rewritten as ut = ∆u 6 +(1−u)u ∞∑ m=0 α (m+1) 0 u m−u(1−u) ∞∑ m=0 α (m+1) 1 (1−u)m+ √ 2u(1− u)Ẇ . Proof. Indeed first use the definition of Fk(a) and collect terms appropriately. Then recall how we rewrote the transition rates in Corollary 4.2.16 and Re- mark 4.2.18 to obtain (4.16) from (4.14). Now analogously rewrite (4.12) as (4.18). Before we start proving the above we need some notation. In what follows we shall consider eλ(x) = exp(λ|x|) for λ ∈ R and we let C = {f : R → [0,∞) cont. with |f(x)eλ(x)| → 0 as |x| → ∞ for all λ < 0} be the set of non-negative continuous functions with slower than exponential growth. Define ‖f ‖λ= sup x |f(x)eλ(x)| and give C the topology generated by the norms (‖·‖λ: λ < 0). Remark 4.2.25. We work on the space C instead of C1 because in Section 4.4 we shall introduce functions 0 ≤ ψzt (x) ≤ CN1/2 and shall show in Lemma 4.5.2(b) that they converge in C to the Brownian transition density p( t3 , z − x). Finally, in Section 4.5 we shall derive estimates on pth-moment differences of Â(ξt)(z) ≡ A(ξt)(z)− < ξ0, ψzt >, where A(ξ0) → u0 in C to finally establish the tightness claim for the sequence of approximate densities A(ξN )(x). Notation 4.2.26. For x ∈ N−1Z, f : N−1Z → R and δ > 0 we shall use D(f, δ)(x) = sup{|f(y)− f(x)| : |y − x| ≤ δ, y ∈ N−1Z}, (4.19) ∆(f)(x) = N − θ(N) 2c(N)N1/2 ∑ y∼x (f(y)− f(x)), where we suppress the dependence on N . 94 4.3 An Approximate Martingale Problem We shall now derive a graphical construction and evolution in time of our ap- proximating processes ξNt . The graphical construction uses independent families of i.i.d. Poisson processes: ( Pt(x; y) : x, y ∈ N−1Z ) i.i.d. P.p. of rate N − θ(N) 2c(N)N1/2 , (4.20) and for m ≥ 2, i, j, k = 0, 1,( Qm,i,j,kt (x; y1, . . . , ym; z) : x, y1, . . . , ym, z ∈ N−1Z ) i.i.d. P.p. of rate q (k,m,N) ij (2c(N))mNm/2 p(N(x− z)). Note that we suppress the dependence on N in the family of Poisson processes Pt(x; y) and Q m,i,j,k t (x; y1, . . . , ym; z). At a jump of Pt(x; y) the voter at x adopts the opinion of the voter at y provided that y has the opposite opinion. At a jump of Qm,i,j,kt (x; y1, . . . , ym; z) the voter at x adopts the opinion 1−k provided that y1 has opinion j, all of y2 . . . , ym have opinion 1 − k and z has opinion i. This yields the following SDE to describe the evolution in time of our ap- proximating processes ξNt : ξNt (x) =ξ N 0 (x) + ∑ y∼x ∫ t 0 { δ0 ( ξNs−(x) ) δ1 ( ξNs−(y) )− δ1(ξNs−(x)) δ0(ξNs−(y))} (4.21) × dPs(x; y) + ∑ k=0,1 (1− 2k) ∑ m≥2,i,j=0,1 ∑ y1,...,ym∼x ∑ z ∫ t 0 δk ( ξNs−(x) ) × δj ( ξNs−(y1) ) m∏ l=2 δ1−k ( ξNs−(yl) ) δi ( ξNs−(z) ) dQm,i,j,ks (x; y1, . . . , ym; z) for all x ∈ N−1Z. We now explain why the above system (4.21) has a unique solution. The problem with (4.21) is that although there is a first flip time for ξNt (x) (the jump rate there is bounded as can be shown using Hypothesis 4.2.19 and as the sum of Poisson processes is a Poisson process again, we have at most one flip at a time), what to do at this time depends on the states of the finite number of “communicating sites” y, z ∈ Z/N, y ∼ x, |z − x| ≤ Cp/N . The equations for each of these sites will in turn involve its communicating sites. If we now try to go backwards in time to determine the configuration at x ∈ Z/N , starting with t > 0, we may encounter an accumulation of jumps before time zero. 95 To see that the described problem cannot occur, we shall use the uniform boundedness of the flip-rates together with the finite interaction range R ≡ dN−1(√N ∨ Cp)e, where dxe denotes the next largest integer. Remark 4.3.1. We have avoided random walk kernels p with infinite range for the fixed kernel interactions to simplify the analysis of the above jump equations (4.21). We show that up to time t, the evolution of ξN· can be divided up into finite random “islands” that do not communicate with each other. Indeed, two regions of Z/N do not interact with each other up to time t, if we can find an intermediate region of length 2R where no flips occur in [0, t]. We can now partition Z/N into such regions. The sums of flips for each region up to time t are independent and can be bounded by i.i.d. Poisson random variables as follows. The flips in the region centered at Z2R,Z ∈ Z can be bounded by PZt ≡ ∑ x∈Z/N, Z2R−R<x≤Z2R+R Pt,x, where Pt,x ≡ ∑ y∼x Pt(x; y) + ∑ m≥2,i,j,k=0,1 ∑ y1,...,ym∼x,|z−x|≤Cp/N Qm,i,j,kt (x; y1, . . . , ym; z). Using (4.20) we obtain that each PZt has mean t2RN ( 2c(N)N1/2 N − θ(N) 2c(N)N1/2 + ∑ m≥2,i,j,k=0,1 ( 2c(N)N1/2 )m q(k,m,N)ij (2c(N))mNm/2   ∑ |z−x|≤Cp/N p(N(x− z)) = t2RN  N − θ(N) + ∑ m≥2,i,j,k=0,1 q (k,m,N) ij   . Thus by Hypothesis 4.2.19, (PZt )Z∈Z is a sequence of i.i.d. Poisson random variables of finite mean. Let XZ be a sequence of random variables with XZ = 1 if PZt > 0 and XZ = 0 if P Z t = 0. Then (XZ)Z∈Z is an i.i.d. sequence of Bernoulli variables with P(X0 = 0) > 0. In particular we can show that with probability one there exists a random sequence . . . , Z−2, Z−1, Z0, Z1, Z2, . . . such that 1 ≤ |Zi −Zi−1| <∞ for all i ∈ Z and XZi = 0. Hence, up until time t, we can partition Z/N = ∪i∈Z(Zi2R,Zi+12R] ∩ Z/N into finite regions that do not communicate with each other up to time t. For all x of each region we can now uniquely solve (4.21) on [0, t]. As the region has finite length, we only need to consider a finite number of sites. To see this, note 96 that as Pt,x is a Poisson random variable of finite mean for each x in the region, we can have at most a finite number of flips in each region up until time t. Now iterate on successive intervals of length t to uniquely solve the entire system for all times. The interested reader is referred to the proof of Proposition 2.1(a) in [4] for how to solve such systems. Remark 4.3.2. For an alternative proof of the partition in non-communicating islands the reader is referred to Theorem 2.1 in Durrett [6]. The ideas of [6] can be applied to our setup but as we only consider dimension d = 1 the more straightforward calculation given above was possible. Having solved the equation (4.21) it remains to ensure that the solution is the spin-flip system with rates c(x, ξN ) given by (4.16). Recall the end of Remark 4.2.2 and the end of Hypothesis 4.2.19. By The- orem I.5.2 in [7] the process ξN constructed from the given rates is the unique in law solution to the martingale problem for Ω, where Ωf ( ξN ) = ∑ x c ( x, ξN ) ( f(ξNx )− f(ξN ) ) with f in the space of finite cylinder functions on {0, 1}Z/N . Hence it remains to show that for the Markov process ξN constructed in (4.21), f(ξNt )− f(ξN0 )− ∫ t 0 Ωf(ξNs )ds = f(ξNt )− f(ξN0 )− ∫ t 0 ∑ x c(x, ξNs ) ( f((ξNs )x)− f(ξNs ) ) ds is a martingale for all f in the space of finite cylinder functions on {0, 1}Z/N . Since f depends on finitely many coordinates, this is an exercise in stochastic calculus for jump processes, see Remark 4.3.4 below. In what follows we shall often drop the superscripts w.r.t. N to simplify notation. We now derive the approximate martingale problem. We take a test function φ : [0,∞)×N−1Z → R with t 7→ φt(x) continuously differentiable and satisfying∫ T 0 < |φs|+ φ2s + |∂sφs|, 1 > ds <∞ (4.22) (this condition ensures that the following integration and summation are well- defined). We apply integration by parts to ξt(x)φt(x), sum over x and multiply 97 by 1N , to obtain for t ≤ T (recall that by Remark 4.2.6 < ξt, φ >= < νt, φ >) < νt, φt > (4.23) =< ν0, φ0 > + ∫ t 0 < νs, ∂sφs > ds + 1 N ∑ x ∑ y∼x ∫ t 0 ξs−(y) (φs(x)− φs(y)) dPs(x; y) (4.24) + 1 N ∑ x ∑ y∼x ∫ t 0 ξs−(x)φs(x) (dPs(y;x)− dPs(x; y)) (4.25) + ∑ k=0,1 (1− 2k) ∑ m≥2,i,j=0,1 1 N ∑ x ∑ y1,...,ym∼x ∑ z ∫ t 0 δk(ξs−(x)) × δj(ξs−(y1)) m∏ l=2 δ1−k(ξs−(yl)) δi(ξs−(z))φs(x)dQm,i,j,ks (x; y1, . . . , ym; z). (4.26) The main ideas for analyzing terms (4.24) and (4.25) will become clear once we analyze term (4.26) in detail. The latter is the only term where calculations changed seriously compared to [9]. Hence, we shall only summarize the results for terms (4.24) and (4.25) in what follows. We break term (4.24) into two parts, an average term and a fluctuation term and after proceeding as for term (3.1) in [9] we obtain (4.24) = ∫ t 0 < νs−,∆(φs) > ds+E (1) t (φ), where E (1) t (φ) ≡ 1 N ∑ x ∑ y∼x ∫ t 0 ξs−(y) (φs(x) − φs(y)) (dPs(x; y)− d〈P (x; y)〉s) . We have suppressed the dependence on N in E (1) t (φ). E (1) t (φ) is a martingale (recall that if N ∼ Pois(λ), then Nt−λt is a martingale with quadratic variation 〈N〉t = λt) with predictable brackets process given by d 〈 E(1)(φ) 〉 t ≤ ∣∣∣∣ ∣∣∣∣D ( φt, 1√ N )∣∣∣∣ ∣∣∣∣ 2 λ < 1, e−2λ > dt. (4.27) Alternatively we also obtain the bound d 〈 E(1)(φ) 〉 t ≤ 4 ‖φt ‖0< |φt| , 1 > dt (4.28) with ‖φt ‖0= supx |φt(x)|. The second term (4.25) is a martingale which we shall denote by M (N) t (φ) (in what follows we shall drop the superscripts w.r.t. N and write Mt(φ)). It 98 can be analyzed as the martingale Zt(φ) of (3.3) in [9]. We obtain in particular that 〈M(φ)〉t = 2N − θ (N) N {∫ t 0 < ξs−, φ2s > ds− ∫ t 0 < A(ξs−φs) , ξs−φs > ds } . (4.29) Using that |A(ξs−φs)(x)| ≡ ∣∣∣∣∣ 12c(N)N1/2 ∑ y∼x ξs−(y)φs(y) ∣∣∣∣∣ ≤ 1 2c(N)N1/2 ∑ y∼x |φs(y)| ≤ sup y∼x |φs(y)|. we can further dominate 〈M(φ)〉t by 〈M(φ)〉t ≤ C(λ) ∫ t 0 (‖φs ‖2λ< 1, e−2λ >) ∧ (‖φs ‖0< ξs−, |φs| >) ds. (4.30) We break the third term (4.26) into two parts, an average term and a fluctua- tion term. Recall Notation 4.2.22 and observe that if we only consider a ∈ {0, 1} we have Fk(a) = δk(a). (4.26) = ∑ k=0,1 (1− 2k) ∑ m≥2,i,j=0,1 1 N ∑ x ∑ y1,...,ym∼x ∑ z ∫ t 0 δk(ξs−(x)) δj(ξs−(y1)) × m∏ l=2 δ1−k(ξs−(yl)) δi(ξs−(z))φs(x) q (k,m,N) ij (2c(N))mNm/2 p(N(x− z))ds +E (3) t (φ) = ∑ k=0,1 (1− 2k) ∑ m≥2,i,j=0,1 q (k,m,N) ij ∫ t 0 1 N ∑ x ( 1 2c(N)N1/2 ∑ y1∼x δj(ξs−(y1)) ) × m∏ l=2 ( 1 2c(N)N1/2 ∑ yl∼x δ1−k(ξs−(yl)) )(∑ z p(N(x− z))δi(ξs−(z)) ) × δk(ξs−(x)) φs(x)ds+E(3)t (φ) = ∑ k=0,1 (1− 2k) ∑ m≥2,i,j=0,1 q (k,m,N) ij ∫ t 0 1 N ∑ x Fj(A(ξs−)(x)) × (F1−k(A(ξs−)(x)))m−1 Fi((pN ∗ ξs−)(x))δk(ξs−(x)) φs(x)ds+E(3)t (φ) = ∑ k=0,1 (1− 2k) ∑ m≥2,i,j=0,1 q (k,m,N) ij ∫ t 0 < (Fj ◦A(ξs−)) × (F1−k ◦A(ξs−))m−1 ( Fi ◦ (pN ∗ ξs−) ) (δk ◦ ξs−) , φs > ds+E(3)t (φ), 99 where for x ∈ Z/N we set ( pN ∗ f) (x) ≡ ∑ z∈Z/N p(N(x− z))f(z) (4.31) and E (3) t (φ) ≡ ∑ k=0,1 (1− 2k) ∑ m≥2,i,j=0,1 1 N ∑ x ∑ y1,...,ym∼x ∑ z ∫ t 0 δk(ξs−(x)) × δj(ξs−(y1)) m∏ l=2 δ1−k(ξs−(yl)) δi(ξs−(z))φs(x) × ( dQm,i,j,ks (x; y1, . . . , ym; z)− q (k,m,N) ij (2c(N))mNm/2 p(N(x− z))ds ) . We have suppressed the dependence on N in E (3) t (φ). Here, E (3) t (φ) is a mar- tingale with predictable brackets process given by 〈 E(3)(φ) 〉 t ≤ ∑ m≥2,i,j,k=0,1 q (k,m,N) ij 1 N2 ∑ x m∏ l=0 (∑ yl∼x 1 2c(N)N1/2 ) (4.32) × (∑ z p(N(x− z)) )∫ t 0 φ2s(x)ds ≤ ∑ m≥2,i,j,k=0,1 q (k,m,N) ij 1 N ∫ t 0 < φ2s , 1 > ds ≤ 1 N ∑ m≥2,i,j,k=0,1 q (k,m,N) ij ∫ t 0 ‖φs ‖2λ< e−2λ, 1 > ds. Taking the above together we obtain the following approximate semimartin- gale decomposition from (4.23). < νt, φt >= < ν0, φ0 > + ∫ t 0 < νs, ∂sφs > ds (4.33) + ∫ t 0 < νs−,∆(φs) > ds+E (1) t (φ) +Mt(φ) + ∑ k=0,1 (1− 2k) ∑ m≥2,i,j=0,1 q (k,m,N) ij ∫ t 0 < (Fj ◦A(ξs−)) × (F1−k ◦A(ξs−))m−1 ( Fi ◦ (pN ∗ ξs−) ) (δk ◦ ξs−) , φs > ds +E (3) t (φ). 100 Remark 4.3.3. Note that this approximate semimartingale decomposition pro- vides the link between our approximate densities and the limiting SPDE in (4.18) for the case with no fixed kernel. Indeed, uniqueness of the limit ut of A(ξ N t ) will be derived by proving that ut solves the martingale problem associated with the SPDE (4.18). Remark 4.3.4. For all f in the space of finite cylinder functions on {0, 1}Z/N the Markov process ξ(= ξN ) constructed in (4.21) yields a martingale f(ξt)−f(ξ0)− ∫ t 0 Ωf(ξs)ds = f(ξt)−f(ξ0)− ∫ t 0 ∑ x c(x, ξs) (f((ξs)x)− f(ξs)) ds. (4.34) Indeed, every finite cylinder function on {0, 1}Z/N , f(ξ) = f(ξ(x1), . . . , ξ(xn)), n ∈ N, ξ(xi) ∈ {0, 1}, xi ∈ Z/N can be rewritten as a linear combination of functions of the form g(ξ) ≡ ξ(xi1 ) · · · ξ(xim ), where m ∈ N,m ≤ n and {i1, . . . , im} ⊂ {1, . . . , n}. By linearity we only need to consider functions of this form. Now rewrite (4.21) as ξt(x) = ξ0(x) + ∫ t 0 c(x, ξs−) (1− 2ξs−(x)) ds+ mart. (4.35) by breaking both integrals in (4.21) into an average term and a fluctuation term. Observe here that we can rewrite the sum of both average terms as in (4.35) by using for example δ0(ξs−(x)) ∑ y∼x δ1(ξs−(y)) N − θ(N) 2c(N)N1/2 = δ0(ξs−(x)) ( N − θ(N) ) f1(x, ξs−) and ∑ z δi(ξs−(z)) p(N(x− z)) = gi(x, ξs−) . Now use the representation of the rates c(x, ξ) from (4.16). Both fluctuation terms turn out to be martingales that only depend on the Poisson processes Pt(x; y) and Q m,i,j,k t (x; y1, . . . , ym; z). Hence, for x 6= x′ the martingale terms are orthogonal. For m = 2, that is for g(ξ) = ξ(x)ξ(x′) with x 6= x′, we can now use the integration by parts formula (cf. Theorem VI.(38.3) in Rogers and Williams [13]), the orthogonality of the martingale terms of ξt(x) and ξt(x ′) and g(ξx)− g(ξ) = (1− ξ(x))ξ(x′)− ξ(x)ξ(x′) = (1− 2ξ(x))ξ(x′) to obtain that (4.34) is a martingale for f = g. Now iterate the above reasoning to obtain the claim for all m ∈ N. 101 4.4 Green’s Function Representation Analogous to [9], define a test function ψzt (x) ≥ 0 for t ≥ 0, x, z ∈ N−1Z as the unique solution, satisfying (4.22) and s.t. ∂ ∂t ψzt = ∆ψ z t , ψz0(x) = N1/2 2c(N) 1(x ∼ z) with ∆ψzt (x) = N − θ(N) 2c(N)N1/2 ∑ y∼x (ψzt (y)− ψzt (x)) (4.36) as in (4.19). Note that ψz0 was chosen s.t. < νt, ψ z 0 >= A(ξt)(z) and that we suppress the dependence on N . Next observe that ∆ is the generator of a simple random walk Xt ∈ N−1Z, jumping at rate N−θ (N) 2c(N)N1/2 ( 2c(N)N1/2 ) = N − θ(N) = (1 + o(1))N with sym- metric steps of variance 1N ( 1 3 + o(1) ) , where we used that c(N) N→∞→ 1. Here o(1) denotes some function that satisfies o(1) → 0 for N →∞. Define ψ̄zt (x) = NP(Xt = x|X0 = z) then < ψz0 , ψ̄ x t > = 1 N ∑ y ψz0(y)ψ̄ x t (y) = 1 N ∑ y N1/2 2c(N) 1(y ∼ z)NP(Xt = y|X0 = x) (4.37) = N1/2 2c(N) ∑ y∼z P(Xt = y|X0 = x) = Ex[ψz0(Xt)] = ψzt (x). As we shall see later in Lemma 4.5.2(b), when linearly interpolated, the functions ψzt (x) and ψ̄ z t (x) converge to p ( t 3 , z − x ) (the proof follows), where pt(x) = 1√ 2pit e− x2 2t is the Brownian transition density. The next Lemma gives some information on the test functions ψ and ψ̄ from above. Later on, this will provide us with estimates necessary for establishing tightness. 102 Lemma 4.4.1. There exists N0 <∞ s.t. for N ≥ N0, T ≥ 0, z ∈ N−1Z, λ ≥ 0, (a) < ψzt , 1 >=< ψ̄ z t , 1 >= 1 and ‖ψzt ‖0≤ CN1/2 for all t ≥ 0. (b) < eλ, ψ z t + ψ̄ z t >≤ C(λ, T )eλ(z) for all t ≤ T , (c) ‖ψzt ‖λ≤ C(λ, T ) ( N1/2 ∧ t−2/3) eλ(z) for all t ≤ T , (d) < ∣∣ψ̄zt − ψ̄zs ∣∣ , 1 >≤ 2N |t− s| for all s, t ≥ 0. If we further restrict ourselves to N ≥ N0, N−3/4 ≤ s < t ≤ T, y, z ∈ N−1Z, |y − z| ≤ 1, we get (e) ‖ψzt − ψyt ‖λ≤ C(λ, T )eλ(z) ( |z − y|1/2t−1 +N−1/2t−3/2 ) , (f) ‖ψzt − ψzs ‖λ≤ C(λ, T )eλ(z) ( |t− s|1/2s−3/2 +N−1/2s−3/2 ) , (g) ∣∣∣∣∣∣D(ψzt , N−1/2) (·)∣∣∣∣∣∣ λ ≤ C(λ, T )eλ(z)N−1/4t−1. Proof. First we shall derive an explicit description for the test functions ψzt and ψ̄zt . We proceed as at the beginning of Section 4 in [9] by using that ∆ as in (4.36) is the generator of a simple random walk. Let (Yi)i=1,2,... be i.i.d. and uniformly distributed on (jN −1 : 0 < |j| ≤ √N). Set ρ(t) = E [ eitY1 ] and Sk = k∑ i=1 Yi. (4.38) Note that E[Y 21 ] = 1+o(1) 3N , where o(1) → 0 for N → ∞. Similarly, E[Y 41 ] = 1+o(1) 5N2 , where o(1) may change from line to line. The relation between the test functions ψzt , ψ̄ z t and Sk is as follows. ψzt (x) = Ex[ψ z 0(Xt)] = ∞∑ k=0 ((N − θ(N))t)k k! e−((N−θ (N))t)NP(Sk+1 = x− z), (4.39) ψ̄zt (x) = NP(Xt = x|X0 = z) = ∞∑ k=0 ((N − θ(N))t)k k! e−((N−θ (N))t)NP(Sk = x− z). Now we can start proving the above Lemma. (a) follows as in the proof of Lemma 3(a), [9], using that P(Sk = x) ≤ CN−1/2 for all x ∈ N−1Z, k ≥ 1. (b) follows as in the proof of Lemma 3(b), [9], where we shall use the bound E [ eµ|Y1| ] ≤ exp { 5µ2 1 N } 103 for all µ ≥ 0 to obtain the claim. (c) Following the proof of Lemma 3(c) in [9], we first show that we have, for k ∈ N and |x| ≥ 1, P(Sk = x) ≤ 1N P(Sk ≥ |x| − 1), which we can use to obtain P(Sk = x) ≤ 1N e−µ(|x|−1) exp { 5kµ2 1N } . Substituting this bound into (4.39) gives for any µ ≥ 0 ψzt (x) ≤ C(µ, T ) exp{−µ|x− z|} (4.40) for all t ≤ T and |x− z| ≥ 1. From (4.39) we further have for N big enough ψzt (x) ≡ E [ p ( (1 + o(1))(Pt + 1) 3N , x− z )] +E(N, t, x − z), where Pt ∼ Pois((N − θ(N))t). Using Corollary B.0.2 we get as in the proof of [9], Lemma 3(c), |E(N, t, x)| ≤ C 1 N ( 1 + t−3/2 ) for N−3/4 ≤ t. Here we used that for P ∼ Pois(r), r > 0 we have E[(P + 1)a] ≤ C(a)ra for all a < 0. (This is obviously true for 0 < r < 1. For r ≥ 1 fixed, prove the claim first for all a ∈ Z. Then extend this result to general a < 0 by an application of Hölder’s inequality.) Using the trivial bound p(t, x) ≤ Ct−1/2 we get from the above ψzt (x) ≤ C(T )t−2/3 for N−3/4 ≤ t ≤ T. Finally, we obtain ‖ψzt ‖λ=sup x { |ψzt (x)| eλ|x| } (4.40) ≤ sup {x:|x−z|≥1} { C(λ, T )e−λ|x−z|eλ|x| } ∨ sup {x:|x−z|<1,N−3/4≤t≤T} { C(T )t−2/3eλ|x| } ∨ sup {x:|x−z|<1,0≤t≤N−3/4} { |ψzt (x)| eλ|x| } (a) ≤ { C(λ, T )eλ|z| } ∨ { 1 ( N−3/4 ≤ t ≤ T ) C(T )t−2/3e1eλ|z| } ∨ { 1 ( 0 ≤ t ≤ N−3/4 ) CN1/2e1eλ|z| } ≤C(λ, T ) ( N1/2 ∧ t−2/3 ) eλ(z) for all t ≤ T. 104 This proves part (c). (d) follows along the lines of the proof of [9], Lemma 3(d). (e) For the remaining parts (e)-(g) we fix N−3/4 ≤ s < t ≤ T, y, z ∈ N−1Z, |y−z| ≤ 1. For part (e) we follow the reasoning of the proof of [9], Lemma 3(e). The only change occurs in the derivation of the last estimate. In summary, we find as in [9] that ‖ψzt − ψyt ‖0≤ C(T ) ( |z − y|t−1 +N−1t−3/2 ) . (4.41) Now recall (4.40) with µ = 2λ to get ψzt (x) +ψ y t (x) ≤ C(λ, T ) exp{−2λ|x− z|} for |x − z| ≥ 1, |x − y| ≥ 1, |y − z| ≤ 1 and thus in particular for |x − z| ≥ 2, |y − z| ≤ 1. This yields ‖ψzt − ψyt ‖λ≤ sup {x:|x−z|<2} ‖ψzt − ψyt ‖0 eλ(x) + sup {x:|x−z|≥2} { C(λ, T ) ‖ψzt − ψyt ‖1/20 e−λ|x−z|eλ(x) } ≤C(λ, T )eλ(z) ( ‖ψzt − ψyt ‖0 + ‖ψzt − ψyt ‖1/20 ) ≤C(λ, T )eλ(z) (( |z − y|t−1 +N−1t−3/2 ) + ( |z − y|t−1 +N−1t−3/2 )1/2) ≤C(λ, T )eλ(z) ( |z − y|1/2t−1 +N−1/2t−3/2 ) . This proves (e). (f) The proof of part (f) follows analogously to the proof of part (e), with changes as suggested in the proof of [9], Lemma 3(f). (g) Finally, to prove part (g), use part (e), ψzt (y) = ψ y t (z) (see (4.39)) and the definition of D ( ψzt , N −1/2 ) (x) = sup { |ψzt (y)− ψzt (x)| : |x− y| ≤ N−1/2, y ∈ N−1Z } to get ‖D ( ψzt , N −1/2 ) (·)‖λ ≤ sup {x:|x−z|<2} { sup y:|x−y|≤N−1/2 {|ψzt (y)− ψzt (x)|} eλ|x| } + sup {x:|x−z|≥2} { sup y:|x−y|≤N−1/2 {|ψzt (y)− ψzt (x)|} eλ|x| } (4.40) ≤ C(λ) sup {x:|x−z|<2} { sup y:|x−y|≤N−1/2 {|ψzt (y)− ψzt (x)|} eλ|z| } + C(λ, T ) sup {x:|x−z|≥2} { sup y:|x−y|≤N−1/2 { |ψzt (y)− ψzt (x)|1/2 } e−λ|x−z|eλ|x| } . 105 Next use that ψat (b) = ψ b t (a) to get as a further upper bound C(λ) sup {x:|x−z|<2} { sup y:|x−y|≤N−1/2 {|ψyt (z)− ψxt (z)|} eλ|z| } + C(λ, T ) sup {x:|x−z|≥2} { sup y:|x−y|≤N−1/2 { |ψyt (z)− ψxt (z)|1/2 } eλ|z| } (4.41) ≤ C(λ, T )eλ(z) sup x,y:|x−y|≤N−1/2 {( |x− y|t−1 +N−1t−3/2 ) + ( |x− y|t−1 +N−1t−3/2 )1/2} ≤ C(λ, T )eλ(z)N−1/4t−1, where we used N−3/4 < t ≤ T . This finishes the proof of (g) and it also finishes the proof of the Lemma. The following Corollary uses the results of Lemma 4.4.1 to obtain estimates that we shall need later. Corollary 4.4.2. There exists N0 < ∞ s.t. for N ≥ N0, 0 ≤ δ ≤ u ≤ t ≤ T and y, z ∈ N−1Z, λ ≥ 0, we have (a) ∫ t u ‖ψzt−s ‖λ ds ≤ C(λ, T )(t− u)1/3eλ(z) and ∫ t 0 ‖ψzt−s ‖2λ ds ≤ C(λ, T )N1/4e2λ(z). (b) For |y − z| ≤ 1 and δ ≤ t−N−3/4 we further have sup0≤s≤δ ‖ψzt−s − ψyt−s ‖λ ≤ C(λ, T )eλ(z) {|z − y|1/2(t− δ)−1 +N−1/2(t− δ)−3/2}. (c) We also have ∫ t δ ‖ψzt−s − ψyt−s ‖λ ds ≤ C(λ, T ) (eλ(z) + eλ(y)) (t− δ)1/3. (d) For N−3/4 ≤ u− δ we have sup0≤s≤δ ‖ψzt−s − ψzu−s ‖λ ≤ C(λ, T )eλ(z) { (t− u)1/2(u− δ)−3/2 +N−1/2(u− δ)−3/2}. (e) Finally, we have ∫ u δ ‖ψzt−s − ψzu−s ‖λ ds ≤ C(λ, T )eλ(z)(u− δ)1/3. Proof. The proof is a combination of the results of Lemma 4.4.1. (a) We have for n = 1, 2 and 0 ≤ u ≤ t by Lemma 4.4.1(c) ∫ t u ∣∣∣∣ψzt−s∣∣∣∣nλ ds ≤ C(λ, T ) ∫ t u Nn/2 ∧ (t− s)−2n/3ds enλ(z). For n = 1 further bound the integrand by (t− s)−2/3, for n = 2 and u = 0 use the above integrand to obtain the claim. (b) follows from Lemma 4.4.1(e). 106 (c) We further have by Lemma 4.4.1(c)∫ t δ ‖ψzt−s − ψyt−s ‖λ ds ≤ C(λ, T ) (eλ(z) + eλ(y)) ∫ t δ (t− s)−2/3ds. (d) follows from Lemma 4.4.1(f). (e) Using Lemma 4.4.1(c) once more, we get∫ u δ ‖ψzt−s − ψzu−s ‖λ ds ≤ C(λ, T )eλ(z) ∫ u δ (t− s)−2/3 + (u− s)−2/3ds, which concludes the proof after some basic calculations. We shall need the following technical Lemma. Lemma 4.4.3. For f : N−1Z → [0,∞) with < f, 1 ><∞, λ ∈ R we have (a) < νs, ψ z t−s >=< A(ξs), ψ̄ z t−s >, (b) |< νt, f > − < A(ξt), f >| ≤ C(λ) ‖D(f,N−1/2)‖λ. Proof. (a) follows easily from < νs, ψ z t−s > =< ξs, ψ z t−s >= 1 N ∑ x ξs(x)ψ z t−s(x) = 1 N ∑ x ξs(x)ψ x t−s(z) (4.37) = 1 N ∑ x ξs(x) < ψ x 0 , ψ̄ z t−s > = 1 N ∑ x ξs(x) 1 N ∑ y N1/2 2c(N) 1(y ∼ x)ψ̄zt−s(y) = 1 N ∑ y {∑ x 1 2c(N)N1/2 1(y ∼ x)ξs(x) } ψ̄zt−s(y) = 1 N ∑ y A(ξs)(y)ψ̄ z t−s(y) =< A(ξs), ψ̄ z t−s > . Part (b) follows as in the proof of Lemma 5(b) in [9]. Observe in particular that < νt, e−λ >≤ C(λ) as will be shown before and in (4.44) below. Taken all together this finishes the proof. Next use the test function φs ≡ ψxt−s for s ≤ t in the semimartingale decomposition (4.33) and observe that φ satisfies (4.22). Here the initial condition is chosen so that < νt, φt >=< νt, ψ x 0 >= A(ξt)(x). The test function chosen in [9] at the beginning of page 526, namely φs = eθc(t−s)ψxt−s was chosen so that the drift term < νs, θcφs > ds of the semi- martingale decomposition (2.9) in [9] would cancel out with the drift term 107 < νs, ∂sφs > ds. As we have multiple coefficients, this is not possible. Also, it turned out that the calculations become easier once we consider time differences in Section 4.5 to follow. With the above choice we obtain, for a fixed value of t, an approximate Green’s function representation for A(ξt), namely A(ξt)(x) = < ν0, ψ x t > + ∫ t 0 < νs, (−∆ψxt−s) > ds (4.42) + ∫ t 0 < νs−,∆ ( ψxt−s ) > ds+E (1) t ( ψxt−· ) +Mt ( ψxt−· ) + ∑ k=0,1 (1− 2k) ∑ m≥2,i,j=0,1 q (k,m,N) ij ∫ t 0 < (Fj ◦A(ξs−)) × (F1−k ◦A(ξs−))m−1 ( Fi ◦ (pN ∗ ξs−) ) (δk ◦ ξs−) , ψxt−s > ds +E (3) t (ψ x t−·). The following Lemma is stated analogously to Lemma 4 of [9]. Parts (a) and (c) will follow easily in our setup and so the only significant statement will be part (b). Lemma 4.4.4. Suppose that the initial conditions satisfy A(ξ0) → u0 in C as N →∞. Then for T ≥ 0, p ≥ 2, λ > 0, (a) E [ supt≤T < νt, e−λ > p ] ≤ C(λ, p). (b) We further have E [∣∣∣E(1)t (ψzt−·)∣∣∣p ∨ ∣∣∣E(3)t (ψzt−·)∣∣∣p] ≤ C(λ, p, T )(1 + Cp/2Q )N−p/16eλp(z) for all t ≤ T and N big enough, where we set CQ ≡ sup N≥N0 ∑ m≥2,i,j,k=0,1 q (k,m,N) ij . (4.43) (c) Finally, ‖E[A(ξt)]‖−λp≤ 1 for all t ≤ T . Proof. First observe that we have ξt ∈ {0, 1}Z/N and 0 ≤ A(ξt) ≤ 1. Therefore, parts (a) and (c) follow immediately. Indeed, for (a) we only need to observe that 0 ≤< νt, e−λ >=< ξt, e−λ|·| >= 1 N ∑ x ξt(x)e −λ|x| ≤ 1 N ∑ x:x∈Z/N e−λ|x| ≤ 2 N ∞∑ j=0 e−λj/N = 2 N 1 1− e−λ/N N→∞→ 2 λ . 108 Note in particular that we showed that < e−λ, 1 >≤ C λ for all λ > 0, N = N(λ) big enough, (4.44) which will prove useful later. For (c) we further have ‖E[A(ξt)]‖−λp= sup x |E[A(ξt)(x)]| e−λp|x| ≤ sup x e−λp|x| ≤ 1. It only remains to show that (b) holds. (b) First observe that CQ <∞ by Hypothesis 4.2.19. We shall apply a Burkholder-Davis-Gundy inequality in the form E [ sup s≤t |Xs|p ] ≤ C(p)E [ 〈X〉p/2t + sup s≤t |Xs −Xs−|p ] (4.45) for a cadlag martingale X with X0 = 0 (this inequality may be derived from its discrete time version, see Burkholder [1], Theorem 21.1). To get an upper bound on the second term of the r.h.s. of (4.45) for the mar- tingales we consider, observe that the largest possible jumps of the martingales E (1) t (ψ z t−·) respectively E (3) t (ψ z t−·) are bounded a.s. by CN −1/2. Indeed, E (1) t (ψ z t−·) = 1 N ∑ x ∑ y∼x ∫ t 0 ξs−(y) ( ψzt−s(x)− ψzt−s(y) ) (dPs(x; y) − d〈P (x; y)〉s) and thus, using Lemma 4.4.1(a), the maximal jump size is bounded by 1 N 2 sup t≤T ‖ψzt ‖0≤ C N1/2 (4.46) (the maximal number of jumps at a fixed time is 1). The bound on the maximal jump size of E (3) t (ψ z t−·) follows analogously. Now choose t ≤ T . We shall start with E(3)t (ψzt−·). By (4.45), (4.46) and (4.32) we have E [∣∣∣E(3)t (ψzt−·)∣∣∣p] ≤ C(p)   ∑ m≥2,i,j,k=0,1 q (k,m,N) ij 1 N ∫ t 0 ∣∣∣∣ψzt−s∣∣∣∣2λ < e−2λ, 1 > ds   p/2 + C(p)N−p/2 (4.44) ≤ C(λ, p)Cp/2Q N−p/2 {(∫ t 0 ∣∣∣∣ψzt−s∣∣∣∣2λ ds )p/2 + 1 } . By Corollary 4.4.2(a) this is bounded from above by C(λ, p, T )C p/2 Q N −p/2 { Np/8eλp(z) + 1 } = C(λ, p, T )C p/2 Q N −3p/8eλp(z). 109 It remains to investigate E (1) t (ψ z t−·). Here (4.45), (4.46), (4.27) and (4.28) yield E [∣∣∣E(1)t (ψzt−·)∣∣∣p] ≤ C(p) (∫ t 0 [‖ψzt−s ‖0< ψzt−s, 1 >] ∧ [∣∣∣∣ ∣∣∣∣D ( ψzt−s, 1√ N )∣∣∣∣ ∣∣∣∣ 2 λ < 1, e−2λ > ] ds )p/2 + C(p)N−p/2. This in turn is bounded from above by C(p) (∫ t 0 [ C(T )(t− s)−2/3 ] ∧ [∣∣∣∣ ∣∣∣∣D ( ψzt−s, 1√ N )∣∣∣∣ ∣∣∣∣ 2 λ C(λ) ] ds )p/2 +C(p)N−p/2, where we used Lemma 4.4.1(a), (c) and (4.44). To apply Lemma 4.4.1(g) to the second part of the integrand, we need to ensure that N−3/4 ≤ t − s. As N−3/4 ≤ N−3/8 we get as a further upper bound C(p) (∫ N−3/8 0 C(T )s−2/3ds+ ∫ t N−3/8∧t ( C(λ, T )eλ(z)N −1/4s−1 )2 C(λ)ds )p/2 + C(p)N−p/2 ≤ C(λ, p, T )eλp(z) {(( N−3/8 )1/3 +N−1/2 ( N−3/8 )−1)p/2 +N−p/2 } ≤ C(λ, p, T )N−p/16eλp(z). This finishes the proof. 4.5 Tightness In what follows we shall derive estimates on pth-moment differences of Â(ξt)(z) ≡ A(ξt)(z)− < ν0, ψzt > . Recall the assumption A(ξ0) → u0 in C from Theorem 4.2.9 resp. Theorem 4.2.23 from the beginning. Also note that Lemma 4.5.2(b) to come will yield that ψzt (x) converges to p ( t 3 , z − x ) . The estimates of Lemma 4.5.1 and the convergence of ψzt taken together will be sufficient to show C-tightness of the approximate densities A(ξt)(z) at the end of this Section. 110 Lemma 4.5.1. For 0 ≤ s ≤ t ≤ T, y, z ∈ N−1Z, |t − s| ≤ 1, |y − z| ≤ 1, λ > 0 and p ≥ 2 we have E [∣∣∣Â(ξt)(z)− Â(ξs)(y)∣∣∣p] ≤ C(λ, p, T ) ( 1 + CpQ ) eλp(z) ( |t− s|p/24 + |z − y|p/24 +N−p/24 ) . Proof. Fix s, t, T, y, z, λ, p as in the statement. We decompose the increment Â(ξt)(z) − Â(ξs)(y) into a space increment Â(ξt)(z) − Â(ξt)(y) and a time in- crement Â(ξt)(y)− Â(ξs)(y). We consider first the space differences. From the Green’s function represen- tation (4.42), the estimates obtained in Lemma 4.4.4(b) for the error terms E(1) and E(3) and the linearity of Mt(φ) and E (1) t (φ), E (3) t (φ) in φ, we get E [∣∣∣Â(ξt)(z)− Â(ξt)(y)∣∣∣p] ≤ C(λ, p, T ) ( 1 + C p/2 Q ) N−p/16eλp(z) + E [∣∣Mt(ψzt−· − ψyt−·)∣∣p] + E [∣∣∣ ∑ k=0,1 (1− 2k) ∑ m≥2,i,j=0,1 q (k,m,N) ij ∫ t 0 < (Fj ◦A(ξs−)) × (F1−k ◦A(ξs−))m−1 ( Fi ◦ (pN ∗ ξs−) ) (δk ◦ ξs−) , ( ψzt−s − ψyt−s ) > ds ∣∣∣p]. Recall definition (4.31) and observe that 0 ≤ (pN ∗ ξs−) (x) ≤ 1 follows from ξs− ∈ {0, 1}Z/N . Use this and 0 ≤ A(ξs−)(x) ≤ 1 together with the definition of Fk from Notation 4.2.22 to get E [∣∣∣Â(ξt)(z)− Â(ξt)(y)∣∣∣p] (4.47) ≤ C(λ, p, T ) ( 1 + C p/2 Q ) N−p/16eλp(z) + E [∣∣Mt(ψzt−· − ψyt−·)∣∣p] + E [( ∑ m≥2,i,j,k=0,1 q (k,m,N) ij × ∫ t 0 < (F1−k ◦A(ξs−)) (δk ◦ ξs−) , ∣∣ψzt−s − ψyt−s∣∣ > ds)p] ≤ C(λ, p, T ) ( 1 + C p/2 Q ) N−p/16eλp(z) + E [∣∣Mt(ψzt−· − ψyt−·)∣∣p] + CpQE [(∫ t 0 < A(ξs−) + ξs−, ∣∣ψzt−s − ψyt−s∣∣ > ds)p]. Note that this is the main step to see why the fixed kernel interaction does not impact our results on tightness. In what follows, we shall employ a similar strategy to the proof of Lemma 6 in [9] to obtain estimates on the above. We nevertheless give full calculations as we proceeded in a different logical order to highlight the ideas for obtaining bounds. Minor changes in the exponents of our bounds ensued, both due to the different logical order and the different setup. 111 Let us first derive a bound on E [∣∣Mt(ψzt−· − ψyt−·)∣∣p]. Using the Burkholder- Davis-Gundy inequality (4.45) from above and observing that the jumps of the martingales Mt(ψ x t−·) are bounded a.s. by CN −1/2 we have for any 0 ≤ δ ≤ t E [∣∣Mt(ψzt−· − ψyt−·)∣∣p] (4.48) (4.30) ≤ C(λ, p)E [(∫ δ 0 ‖ψzt−s − ψyt−s ‖2λ< 1, e−2λ > ds + ∫ t δ ‖ψzt−s − ψyt−s ‖0< ξs−, ∣∣ψzt−s − ψyt−s∣∣ > ds )p/2] + C(p)N−p/2 (4.44) ≤ C(λ, p)E [( T sup 0≤s≤δ ‖ψzt−s − ψyt−s ‖2λ 1 λ + ∫ t δ ‖ψzt−s − ψyt−s ‖0< ξs−, ∣∣ψzt−s − ψyt−s∣∣ > ds )p/2] + C(p)N−p/2. Now observe that by Lemma 4.4.3(a) and Lemma 4.4.1(a), < ξs−, ∣∣ψzt−s − ψyt−s∣∣ > ≤< ξs−, ψzt−s + ψyt−s > (4.49) =< A(ξs−), ψ̄zt−s + ψ̄ y t−s > ≤< 1, ψ̄zt−s + ψ̄yt−s > = 2. We can therefore apply the estimates from Corollary 4.4.2(b) to the first term in (4.48) and Corollary 4.4.2(c) to the second term, assuming δ ≤ (t−N−3/4)∨ 0 and using |y − z| ≤ 1 to obtain E [∣∣Mt(ψzt−· − ψyt−·)∣∣p] ≤ C(λ, p, T )eλp(z) { |z − y|p/2(t− δ)−p +N−p/2(t− δ)−3p/2 + (t− δ)p/6 } + C(p)N−p/2. Now set δ = t− (( |z − y|1/4 ∨N−1/4 ) ∧ t ) and observe that δ ≤ (t−N−3/4) ∨ 0 follows. We obtain t− δ = (|z − y|1/4 ∨N−1/4) ∧ t and |z − y|1/4 ≤ N−1/4 ⇒ |z − y|p/2(t− δ)−p +N−p/2(t− δ)−3p/2 + (t− δ)p/6 (4.50) ≤ |z − y|p/2|z − y|−p/4 +N−p/2N3p/8 +N−p/24 = |z − y|p/4 +N−p/8 +N−p/24, |z − y|1/4 > N−1/4 ⇒ |z − y|p/2(t− δ)−p +N−p/2(t− δ)−3p/2 + (t− δ)p/6 ≤ |z − y|p/2|z − y|−p/4 +N−p/2N3p/8 + |z − y|p/24 = |z − y|p/4 +N−p/8 + |z − y|p/24. 112 Plugging this back in the above estimate we finally have E [∣∣Mt(ψzt−· − ψyt−·)∣∣p] ≤ C(λ, p, T )eλp(z){|z − y|p/24 +N−p/24} . Next we shall get a bound on the last term of (4.47). Recall that < ξt, φ >= < νt, φ >. We get E [(∫ t 0 < A(ξs−) + ξs−, ∣∣ψzt−s − ψyt−s∣∣ > ds)p] ≤ C(p) { E [(∫ δ 0 < A(ξs−) + νs−, e−λ > ds sup 0≤s≤δ ‖ψzt−s − ψyt−s ‖λ )p] +E [(∫ t δ < A(ξs−) + νs−, e−λ >‖ψzt−s − ψyt−s ‖λ ds )p]} . Now use that < A(ξs−) + νs−, e−λ >=< A(ξs−) + ξs−, e−λ >≤< 2, e−λ > (4.44) ≤ C(λ) (4.51) to obtain that the above is bounded by C(p) {( TC(λ) sup 0≤s≤δ ‖ψzt−s − ψyt−s ‖λ )p + (∫ t δ C(λ) ‖ψzt−s − ψyt−s ‖λ ds )p} ≤ C(λ, p, T )eλp(z) { |z − y|p/2(t− δ)−p +N−p/2(t− δ)−3p/2 +(t− δ)p/3 } , where we used Corollary 4.4.2(b),(c) and |y − z| ≤ 1. Here we assumed δ ≤( t−N−3/4) ∨ 0 when we applied Corollary 4.4.2(b). Now choose δ = t− ((|z − y|1/4 ∨N−1/4) ∧ t) ≤ (t−N−3/4)∨ 0 as before. Reasoning as in (4.50), we get C(λ, p, T )eλp(z) ( N−p/8 + |z − y|p/12 ) as an upper bound. Now we can take all the above bounds together and plug them back into (4.47) to obtain (recall that |z − y| ≤ 1) E [∣∣∣Â(ξt)(z)− Â(ξt)(y)∣∣∣p] ≤ C(λ, p, T ) ( 1 + C p/2 Q ) N−p/16eλp(z) + C(λ, p, T ) ( 1 + CpQ ) eλp(z) { |z − y|p/24 +N−p/24 } ≤ C(λ, p, T ) ( 1 + C p/2 Q + C p Q ) eλp(z) ( |z − y|p/24 +N−p/24 ) . 113 Next we derive a similar bound on the time differences. We start by sub- tracting the two Green’s function representations again, this time for the time differences, using (4.42) and Lemma 4.4.4(b) for the error terms. E [∣∣∣Â(ξt)(z)− Â(ξu)(z)∣∣∣p] (4.52) ≤ C(λ, p, T ) ( 1 + C p/2 Q ) N−p/16eλp(z) + E [∣∣Mt(ψzt−·)−Mu(ψzu−·)∣∣p] + E [∣∣∣ ∑ k=0,1 (1− 2k) ∑ m≥2,i,j=0,1 q (k,m,N) ij × {∫ t 0 < (Fj ◦A(ξs−)) (F1−k ◦A(ξs−))m−1 ( Fi ◦ (pN ∗ ξs−) ) × (δk ◦ ξs−) , ψzt−s > ds − ∫ u 0 < (Fj ◦A(ξs−)) (F1−k ◦A(ξs−))m−1 ( Fi ◦ (pN ∗ ξs−) ) × (δk ◦ ξs−) , ψzu−s > ds }∣∣∣p] ≤ C(λ, p, T ) ( 1 + C p/2 Q ) N−p/16eλp(z) + E [∣∣Mt(ψzt−·)−Mu(ψzu−·)∣∣p] + E [( ∑ m≥2,i,j,k=0,1 q (k,m,N) ij × {∫ t u < (Fj ◦A(ξs−)) (F1−k ◦A(ξs−))m−1 ( Fi ◦ (pN ∗ ξs−) ) × (δk ◦ ξs−) , ψzt−s > ds + ∫ u 0 < (Fj ◦A(ξs−)) (F1−k ◦A(ξs−))m−1 ( Fi ◦ (pN ∗ ξs−) ) × (δk ◦ ξs−) , ∣∣ψzt−s − ψzu−s∣∣ > ds})p] ≤ C(λ, p, T ) ( 1 + C p/2 Q ) N−p/16eλp(z) + E [∣∣Mt(ψzt−·)−Mu(ψzu−·)∣∣p] + E [( ∑ m≥2,i,j,k=0,1 q (k,m,N) ij {∫ t u < (F1−k ◦A(ξs−)) (δk ◦ ξs−) , ψzt−s > ds + ∫ u 0 < (F1−k ◦A(ξs−)) (δk ◦ ξs−) , ∣∣ψzt−s − ψzu−s∣∣ > ds })p] ≤ C(λ, p, T ) ( 1 + C p/2 Q ) N−p/16eλp(z) + E [∣∣Mt(ψzt−·)−Mu(ψzu−·)∣∣p] + CpQE [(∫ t u < A(ξs−) + ξs−, ψzt−s > ds + ∫ u 0 < A(ξs−) + ξs−, ∣∣ψzt−s − ψzu−s∣∣ > ds)p]. For the martingale term we further get via the Burkholder-Davis-Gundy in- 114 equality (4.45) E [∣∣Mt(ψzt−·)−Mu(ψzt−·)∣∣p] ≤ C(p){E[∣∣Mt(ψzt−·)−Mu(ψzt−·)∣∣p]+ E[∣∣Mu(ψzt−·)−Mu(ψzu−·)∣∣p]} ≤ C(p)E [∣∣〈M·(ψzt−·) 〉t − 〈M·(ψzt−·) 〉u∣∣p/2 ] + C(p)E [∣∣〈M·(ψzt−· − ψzu−·) 〉u∣∣p/2 ] + C(p)N−p/2 ≤ C(λ, p)E [(∫ t u ‖ψzt−s ‖0< ξs−, ψzt−s > ds )p/2] + C(λ, p) (∫ δ∧u 0 ‖ψzt−s − ψzu−s ‖2λ< 1, e−2λ > ds )p/2 + C(λ, p)E [(∫ u δ∧u ‖ψzt−s − ψzu−s ‖0< ξs−, ∣∣ψzt−s − ψzu−s∣∣ > ds )p/2] + C(p)N−p/2, where we used equation (4.30) to bound the first and second term. Using (4.44) and reasoning as in (4.49) the above can further be bounded by E [∣∣Mt(ψzt−·)−Mu(ψzu−·)∣∣p] ≤ C(λ, p) (∫ t u ‖ψzt−s ‖0 ds )p/2 + C(λ, p, T ) sup 0≤s≤δ∧u ‖ψzt−s − ψzu−s ‖pλ + C(λ, p) (∫ u δ∧u ‖ψzt−s − ψzu−s ‖0 ds )p/2 + C(p)N−p/2. Under the assumption N−3/4 ∧ u ≤ u − (δ ∧ u) we obtain from Corol- lary 4.4.2(a), (d), (e) that E [∣∣Mt(ψzt−·)−Mu(ψzu−·)∣∣p] (4.53) ≤ C(λ, p, T )eλp(z) { (t− u)p/6 + ( |t− u|p/2 +N−p/2 ) (u− (δ ∧ u))−3p/2 + (u− (δ ∧ u))p/6 +N−p/2 } . Finally observe that with δ = u− (( |t− u|1/4 ∨N−1/4 ) ∧ u ) we get N−3/4 ∧ u ≤ u− δ and by proceeding as in (4.50) we obtain E [∣∣Mt(ψzt−·)−Mu(ψzu−·)∣∣p] ≤ C(λ, p, T )eλp(z) { (t− u)p/6 + |t− u|p/24 +N−p/24 +N−p/2 } . 115 Finally, we can bound the last expectation of the last line of (4.52) by using < A(ξt−s) + ξs−, ψzt−s > ≤ < 1 + 1, ψzt−s >= 2. Here the last equality followed from Lemma 4.4.1(a). We thus obtain as an upper bound on the last expectation of the last line of (4.52), C(p) { |t− u|p + E [(∫ u 0 < A(ξs−) + νs−, ∣∣ψzt−s − ψzu−s∣∣ > ds )p]} . We further have for the second term E [(∫ u 0 < A(ξs−) + νs−, ∣∣ψzt−s − ψzu−s∣∣ > ds)p] ≤ C(p) { E [(∫ δ∧u 0 < A(ξs−) + νs−, e−λ > ds sup 0≤s≤δ ‖ψzt−s − ψzu−s ‖λ )p] +E [(∫ u δ∧u < A(ξs−) + νs−, e−λ >‖ψzt−s − ψzu−s ‖λ ds )p]} (4.51) ≤ C(λ, p, T ) {( sup 0≤s≤δ∧u ‖ψzt−s − ψzu−s ‖λ )p + (∫ u δ∧u ‖ψzt−s − ψzu−s ‖λ ds )p} ≤ C(λ, p, T )eλp(z) { (t− u)p/2(u− (δ ∧ u))−3p/2 +N−p/2(u− (δ ∧ u))−3p/2 +(u− (δ ∧ u))p/3 } , where we assumed N−3/4 ∧ u ≤ u− (δ ∧ u) when we applied Corollary 4.4.2(d) together with Corollary 4.4.2(e) in the last line. Now reason as from (4.53) on to obtain C(λ, p, T )eλp(z) { |t− u|p/24 +N−p/24 } as an upper bound. Taking all bounds together we have for the time differences from (4.52) E [∣∣∣Â(ξt)(z)− Â(ξu)(z)∣∣∣p] ≤ C(λ, p, T ) ( 1 + C p/2 Q ) N−p/16eλp(z) + C(λ, p, T )eλp(z) { (t− u)p/6 + |t− u|p/24 +N−p/24 +N−p/2 } + C(p)CpQ { |t− u|p + C(λ, p, T )eλp(z) ( |t− u|p/24 +N−p/24 )} ≤ C(λ, p, T ) ( 1 + C p/2 Q + C p Q ) eλp(z) { |t− u|p/24 +N−p/24 } . The bounds on the space difference and the time difference taken together complete the proof. 116 We now show that these moment estimates imply C-tightness of the approx- imate densities. We shall start including dependence on N again to clarify the tightness argument. First define Ã(ξNt )(z) = Â(ξ N t )(z) on the grid z ∈ N−1Z, t ∈ N−2N0. Linearly interpolate first in z and then in t to obtain a continuous C-valued process. Note in particular that we can use Lemma 4.5.1 to show that for 0 ≤ s ≤ t ≤ T, |t− s| ≤ 1 and y, z ∈ R, |y − z| ≤ 1, E [∣∣∣Ã(ξNt )(z)− Ã(ξNs )(y)∣∣∣p] ≤ C(λ, p, T ) ( 1 + CpQ ) eλp(z) ( |t− s|p/48 + |z − y|p/24 ) for λ > 0, p ≥ 2 arbitrarily fixed. The next Lemma shows that Ã(ξNt ) and Â(ξ N t ) remain close. The advantage of using Ã(ξNt ) is that it is continuous. Using Kolmogorov’s continuity theorem (see for instance Corollary 1.2 in Walsh [15]) on compacts R (i1,i2) 1 ≡ {(t, x) ∈ R+ × R : (t, x) ∈ (i1, i2) + [0, 1]2} for i1 ∈ N0, i2 ∈ Z we obtain tightness of Ã(ξNt )(x) in the space of contin- uous functions on { (t, x) : (t, x) ∈ R(i1,i2)1 } . Indeed, we can use the Arzelà- Ascoli theorem. With arbitrary high probability, part (ii) of Corollary 1.2 of [15] provides a uniform (in N) modulus of continuity for all N ≥ N0. Point- wise boundedness follows from the boundedness of A(ξNt )(x) together with Lemma 4.5.2(b) below. Now use a diagonalization argument to obtain tight- ness of (Ã(ξNt )(x) : t ∈ R+, x ∈ R)N∈N in the space of continuous functions from R+ ×R to R equipped with the topology of uniform convergence on com- pact sets. Next observe that if we consider instead the space of continuous functions from R+ to the space of continuous functions from R to R, both equipped with the topology of uniform convergence on compact sets, tightness of (Ã(ξNt )(x) : t ∈ R+, x ∈ R)N∈N in the former space is equivalent to tightness of (Ã(ξNt )(·) : t ∈ R+)N∈N in the latter. Finally, tightness of (A(ξNt ) : t ∈ R+)N∈N as cadlag C1-valued processes (recall that 0 ≤ A(ξNt )(x) ≤ 1 by construction) and also the continuity of all weak limit points follow from the next Lemma. Lemma 4.5.2. For any λ > 0, T <∞ we have (a) P ( supt≤T ‖Ã(ξNt )− Â(ξNt )‖−λ≥ 7N−1/4 ) → 0 as N →∞. (b) supt≤T ‖< νN0 , ψ·t > −Pt/3u0 ‖−λ→ 0 as N →∞. Proof. The proof is very similar to the proof of Lemma 7 in [9]. We shall only give some additional steps for part (a) to complement the proof of the given reference. 117 (a) For 0 ≤ s ≤ t we have ‖< νN0 , ψ·t > − < νN0 , ψ·s >‖−λ = sup z ∣∣< A(ξN0 ), ψ̄zt − ψ̄zs >∣∣ e−λ|z| ≤ sup z < 1, ∣∣ψ̄zt − ψ̄zs ∣∣ > ≤ 2N |t− s|. Here we used Lemma 4.4.3(a) in the first line, 0 ≤ A(ξN0 ) ≤ 1 in the second line and Lemma 4.4.1(d) in the last. Hence, this only changes by O(N−1) between the (time-)grid points in N−2N0. We obtain that P ( sup t≤T ‖Ã(ξNt )− Â(ξNt )‖−λ≥ 7N−1/4 ) = P (∃t ∈ [0, T ] ∩N−2N0, s ∈ [0, T ], |s− t| ≤ N−2 s.t. ‖Â(ξNt )− Â(ξNs )‖−λ≥ 7N−1/4 ) ≤ P(∃t ∈ [0, T ] ∩N−2N0, s ∈ [0, T ], |s− t| ≤ N−2 s.t. ‖A(ξNt )−A(ξNs )‖−λ + ∣∣∣∣< νN0 , ψ·t − ψ·s >∣∣∣∣−λ ≥ 7N−1/4 ) ≤ P(∃t ∈ [0, T ] ∩N−2N0, s ∈ [0, T ], |s− t| ≤ N−2 s.t. ‖A(ξNt )−A(ξNs )‖−λ≥ 6N−1/4 ) for N big enough. Next note that the value of A(ξNt )(x) changes only at jump times of Pt(x; y) or Qm,i,j.kt (x; y1, . . . , ym; z), i, j, k = 0, 1,m ≥ 2 for some y ∼ x respectively for some y1, . . . , ym ∼ x and arbitrary z ∈ N−1Z and that each jump of A(ξNt ) is by definition of A(ξNt ) bounded by N −1/2. Then, writing P(a) for a Poisson variable with mean a, we get as a further bound on the above∑ l∈Z P (∃z ∈ N−1Z ∩ (l, l+ 1], ∃t ∈ [0, T ] ∩N−2N0, ∃s ∈ [t, t+N−2] with {∣∣A(ξNt )(z)−A(ξNs )(z)∣∣ ∧ ∣∣A(ξNt+N−2 (z)−A(ξNs )(z)∣∣} ≥ N−1/4eλ(|l|−1)) ≤ ∑ l∈Z N(N2T )P ( CN−1/2 (∑ y∼0 PN−2(0; y) + ∑ i,j,k=0,1,m≥2 ∑ y1,...,ym∼0 ∑ u Qm,i,j,kN−2 (0; y1, . . . , ym;u)   ≥ N−1/4eλ(|l|−1)   ≤ ∑ l∈Z C(T )N3P ( CN−1/2P ( N−2 (( N − θ(N) ) + CQ )) ≥ N−1/4eλ(|l|−1) ) ≤ ∑ l∈Z C(T )N3P ((P(N−2 (N + CQ)))p ≥ CNp/4eλp(|l|−1)) 118 for some p > 0. Now apply Chebyshev’s inequality. Choose p > 0 such that 3− p/4 < 0. Then the resulting sum is finite and goes to zero for N →∞. (b) The proof of part (b) follows as the proof of Lemma 7(b) of [9]. 4.6 Characterizing Limit Points To conclude the proof of Theorem 4.2.9, Theorem 4.2.14 and Theorem 4.2.23 we can proceed as in Section 4 in [9], except for the proof of weak uniqueness of (4.12) respectively (4.18). We shall give a short overview in what follows. The interested reader is referred to [9] for complete explanations. In short, Lemma 4.4.3(b) implies for all φ ∈ Cc that sup t ∣∣< νNt , φ > − < A(ξNt ), φ >∣∣ ≤ C(λ) ‖D(φ,N−1/2)‖λN→∞→ 0. (4.54) The C-tightness of (A(ξNt ) : t ≥ 0) in C1 follows from the results of Section 4.5. This in turn implies the C-tightness of (νNt : t ≥ 0) as cadlag Radon measure valued processes with the vague topology. Indeed, let ϕk , k ∈ N be a sequence of smooth functions from R to [0, 1] such that ϕk(x) is 1 for |x| ≤ k and 0 for |x| ≥ k + 1. Then a diagonalization argument shows that C-tightness of (νNt : t ≥ 0) as cadlag Radon measure valued processes with the vague topology holds if and only if C-tightness of (ϕkdν N t : t ≥ 0) as cadlag MF ([−(k + 1), k + 1])-valued processes with the weak topology holds. Here, MF ([−(k + 1), k + 1]) denotes the space of finite measures on [−(k + 1), k + 1]. Now use Theorem II.4.1 in [11] to obtain C-tightness of (ϕkdν N t : t ≥ 0) in D(MF ([−(k+ 1), k+ 1])). The compact containment condition (i) in [11] is obvious. The second condition (ii) in [11] follows from (4.54) and the C-tightness of (A(ξNt ) : t ≥ 0) in C1 together with Lemma 4.4.4(a). Observe in particular, that (4.54) implies the existence of a subsequence( A(ξNkt ), ν Nk t ) that converges to (ut, νt). Hence, we can define variables with the same distributions on a different probability space such that with probability one, for all T <∞, λ > 0, φ ∈ Cc, sup t≤T ∣∣∣∣∣∣A(ξNkt )− ut∣∣∣∣∣∣−λ → 0 as k →∞, sup t≤T ∣∣∣< φ, νNkt > − < φ, νt >∣∣∣→ 0 as k →∞, where we used 0 ≤ A(ξNkt ) ≤ 1 and thus 0 ≤ ut(x) ≤ 1 a.s. for the first limit. We obtain in particular νt(dx) = ut(x)dx for all t ≥ 0. It remains to investigate ut in the special case, i.e. with no fixed kernel, i.e. where q (k,m,N) 0j = q (k,m,N) 1j , j = 0, 1. 119 Take φt ≡ φ ∈ C3c in (4.33). We get M (N) t (φ) =< ν N t , φ > − < νN0 , φ > − ∫ t 0 < νNs−,∆(φ) > ds−E(1)t (φ) (4.55) − ∑ k=0,1 (1− 2k) ∑ m≥2,j=0,1 q (k,m,N) 0j ∫ t 0 < ( Fj ◦A ( ξNs− )) × (F1−k ◦A(ξNs−))m−1 (δk ◦ ξNs−) , φ > ds−E(3)t (φ). From (4.27) and (4.32) and the Burkholder-Davis-Gundy inequality (4.45) we obtain that the error terms converge to zero for all 0 ≤ t ≤ T almost surely. Taylor’s theorem further shows that (replace Nk by N for notational ease) ∆(φ) (xN ) = N − θ(N) 2c(N)N1/2 ∑ y∼xN (φ(y) − φ(xN )) = N − θ(N) c(N)N √ N 2 ∑ y∼xN (φ(y) − φ(xN )) → ∆φ 6 (x) as xN → x and N →∞ on the support of φ. Using this in (4.55) we can show that M (N) t (φ) converges to a continuous martingale Mt(φ) satisfying Mt(φ) = ∫ φ(x)ut(x)dx − ∫ φ(x)u0(x)dx − ∫ t 0 ∫ ∆φ(x) 6 us(x)dxds (4.56) − ∑ k=0,1 (1− 2k) ∑ m≥2,j=0,1 q (k,m) 0j ∫ t 0 ∫ Fj(us(x)) × (F1−k(us(x)))m−1 Fk(us(x))φ(x)dxds. To exchange the limit in N → ∞ with the infinite sum we used [12], Propo- sition 11.18 together with Hypothesis 4.2.20. Recall in particular, that 0 ≤ Fl(us(x)) ≤ 1 for l = 0, 1. To show that Mt(φ) is indeed a martingale we used in particular (4.30) to see that 〈M (N)(φ)〉t ≤ C(λ)t ‖ φ ‖2λ< 1, e−2λ > is uniformly bounded. Therefore, (M (N) t (φ) : N ≥ N0) and all its moments are uniformly integrable, using the Burkholder-Davis-Gundy inequality of the form (4.45) once more. We can further calculate its quadratic variation by making use of (4.29) for N →∞ together with the uniform integrability of ((M (N)t (φ))2 : N ≥ N0). Use our results for φ ∈ C3c , note that C3c is dense in C2c with respect to the norm ‖f ‖≡‖f ‖∞ + ‖f ′ ‖∞ + ‖∆f ‖∞, and use (4.56) to see that ut solves the martingale problem associated with the SPDE (4.18). It is now straightforward to show that, with respect to some white noise, ut is actually a solution to (4.18) (see [13], V.20 for the similar argument in the case of SDEs). 120 4.7 Uniqueness in Law To show uniqueness of all limit points of Section 4.6 in the case with no fixed kernel and with < u0, 1 >< ∞, we need to show uniqueness in law of [0, 1]- valued solutions to either (4.12) or (4.18) (recall Corollary 4.2.24). Indeed, as 0 ≤ A(ξNt )(x) ≤ 1 by definition, any limit point has to satisfy ut(x) ∈ [0, 1]. We shall choose to prove weak uniqueness of (4.18), i.e. of ∂u ∂t = ∆u 6 + ∑ k=0,1 (1− 2k) ∑ m≥2,j=0,1 q (k,m) 0j Fj(u) (F1−k(u)) m−1 Fk(u) (4.57) + √ 2u(1− u)Ẇ = ∆u 6 + u(1− u) ∑ k=0,1 (1− 2k) ∑ m≥2,j=0,1 q (k,m) 0j Fj(u) (F1−k(u)) m−2 + √ 2u(1− u)Ẇ ≡∆u 6 + u(1− u)Q(u) + √ 2u(1− u)Ẇ with initial condition u0 in what follows. Observe that |Q(us(x))| ≤ CQ with CQ as in (4.43) because 0 ≤ us(x) ≤ 1. To check uniqueness in law of [0, 1]-valued solutions we shall apply a version of Dawson’s Girsanov Theorem, see Theorem IV.1.6 in [11], p. 252. Let Pu denote the law of a solution to the SPDE (4.57) and Pv denote the unique law of the [0, 1]-valued solution to the SPDE ∂v ∂t = ∆v 6 + √ 2v(1− v)Ẇ (4.58) with v0 = u0. Reasons for existence and uniqueness of a [0, 1]-valued solution to the latter can be found in Shiga [14], Example 5.2, p. 428. Note in particular that the solution vt takes values in C1. To prove weak uniqueness, we shall follow the reasoning of the proof of Theorem IV.1.6(a),(b) in [11] in a univariate setup. To follow the reasoning from [11], we need to show the following Lemma first. Lemma 4.7.1. Given u0 = v0 satisfying < u0, 1 ><∞, we have Pu-a.s.∫ t 0 < us, 1 > ds < ∞ for all t ≥ 0 and Pv-a.s. ∫ t 0 < vs, 1 > ds < ∞ for all t ≥ 0. Proof. We shall prove the claim for Pu. The other claim then follows by consid- ering the special case Q ≡ 0. As a first step we shall use a generalization of the weak form of (4.57) to functions in two variables. In the proof of Theorem 2.1 121 on p. 430 of [14] it is shown that for every ψ ∈ D2rap(T ) and 0 < t < T we have < ut, ψt >= < u0, ψ0 > + ∫ t 0 < us, ( ∂ ∂s + ∆ 6 ) ψs > ds (4.59) + ∫ t 0 < us(1− us)Q(us), ψs > ds + ∫ t 0 ∫ √ 2us(x)(1− us(x))ψs(x)dW (x, s). Here we have for T > 0, C(R) = {f : R → R continuous} , Crap = { f ∈ C(R) s.t. sup x eλ|x||f(x)| <∞ for all λ > 0 } , C2rap = { ψ ∈ C2(R) s.t. ψ, ψ′, ψ′′ ∈ Crap } , D2rap(T ) = { ψ ∈ C1,2([0, T )× R) s.t. ψ(t, ·) is C2rap-valued continuous and ∂ψ ∂t (t, ·) is Crap-valued continuous in 0 ≤ t < T } . Also observe that the condition (2.2) of [14] is satisfied as we have 0 ≤ us(x) ≤ 1 and therefore |Q(us(x))| ≤ CQ. Now recall that the Brownian transition density is ps(x) = 1√ 2pis e− x2 2s . Let (Psφ)(x) = ∫ p s 3 (y − x)φ(y)dy with φ ∈ C∞c , φ ≥ 0 and let ψ(s, x) = ψs(x) = e CQ(T−s) (PT−sφ)(x) and thus ψ ∈ D2rap(T ). Note that ∂∂s (PT−sφ)(x) = −∆6 (PT−sφ)(x), where we used that ∂∂sps(x) = 1 2∆ps(x). We obtain for the drift term in (4.59) that < us, ( ∂ ∂s + ∆ 6 ) ψs > + < us(1− us)Q(us), ψs > =< us,−CQψs − ∆ 6 ψs + ∆ 6 ψs > + < us(1− us)Q(us), ψs > ≤ 0 using that ψ(s, x) ≥ 0 for φ ≥ 0. Additionally, the local martingale in (4.59) is a true martingale as〈∫ · 0 ∫ √ 2us(x)(1− us(x))ψs(x)dW (x, s) 〉 t = ∫ t 0 < 2us(1− us), ψ2s > ds ≤ 2e2CQT ∫ t 0 < 1, (PT−sφ) 2 > ds ≤ 2e2CQT ‖φ‖0 ∫ t 0 < 1, PT−sφ > ds = 2e2CQT ‖φ‖0< 1, φ > t <∞. 122 Hence we obtain from (4.59) for all 0 < t < T after taking expectations E[< ut, ψt >] ≤< u0, ψ0 >, i.e. eCQ(T−t)E[< ut, (PT−tφ) >] ≤ eCQT < u0, (PTφ) > . Now choose an increasing sequence of non-negative functions φn ∈ C∞c such that φn ↑ 1 for n → ∞. Using the monotone convergence theorem, we obtain from the above eCQ(T−t)E[< ut, 1 >] = lim n→∞ e CQ(T−t)E[< ut, (PT−tφn) >] ≤ lim n→∞ e CQT < u0, (PTφ n) >= eCQT < u0, 1 > . Hence by the Fubini-Tonelli theorem, E [∫ t 0 < us, 1 > ds ] ≤< u0, 1 > ∫ t 0 eCQsds <∞ for all t ≥ 0, which proves the claim. Lemma 4.7.2. If < u0, 1 >< ∞ the weak [0, 1]-valued solution to (4.57) is unique in law. If we let Rt ≡ exp {∫ t 0 ∫ Q(vs(x)) 2 √ 2vs(x)(1− vs(x))dW (x, s) −1 2 ∫ t 0 ∫ (1− vs(x)) (Q(vs(x)))2 2 vs(x)dxds } , then dPu dPv ∣∣∣∣ Ft = Rt for all t > 0, (4.60) where Ft is the canonical filtration of the process v(t, x). Proof. We proceed analogously to the proof of Theorem IV.1.6(a),(b) in [11]. Observe in particular that we take Tn = inf { t ≥ 0 : ∫ t 0 ∫ (1− us(x)) (Q(us(x)))2 2 us(x)dx + 1 ds ≥ n } . Lemma 4.7.1 shows that under Pu∫ t 0 ∫ (1− us(x)) (Q(us(x)))2 2 us(x)dxds ≤ (CQ) 2 2 ∫ t 0 < us, 1 > ds <∞ for all t > 0 Pu−a.s. and so Tn ↑ ∞ Pu-a.s. As in Theorem IV.1.6(a) of [11] this gives uniqueness of the law Pu of a solution to (4.57). As in Theorem IV.1.6(b) of [11] the fact that Tn ↑ ∞ Pv-a.s. (from Lemma 4.7.1) shows that (4.60) defines a probability Pu which satisfies (4.57). 123 Bibliography [1] Burkholder, D.L. Distribution function inequalities for martingales. Ann. Probab. (1973) 1, 19–42. MR0365692 [2] Cox, J.T. and Durrett, R. and Perkins, E.A. Rescaled voter mod- els converge to super-Brownian motion. Ann. Probab. (2000) 28, 185–234. MR1756003 [3] Cox, J.T. and Perkins, E.A. Rescaled Lotka-Volterra models converge to super-Brownian motion. Ann. Probab. (2005) 33, 904–947. MR2135308 [4] Cox, J.T. and Perkins, E.A. Survival and coexistence in stochastic spa- tial Lotka-Volterra models. Probab. Theory Related Fields (2007) 139, 89– 142. MR2322693 [5] Dawson, D.A. Measure-valued Markov processes. École d’éte de proba- bilités de Saint Flour, XXI (1991), 1–260, Lecture Notes in Math., 1541, Springer, Berlin, 1993. MR1242575 [6] Durrett, R. Ten lectures on particle systems. Lectures on probability theory (Saint-Flour, 1993), 97–201, Lecture Notes in Math., 1608, Springer, Berlin, 1995. MR1383122 [7] Liggett, T.M. Interacting Particle Systems, Reprint of the 1985 original. Classics in Mathematics, Springer, Berlin, 2005. MR2108619 [8] Liggett, T.M. Stochastic interacting systems: contact, voter and exclu- sion processes. 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Two contrasting properties of solutions for one-dimensional stochastic partial differential equations. Canad. J. Math. (1994) 46, 415– 437. MR1271224 [15] Walsh, J.B. An Introduction to Stochastic Partial Differential Equations. École d’éte de probabilités de Saint Flour, XIV (1984), 265–439, Lecture Notes in Math., 1180, Springer, Berlin, 1986. MR0876085 125 Chapter 5 Conclusion The last three Chapters investigate different models for interacting multi-type populations from biology. All models under consideration are parameter-depen- dent and the behaviour of the parameters determines the behaviour of the sys- tem with respect to weak uniqueness or survival and coexistence results of types. 5.1 Overview of Results and Future Perspectives of the Manuscripts 5.1.1 Degenerate stochastic differential equations for catalytic branching networks Chapter 2 establishes weak uniqueness for systems of SDEs, modelling cat- alytic branching networks. These networks can be obtained as limit points of branching particle systems. Weak uniqueness of the solutions to the SDE shows uniqueness of the limit points and hence implies convergence of the above ap- proximations. It also makes certain additional tools available for analysis of the solution, as can be seen in the application of the results of Chapter 2 in Chapter 3. For instance, in the proof of existence of a stationary distribution for the normalized processes in Subsection A.2, weak uniqueness yields that the generator A satisfies the positive maximum principle. This paper is an extension of Dawson and Perkins [6] (where the networks were essentially trees or cycles) to arbitrary networks. The additional depen- dency among catalysts led to a change of perspective from reactants to catalysts. In [6] every reactant j had one catalyst cj only but as it turned out for networks it is more effective to consider every catalyst i with the set Ri of its reactants. In particular, the restriction from Ri to R̄i, including only reactants whose cata- lysts are all zero, turns out to be crucial. As a result, this paper introduces new ideas on how to handle networks where there exist catalytic interlinks between vertices. As already mentioned in Subsection 1.1 of the introductory Chapter 1, the extension to networks becomes necessary for example in dimensions d ≥ 3 in the renormalization analysis of hierarchically interacting multi-type branching models treated in Dawson, Greven, den Hollander, Sun and Swart [5]. The consideration of successive block averages leads to a renormalization transfor- mation on the diffusion functions of a system of SDEs similar to (2.1), (2.2). Unfortunately, [5] only show preservation of the continuity of the coefficients of 126 the SDE under this transformation but not preservation of Hölder-continuity. In [2], Bass and Perkins prove similar results to [6], i.e. they restrict themselves to networks with at most one catalyst per reactant, but drop the requirement of Hölder-continuity of the coefficients and replace it by continuity only. A future challenge would be to investigate how the ideas of my paper can be applied to extend the results of [2] to arbitrary networks. As a first step, [2] views the sys- tem of SDEs as a perturbation of a well-behaved system of SDEs with generator A0. As part of my paper I found an explicit representation of the semigroup Pt corresponding to A0 in the extension to the [6] setup. This representation directly carries over to the setting of [2]. The modification of the remaining rea- soning in [2] to arbitrary networks remains to be done. Here we note that the extension to general graphs in Chapter 2 led to a number of technical problems in the approach of [6] even after the structure of the generator was resolved. 5.1.2 Long-term behaviour of a cyclic catalytic branching system The results of Chapter 2 are used in Chapter 3 to establish weak uniqueness of the system of SDEs under consideration. The system involves catalytic branch- ing and mutation between types. Questions for survival and coexistence of types in the time-limit arise. Such questions naturally arise in biological competition models. Recall that Fleis- chmann and Xiong [7] investigated a cyclically catalytic super-Brownian mo- tion. They showed global segregation (noncoexistence) of neighbouring types in the limit and other results on the finite time survival-extinction but they were not able to determine, if the overall sum dies out in the limit or not. I was able to show that in my setup the sum of all coordinates converges to zero but does not hit zero in finite time a.s. By changing my focus to the normalized processes I showed in particular that the normalized processes converge weakly to a unique stationary distribution that does not charge the set where at least one of the coordinates is zero. The weakness of this manuscript lies in the restriction to constant positive coefficients γi and qji in the SDEs (3.1). It would be of interest to allow the coefficients to depend on the state of the system. As long as they are uniformly bounded away from zero and infinity, I conjecture a similar behaviour of the system but it would be of particular interest to see how the failure of uniform boundedness impacts questions on survival and coexistence. If one considers coefficients satisfying Hypothesis 2.1.2, the results of Chapter 2 can be applied to obtain weak uniqueness of the new system of SDEs as a first step. Finally, following the approach in Section III.12.2 of [8], one could consider dilution flows instead of the normalized processes, that is, one removes the excess concentration so that the total concentration remains constant. 127 5.1.3 Convergence of rescaled competing species processes to a class of SPDEs In Chapter 4, the tightness results obtained yield the relative compactness of the approximating particle systems. Limits along subsequences therefore exist for combinations of long-range kernel and fixed kernel interactions in the per- turbations, where a wide class of admissible perturbations was found, including analytic functions as they appear in the spatial versions of Lotka-Volterra models of Neuhauser and Pacala [11]. It was particularly interesting to see that adding fixed kernel perturbations to the long-range case does not impact tightness. In the long-range case I obtain that all subsequential limits satisfy a certain SPDE. It remains open to find the form of the limiting equations in the case of long-range dispersal in the presence of short-range, i.e. fixed kernel competition. If I additionally assume finite initial mass in the long-range case, weak uniqueness of the limiting SPDE follows. As a consequence of this, the weak uniqueness of the limits of the approximating particle systems follows. It would be of interest to find necessary and sufficient conditions for weak uniqueness of the limiting SPDE. I conjecture ∫ u0(x)(1−u0(x))dx <∞ to be a sufficient con- dition. If this condition is preserved by the dynamics it suffices for the Girsanov Lemma 4.7.2 to hold. In the special case under investigation in Mueller and Sowers [9] (see below), ∫ ut(x)(1− ut(x))dx <∞, ∀t > 0 a.s. follows and so we expect this condition to define a state space for the solutions of the equations. The class of SPDEs that I obtain as the limits of sequences of rescaled long- range perturbations of the voter model under some conditions on the parameters α (m) i ∈ R, i = 0, 1,m ∈ N and their counterparts in the approximating models are ∂u ∂t = ∆u 6 +(1−u)u { ∞∑ m=0 α (m+1) 0 u m − ∞∑ m=0 α (m+1) 1 (1− u)m } + √ 2u(1− u)Ẇ . As I work at the critical range √ N , while Cox and Perkins [3] works with longer range interactions, I obtain a wide class of non-linear drifts. This opens up the possibility to interpret the limiting SPDEs and their behaviour via their approximating long-range particle systems and vice versa. For instance, a future challenge would be to use properties of the SPDE to obtain results on the approximating particle systems, following the ideas of [3] and Cox and Perkins [4]. As an example, recall Remark 4.2.15 where I obtained the SPDE ∂u ∂t = ∆u 6 + (1− u)u {λ′ − α10 + u (α01 + α10)}+ √ 2u(1− u)Ẇ with parameters λ′, α10, α01 ∈ R as the limit of spatial versions of the Lotka- Volterra model with competition and fecundity parameters near one. We can rewrite this SPDE to ∂u ∂t = ∆u 6 + θ0(1− u)u2 − θ1u(1− u)2 + √ 2u(1− u)Ẇ , (5.1) u(0, x) ≡ u0(x) ≥ 0, 128 where θi ∈ R, i = 0, 1. For θ1 = −θ0 < 0 one obtains after rescaling the Kolmogorov-Petrovskii-Piscuinov (KPP) equation driven by Fisher-Wright noise. This SPDE has already been investigated in Mueller and Sowers [9] in detail, where the existence of travelling waves was shown for θ0 big enough. A major question is how the change in the drift, in particular, the possibly additional zero at θ1θ0+θ1 ∈ (0, 1), impacts the set of parameters for survival (i.e. lim supt→∞ < ut, 1 >> 0 with positive probability), coexistence (i.e. there exists a stationary distribution giving zero mass to the configurations 0 and 1) and extinction (i.e. lim supt→∞ < ut, 1 >= 0 with probability 1) and if there exist phase transitions. Aronson and Weinberger [1] showed for instance in Corollary 3.1(ii) that for θ0 < 0, θ1 < 0, the corresponding deterministic PDE converges to the intermediate zero θ1θ0+θ1 of the drift term uniformly on bounded sets if u0 6≡ 0, 1. The author conjectures that there are parameter regions for (5.1) that yield survival and others that yield extinction. To prove survival the author envisions to apply methods of Mueller and Tribe [10] to the SPDE (5.1). In [10] and Tribe [12], rescaled versions of the SPDE ∂u ∂t = ∆u 6 + θu− u2 +√uẆ , θ > 0 (5.2) were under investigation. Results on the existence of a phase transition between extinction and survival in terms of θ are obtained in the former paper and the existence of travelling wave solutions in the latter. Unfortunately, the proof of extinction in [10] and the proof of existence of travelling waves in [12] relies on the additive properties of the fluctuation term in (5.2) (also recall the discussion of additive properties of super-Brownian motion in the beginning of the intro- ductory Chapter 1), which makes the application of their methods to SPDEs of the form (5.1) difficult. On the other hand, [10] shows that for θ large, the drift term of the SPDE (5.2) outcompetes the fluctuation term. The proof for survival then uses a comparison of u(t, x) to oriented site percolation to prove survival for θ big. It should be possible to apply their reasoning together with the results of [1] for the corresponding deterministic PDE to the SPDE (5.1) to show survival of types in certain parameter-regions. Extinction on the other hand seems much more delicate to prove. 129 Bibliography [1] Aronson, D.G. and Weinberger, H.F. Multidimensional Nonlinear Diffusion Arising in Population Genetics. Adv. Math. (1978) 30, 33–76. MR0511740 [2] Bass, R.F. and Perkins, E.A. Degenerate stochastic differential equa- tions arising from catalytic branching networks. Electron. J. Probab. (2008) 13, 1808–1885. MR2448130 [3] Cox, J.T. and Perkins, E.A. Rescaled Lotka-Volterra Models converge to Super-Brownian Motion. Ann. Probab. (2005) 33, 904–947. MR2135308 [4] Cox, J.T. and Perkins, E.A. Survival and coexistence in stochastic spa- tial Lotka-Volterra models. Probab. Theory Related Fields (2007) 139, 89– 142. MR2322693 [5] Dawson, D.A. and Greven, A. and den Hollander, F. and Sun, R. and Swart, J.M. The renormalization transformation for two-type branch- ing models. Ann. Inst. H. Poincaré Probab. Statist. (2008) 44, 1038–1077. MR2469334 [6] Dawson, D.A. and Perkins, E.A. On the uniqueness problem for cat- alytic branching networks and other singular diffusions. Illinois J. Math. (2006) 50, 323–383 (electronic). MR2247832 [7] Fleischmann, K. and Xiong, J. A cyclically catalytic super-brownian motion. Ann. Probab. (2001) 29, 820–861. MR1849179 [8] Hofbauer, J. and Sigmund, K. The Theory of Evolution and Dynamical Systems. London Math. Soc. Stud. Texts, vol. 7, Cambridge Univ. Press, Cambridge, 1988. MR1071180 [9] Mueller, C. and Sowers, R.B. Random Travelling Waves for the KPP Equation with Noise. J. Funct. Anal. (1995) 128, 439–498. MR1319963 [10] Mueller, C. and Tribe, R. A phase transition for a stochastic PDE related to the contact process. Probab. Theory Related Fields (1994) 100, 131–156. MR1296425 [11] Neuhauser, C. and Pacala, S.W. An explicitly spatial version of the Lotka-Volterra model with interspecific competition. Ann. Appl. Probab. (1999) 9, 1226–1259. MR1728561 [12] Tribe, R. A travelling wave solution to the Kolmogorov equation with noise. Stochastics Stochastics Rep. (1996) 56, 317–340. MR1396765 130 Appendix A Appendix for Chapter 3 A.1 ã is non-singular Corollary A.1.1. The matrix ã is non-singular for all x ∈ S̃, where S̃ = ( [0, 1]d−1 ∩ { d−1∑ i=1 xi ≤ 1 }) \ { x : ∃i : xi = 0 or d−1∑ i=1 xi = 1 } . Proof. Recall that σ ∈ M(d, d) (the space of d × d-matrices) and a = σσT ∈ M(d, d). Let σ̄ ∈ M(d − 1, d) be constructed from σ by deleting the last line of the matrix (i.e. by deleting the last equation for Y dt of our system of SDEs). Then ã = σ̄σ̄T ∈ M(d−1, d−1). Further let σ̃ ∈M(d−1, d−1) be the matrix obtained from σ̄ by deleting the last column. We claim that if σ̃ is non-singular, then ã is non-singular as well. Indeed, let v ∈ M(d− 1, 1) denote the last column of σ̄ and suppose σ̃ is non-singular, then det(ã) = det(σ̄σ̄T ) = det(σ̃σ̃T + vvT ) = det ( σ̃σ̃T ) ( 1 + vT (σ̃σ̃T )−1v ) = det ( σ̃σ̃T ) ( 1+ ‖ σ̃−1v ‖2) = (det(σ̃))2 (1+ ‖ σ̃−1v ‖2) > 0. Recall that for i, j ∈ {1, . . . , d−1} we have σ̃ii(x) = (1−xi) √ 2γixixi+1 and σ̃ij(x) = −xi √ 2γjxjxj+1 if i 6= j, where we set xd ≡ 1 − ∑d−1 i=1 xi. Suppose that xi > 0 for all i ∈ {1, . . . , d − 1} and ∑d−1 i=1 xi < 1. We shall show that in this case σ̃ is non-singular. As a first step we divide the ith line of σ̃ by xi for i = 1, . . . , d− 1. We obtain det(σ̃(x))∏d−1 i=1 xi = det   d1 a2 a3 . . . ad−1 a1 d2 a3 . . . ad−1 a1 a2 d3 . . . ad−1 . . . . . . . . . . . . . . . a1 a2 a3 . . . dd−1   = det   d1 − a1 a2 − d2 0 . . . 0 0 0 d2 − a2 a3 − d3 . . . 0 0 0 0 d3 − a3 . . . 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 . . . dd−2 − ad−2 ad−1 − dd−1 a1 a2 a3 . . . ad−2 dd−1   ≡ detA1, 131 where we used the line operations i→ i− (i+1) on all but the last line and set di − ai = (1− xi) √ 2γi xi+1 xi + √ 2γixixi+1 = √ 2γi xi+1 xi = −ai xi . (A.1) Based on the first column we can calculate the determinant of σ̃. det(A1) ≡ (d1 − a1) det(A2) + (−1)da1 d−1∏ i=2 (ai − di), (A.2) where we obtain the matrix A2 by crossing out the first row and column of A1. We obtain recursively for k = 1, . . . , d− 3, using (A.1) that det(Ak) = −ak xk det(Ak+1) + ak(−1)d−k+1 d−1∏ i=k+1 ai xi . (A.3) By using (A.3) recursively in (A.2) we get det(A1) = (−1)d−3 d−3∏ i=1 ai xi det(Ad−2) + (−1)d d−1∏ i=1 ai d−3∑ i=1 ∏ j=1,...,d−1,j 6=i 1 xj . Finally, det(Ad−2) = det ( dd−2 − ad−2 ad−1 − dd−1 ad−2 dd−1 ) (A.1) = −ad−2dd−1 xd−2 − ad−2ad−1 xd−1 and thus (recall that xi 6= 0 and thus ai 6= 0 for i = 1, . . . , d) det(σ̃(x)) = 0 ⇐⇒ det(A1) = 0 ⇐⇒ d−2∑ i=1 ∏ j=1,...,d−1,j 6=i 1 xj + dd−1 ad−1 ∏ j=1,...,d−1,j 6=d−1 1 xj = 0 ⇐⇒ d−2∑ i=1 ∏ j=1,...,d−1,j 6=i 1 xj − 1− xd−1 xd−1 ∏ j=1,...,d−1,j 6=d−1 1 xj = 0 ⇐⇒ d−2∑ i=1 xi − (1− xd−1) = 0 ⇐⇒ d−1∑ i=1 xi = xd = 1, which is a contradiction to x ∈ S̃. Hence det(σ̃(x)) 6= 0 for all x ∈ S̃. 132 A.2 Proof of Proposition 3.2.18 Proof. The main part of the proof is taken from Dawson, Greven, den Hollan- der, Sun and Swart [1], Section 3.1 and adjusted to our setting. Existence. Denote the distribution of Yt ∈ [0, 1]d by µt, with µ0 = δy for some arbitrary y ∈ S. As the state space [0, 1]d is compact, {νt : νt ≡ 1t ∫ t 0 µsds}t≥0 forms a tight family of distributions. In this case Theorem III.2.2 in Ethier and Kurtz [2] implies that {νt}t≥0 is relatively compact for the Prohorov metric. As [0, 1]d is Polish, Theorem III.1.7 in [2] gives the existence of a limit. Also note that the convergence of a sequence of {νt}t≥0 is equivalent to weak convergence by Theorem III.3.1 of [2]. Taking this together we find a sequence (tn) tending to infinity such that νtn converges weakly to a limiting distribution ν. The goal will be now to apply Theorem IV.9.17 of [2], where Ĉ([0, 1]d) = C([0, 1]d) by definition (cf. definition before Lemma IV.2.1) due to the compactness of our state space. To this purpose note first that the generator corresponding to (3.15) is given by Af(x) = d∑ i=1 bi(x)∂if(x) + 1 2 d∑ i,j=1 aij(x)∂ijf(x), where a = σσT and b are as in (3.19) and D(A) = C2([0, 1]d). Note further that our system of SDEs has a solution that is unique in law (cf. Proposition 3.2.13). Hence Theorem V.(21.2) and Remark V.(21.9) in Rogers and Williams [3] yield that the solution is strong Markov. Hence A is the generator of a strong Markov process and thus satisties the positive maximum principle by [2], Theorem IV.2.2. Moreover, if ν is the limit of νtn then for any f ∈ C2([0, 1]d) we have Af ∈ C([0, 1]d) and Eν [(Af)(Y0)] = lim n→∞Eνtn [(Af)(Y0)] = lim n→∞ 1 tn ∫ tn 0 Eµs [(Af)(Y0)] ds = lim n→∞ 1 tn ∫ tn 0 Eµ0 [(Af)(Ys)] ds = lim n→∞ 1 tn Eµ0 [∫ tn 0 (Af)(Ys)ds ] = lim n→∞ 1 tn Eµ0 [f(Ytn)− f(Y0)] = 0. Here we used νtn ⇒ ν for Af ∈ C([0, 1]d) in the first equality, the definition of νt in the second, Af ∈ C([0, 1]d) in the fourth equality, that Y solves the martingale problem for A in the fifth equality and f bounded in the last. 133 Finally observe that C2([0, 1]d) forms an algebra of functions dense in C([0, 1]d). Taken the above together, we can apply Theorem IV.9.17 of [2] and obtain the existence of a stationary solution. Uniqueness. Let Yt be the unique strong Markov solution to (3.15). Recall that we already showed that every equilibrium distribution for A doesn’t put mass on N = {y : ∃i : yi = 0} in Proposition 3.2.17 and that ∑ i Y i t = 1 for all t ≥ 0 in Proposition 3.2.13. Hence, if Yt has two distinct equilibrium distributions, concentrated on [0, 1]d, or to be more precise, on S ≡ ([0, 1]d\∂[0, 1]d) ∩ { y : ∑ i yi = 1 } , then we can find two extremal equilibrium distributions µ and ν that are singular with respect to each other (see for instance Exercise 6.9 in Varadhan [5]). As µ(B) = Pµ(Yt ∈ B) = ∫ pt(x,B)dµ(x), there have to exist x, y ∈ S such that the transition kernels pt(x, dz) and pt(y, dz) are mutually singular for all t > 0 as well. Also, as µ respectively ν do not put mass on N the same holds for pt(x, dz) respectively pt(y, dz). In what follows we shall consider the process Ỹt ≡ (Y 1t , . . . , Y d−1t ) ∈ [0, 1]d−1 with transition kernels p̃t instead. The martingale problem for the resulting SDE for Ỹ is consequently well-posed as the corresponding martingale problem for Y is well-posed. Let p̃t(x̃, dz̃) and p̃t(ỹ, dz̃) be the resulting transition kernels corresponding to pt(x, dz) and pt(y, dz). For x̃ ∈ S̃ ≡ ( [0, 1]d−1 ∩ { d−1∑ i=1 x̃i ≤ 1 }) \ { x̃ : ∃i : x̃i = 0 or d−1∑ i=1 x̃i = 1 } = ( [0, 1]d−1\∂[0, 1]d−1) ∩ { x̃ : 0 < d−1∑ i=1 x̃i < 1 } , ãij(x̃) is non-singular by Corollary A.1.1 of the Appendix. Now we can ap- peal to Theorem B.4 (“Support theorem for uniformly elliptic diffusions”) of [1] with x̃, ỹ ∈ D, D̄ ⊂ S̃, where D is an arbitrarily fixed ball. Here observe that S̃ is an open subset of [0, 1]d−1. The Theorem allows us to transport the diffusions started at x̃ respectively ỹ to a common small neighbourhood with positive probability. Subsequently we can apply Corollary B.3 (“Transition den- sity for diffusions restricted to bounded domains”) of [1] to see that p̃t(x̃, dz̃) and p̃t(ỹ, dz̃) and hence pt(x, dz) and pt(y, dz) cannot be mutually singular for all t > 0. Convergence. Once more, we shall consider the process Ỹ instead of Y and the state space S̃ instead of S. Let p̃i be the unique equilibrium distribution of Ỹ corresponding to the unique equilibrium distribution pi of Y . 134 Firstly, note that by Theorem B.4 of [1] the equilibrium distribution pi assigns positive measure to every open subset of S̃. Secondly we shall show that L(Ỹt|Ỹ0 = x) ⇒ pi for all x ∈ S̃ in three steps, namely by showing first that it holds for almost all x ∈ S̃ w.r.t. p̃i. Then we shall extend this result to Lebesgue almost every x ∈ S̃ and finally we shall conclude that this implies that it holds for all x ∈ S̃. To prove the first step, we first choose x∗ ∈ S̃ arbitrary but fixed and let D be open such that x∗ ∈ D, D̄ ⊂ S̃. Recall that ã is non-singular on S̃ and S̃ is an open subset of [0, 1]d−1. Let Ỹt, Z̃t be two independent copies of the process on [0, 1]d−1. Then the joint process (Ỹt, Z̃t) is strong Markov and has a unique equilibrium distribution given by the product measure pi ⊗ pi. By Theorem 6.9 in [5] and the following Remarks the process (Ỹt, Z̃t) started in equilibrium, i.e. with L((Ỹt, Z̃t)) = pi ⊗ p̃i is ergodic. As the equilibrium distribution pi assigns positive measure to every open subset of S̃, p̃i ⊗ p̃i assigns positive measure to B(x∗) × B(x∗) ⊂ D ×D for  small enough. Therefore (Ỹt, Z̃t) visits the set B(x ∗) ×B(x∗) after any finite time T a.s. by the ergodic theorem. We obtain in particular that for almost all (x, x′) w.r.t. pi ⊗ p̃i, (Ỹt, Z̃t) started at (x, x′) visits B(x∗) × B(x∗) after any finite time T a.s. Fix such a (x, x′). In what follows we shall start two independent processes Ỹt and Z̃t with initial conditions x respectively x′ as above and denote their laws by Px respec- tively Px ′ . Let the first exit time from D be τD(ω) ≡ inf{t ≥ 0 : ω(t) 6∈ D}. By Corollary B.3, for each δ > 0 and z ∈ D, the measure µDδ (z, ·) ≡ Pz(ω : δ < τD(ω), ω(δ) ∈ ·) admits a density pDδ (z, ·) with respect to Lebesgue measure. Moreover, (B.2) of the Corollary yields that for , δ sufficiently small we have uniformly for y, y′ ∈ B(x∗), ∫ pDδ (y, u) ∧ pDδ (y′, u)du ≥ 1 2 , (A.4) where we used that a+ b− (a∨ b−a∧ b) = a∨ b+a∧ b− (a∨ b−a∧ b)≥ 1 ⇐⇒ a ∧ b ≥ 1/2. We obtain in particular that for y ∈ B(x∗) (and analogously for y′ ∈ B(x∗)) P y(ω : ω(δ) ∈ du) ≥ Py(ω : δ < τD(ω), ω(δ) ∈ du) (A.5) = pDδ (y, u)du ≥ pDδ (y, u) ∧ pDδ (y′, u)du. For y, y′ ∈ B(x∗) fixed let µ1(y,y′) be the measure on [0, 1]d−1 × [0, 1]d−1 defined by µ1(y,y′)(A×B) ≡ ∫ A×B ( pDδ (y, u) ∧ pDδ (y′, u) ) ( pDδ (y, v) ∧ pDδ (y′, v) ) dudv 135 for A,B ∈ B([0, 1]d−1) and observe that µ1(y,y′)(A×B) = µ1(y,y′)(B ×A). (A.6) Let µ2(y,y′)(A×B) ≡ Py(ω : ω(δ) ∈ A)Py ′ (ω : ω(δ) ∈ B)− µ1(y,y′)(A×B), which is non-negative by (A.5) and note that by (A.4), µ2(y,y′) ( [0, 1]d−1 × [0, 1]d−1) ≤ 3 4 for all y, y′ ∈ B(x∗). (A.7) In what follows we shall give the motivation for the later rigorous definitions and calculations. We obtain that whenever (Ỹt, Z̃t), starting at (x, x ′) entersB(x∗)×B(x∗) at a random time, say T1 and through some random point (y, y ′) ∈ B(x∗)×B(x∗) we can decompose the conditional law of (Ỹt, Z̃t) at time T1 + δ as follows: P y(ω : ω(δ) ∈ A)Py′(ω : ω(δ) ∈ B) = µ1(y,y′)(A×B) + µ2(y,y′)(A×B). (A.8) Now we shall successively decompose the law of the process (Ỹt, Z̃t) at times Tk + δ, k ∈ N, where Tk is the first time after Tk−1 + δ that (Ỹt, Z̃t) enters B(x ∗) × B(x∗) (see (A.9) below). For instance, if at time T1, (Ỹt, Z̃t) enters B(x ∗) × B(x∗) through some random point (y, y′) ∈ B(x∗)×B(x∗), we decompose the law of (Ỹt, Z̃t) at time T1 + δ into µ 1 (y,y′), µ 2 (y,y′) as above. Next we consider (ỸT1+δ+·, Z̃T1+δ+·), starting in µ1(y,y′) resp. µ 2 (y,y′). As µ 1 (y,y′) yields a common part, we do not decompose the law of (ỸT1+δ+·, Z̃T1+δ+·) starting in µ1(y,y′) any further. On the other hand, starting in µ 2 (y,y′), we wait until (ỸT1+δ+·, Z̃T1+δ+·) enters B(x ∗)×B(x∗) again, say at time T ′1 = T2− (T1 + δ) through the point (u, u′) ∈ B(x∗)×B(x∗). T ′1 is finite µ2(y,y′)-a.s. as µ2(y,y′) is absolutely continuous with respect to Py(ω : ω(δ) ∈ ·) ⊗ Py′(ω : ω(δ) ∈ ·) by (A.8). Here recall that (y, y′) was the first entrance point of B(x∗) × B(x∗) by (Ỹt, Z̃t) under P x⊗Px′ and that (Ỹt, Z̃t) started at (x, x′) visits B(x∗)×B(x∗) after any finite time T a.s. Hence (Ỹt, Z̃t) started at (x, x ′) visits B(x∗)×B(x∗) again at some finite random time T2 = T1 + δ + T ′ 1. Thus, starting in µ 2 (y,y′), we can decompose (ỸT1+δ+T ′1+δ , Z̃T1+δ+T ′1+δ) in µ 1 (u,u′) and µ 2 (u,u′). Now iterate the above. To be more explicit, let U0 = 0 and define stopping times Tk = inf { t ≥ Uk−1 : ( Ỹt, Z̃t ) ∈ B(x∗)×B(x∗) } and Uk = Tk + δ (A.9) for k ∈ N and δ > 0. Then almost surely Tk <∞ for all k ∈ N. 136 By the strong Markov property of the process (Ỹt, Z̃t) we can condition on FU1 and obtain for n ∈ N arbitrarily fixed,( P x ⊗ Px′ )( Ỹt ∈ A, Z̃t ∈ B ) = ( P x ⊗ Px′ )( Ỹt ∈ A, Z̃t ∈ B, t < Un ) + ( P x ⊗ Px′ )[( P ỸT1(ω) ⊗ PZ̃T1(ω) )( Ỹt−T1(ω) ∈ A, Z̃t−T1(ω) ∈ B, t ≥ U1(ω) + Un−1 ◦ θδ ) 1(t ≥ U1(ω)) ] . Here θδ denotes the shift-operator θδ(ω(·)) = ω(δ + ·). Using (A.8) we can rewrite this as( P x ⊗ Px′ )( Ỹt ∈ A, Z̃t ∈ B ) (A.10) = ( P x ⊗ Px′ )( Ỹt ∈ A, Z̃t ∈ B, t < Un ) + ( P x ⊗ Px′ )[ µ1 (ỸT1(ω),Z̃T1(ω)) (dw, dw′) ( P w ⊗ Pw′ )( Ỹt−U1(ω) ∈ A, Z̃t−U1(ω) ∈ B, t ≥ U1(ω) + Un−1 ) 1(t ≥ U1(ω)) ] + ( P x ⊗ Px′ )[ µ2(ỸT1(ω),Z̃T1(ω)) (dw, dw′) ( P w ⊗ Pw′ )( Ỹt−U1(ω) ∈ A, Z̃t−U1(ω) ∈ B, t ≥ U1(ω) + Un−1 ) 1(t ≥ U1(ω)) ] . Using (A.6) and the symmetry of Tk and Uk in (Ỹ , Z̃) we get in particular that∣∣∣(Px ⊗ Px′)(Ỹt ∈ A, Z̃t ∈ B)− (Px ⊗ Px′)(Ỹt ∈ B, Z̃t ∈ A)∣∣∣ ≤ 2 ( P x ⊗ Px′ ) (t < Un) + ( P x ⊗ Px′ )[ µ2(ỸT1(ω),Z̃T1(ω)) (dw, dw′)( P w ⊗ Pw′ )( Ỹt−U1(ω) ∈ [0, 1]d−1, Z̃t−U1(ω) ∈ [0, 1]d−1 )] ≤ 2 ( P x ⊗ Px′ ) (t < Un) + 3 4 , the last by (A.7). If n ≥ 2 we can further condition the inner probability on FU1 and decompose 137 the last term in (A.10) into( P x ⊗ Px′ )[ µ2(ỸT1(ω),Z̃T1(ω)) (dw, dw′) ( P w ⊗ Pw′ )( Ỹt−U1(ω) ∈ A, Z̃t−U1(ω) ∈ B, t ≥ U1(ω) + Un−1 ) 1(t ≥ U1(ω)) ] = ( P x ⊗ Px′ )[ µ2(ỸT1(ω),Z̃T1(ω)) (dw, dw′) ( P w ⊗ Pw′ )[ µ1(ỸT1(ω′),Z̃T1(ω′)) (dz, dz′)( P z ⊗ Pz′ )( Ỹt−(U1(ω)+U1(ω′)) ∈ A, Z̃t−(U1(ω)+U1(ω′)) ∈ B, t ≥ U1(ω) + U1(ω′) + Un−2 ) 1(t ≥ U1(ω) + U1(ω′)) ] 1(t ≥ U1(ω)) ] + ( P x ⊗ Px′ )[ µ2(ỸT1(ω) ,Z̃T1(ω)) (dw, dw′) ( P w ⊗ Pw′ )[ µ2(ỸT1(ω′),Z̃T1(ω′)) (dz, dz′)( P z ⊗ Pz′ )( Ỹt−(U1(ω)+U1(ω′)) ∈ A, Z̃t−(U1(ω)+U1(ω′)) ∈ B, t ≥ U1(ω) + U1(ω′) + Un−2 ) 1(t ≥ U1(ω) + U1(ω′)) ] 1(t ≥ U1(ω)) ] . Using this in (A.10) we obtain with (A.6)∣∣∣(Px ⊗ Px′)(Ỹt ∈ A, Z̃t ∈ B)− (Px ⊗ Px′)(Ỹt ∈ B, Z̃t ∈ A)∣∣∣ ≤ 2 ( P x ⊗ Px′ ) (t < Un) + ( P x ⊗ Px′ )[ µ2 (ỸT1(ω),Z̃T1(ω)) (dw, dw′) ( P w ⊗ Pw′ )[ µ2 (ỸT1(ω′),Z̃T1(ω′)) (dz, dz′)( P z ⊗ Pz′ )( Ỹt−(U1(ω)+U1(ω′)) ∈ [0, 1]d−1, Z̃t−(U1(ω)+U1(ω′)) ∈ [0, 1]d−1 )]] ≤ 2 ( P x ⊗ Px′ ) (t < Un) + ( 3 4 )2 , the last by (A.7). By iterating the above decomposition we obtain for n ∈ N fixed,∣∣∣(Px ⊗ Px′)(Ỹt ∈ A, Z̃t ∈ B)− (Px ⊗ Px′)(Ỹt ∈ B, Z̃t ∈ A)∣∣∣ ≤ 2 ( P x ⊗ Px′ ) (t < Un) + ( 3 4 )n . Recall that we have for almost all (x, x′) w.r.t. p̃i⊗ p̃i that almost surely Tk <∞ for all k ∈ N. Hence, to given  > 0 we can choose n ∈ N such that ( 34)n < 2 and then choose T > 0 such that ( P x ⊗ Px′ ) (t < Un) <  4 for all t ≥ T . We obtain ∣∣∣(Px ⊗ Px′)(Ỹt ∈ A, Z̃t ∈ B)− (Px ⊗ Px′)(Ỹt ∈ B, Z̃t ∈ A)∣∣∣ <  138 for all t ≥ T and thus that lim t→∞ supA,B∈B([0,1]d−1) ∣∣∣(Px ⊗ Px′)(Ỹt ∈ A, Z̃t ∈ B) − ( P x ⊗ Px′ )( Ỹt ∈ B, Z̃t ∈ A )∣∣∣ = 0. Choosing A = [0, 1]d−1 yields lim t→∞ supB∈B([0,1]d−1) ∣∣∣Px′(Z̃t ∈ B)− Px(Ỹt ∈ B)∣∣∣ = 0. A simple tightness-argument completes the proof of our first step. Next we shall extend our result L(Ỹt|Ỹ0 = x) ⇒ p̃i as t → ∞ for all x ∈ S̃, p̃i-a.s. to Lebesgue almost every x ∈ S̃. The proof goes by contradiction. Let A = {x ∈ S̃ : L(Ỹt|Ỹ0 = x) 6⇒ p̃i}. We first claim that A is Borel-measurable. Indeed, the martingale problem for Ỹ is well-posed and thus the process Ỹ is Feller continuous (see for example Stroock and Varadhan [4], Corollary 11.1.5). By using that the corresponding semigroup is a contraction we obtain the claim. Suppose by contradiction that A has positive Lebesgue measure. In this case there exists a simply connected bounded open domain D ⊂ S̃ with smooth boundary such that A ∩ D has positive Lebesgue measure. As p̃i(A) = 0 by the step above, pi(A ∩ D) = 0 follows. If Z̃t is the stationary solution of the SDE in [0, 1]d−1 started with initial law p̃i, then E [∫ T 0 1(Z̃t ∈ A ∩D)dt ] = 0 for all T > 0. On the other hand, by Theorem B.5 (“Occupation time measure for uniformly elliptic diffusions”) of [1], we have for every x ∈ D, E [∫ τD 0 1(Ỹt ∈ A ∩D)dt | Ỹ0 = x ] > 0, where τD = inf{t ≥ 0 : Ỹt 6∈ D}. As p̃i assigns positive probability to D, we have∫ D E [∫ τD 0 1(Ỹt ∈ A ∩D)dt | Ỹ0 = x ] p̃i(dx) > 0. By the monotone convergence theorem, we can choose T sufficiently large such that ∫ D E [∫ τD∧T 0 1(Ỹt ∈ A ∩D)dt | Ỹ0 = x ] p̃i(dx) > 0. But the l.h.s. is dominated by E [∫ T 0 1(Z̃t ∈ A ∩D)dt ] = 0, which is a contra- diction. Therefore A has Lebesgue measure zero. It remains to show that L(Ỹt|Ỹ0 = x) ⇒ pi for all x ∈ S̃. Indeed, for x ∈ S̃, let  > 0 be such that B(x) ⊂ S̃. By Corollary B.3 applied to D = B(x), the transition kernel µ B(x) t (x, ·) is absolutely continuous w.r.t. Lebesgue measure. As shown above, for Lebesgue almost every y ∈ B(x), L(Ỹt+s|Ỹt = y) ⇒ p̃i as s→ ∞. By observing that µB(x)t (x,B(x)) ↑ 1 as t → 0 (see (B.3)), we finally get L(Ỹt|Ỹ0 = x) ⇒ pi for arbitrary x ∈ S̃, which completes our proof. 139 Bibliography [1] Dawson, D.A. and Greven, A. and den Hollander, F. and Sun, R. and Swart, J.M. The renormalization transformation for two-type branch- ing models. Ann. Inst. H. Poincaré Probab. Statist. (2008) 44, 1038–1077. MR2469334 [2] Ethier, S.N. and Kurtz, T.G. Markov Processes: Characterization and Convergence. Wiley and Sons, Inc., Hoboken, New Jersey , 2005. MR0838085 [3] Rogers, L.C.G. and Williams, D. Diffusions, Markov Processes, and Martingales, vol. 2, Reprint of the second (1994) edition. Cambridge Math- ematical Univ. Press, Cambridge, 2000. MR1780932 [4] Stroock, D.W. and Varadhan, S.R.S. Multidimensional Diffusion Pro- cesses. Grundlehren Math. Wiss., vol. 233, Springer, Berlin-New York, 1979. MR532498 [5] Varadhan, S.R.S. Probability Theory. Courant Lect. Notes Math., 7, New York; Amer. Math. Soc., Providence, Rhode Island, 2001. MR1852999 140 Appendix B Appendix for Chapter 4 The following Lemma and Corollary are necessary to prove Lemma 4.4.1 of Chapter 4. Lemma B.0.1. There exists N0 <∞ such that for all N ≥ N0, k ≥ 1 (a) ∣∣∣ρk(t)− exp(−(1 + o(1)) kt26N )∣∣∣ ≤ C 1k exp(−(1 + o(1)) kt212N ) for t ≤ (1 + o(1)) √ N 3 , (b) |ρ(t)| ≤ exp ( −C t212N ) for t ≤ ( 6N (1+o(1)) )1/2 , (c) There exists δ > 0 such that |ρ(t)| ≤ 1− δ for t ∈ [( 6N (1+o(1)) )1/2 , piN ] . Proof. The proof mainly follows along the lines of the proof of Lemma 8 in Mueller and Tribe [3]. Some small changes ensued due to the different setup. Recall the definition of ρ(t) from equation (4.38). For (b), we could not find the reference mentioned in [3] but the following reasoning in [3] based on applying Taylor’s theorem at t = 0 works well without it. For (a), first observe that ρk(t) = E [ eitSk ] and use Bhattacharya and Rao [1], (8.11), (8.13) and [1], Theorem 8.5. as suggested in [3]. We used that E[Y1] = E[Y 3 1 ] = 0. It remains to prove (c). We have to change the proof of [3], Lemma 8(c) slightly, as we used x 6∼ x. We get |ρ(t)| = ∣∣∣∣∣∣ 1 2c(N)N1/2 ∑ 0<|j|≤c(N)√N eit j N ∣∣∣∣∣∣ = ∣∣∣∣∣∣ 1 2c(N)N1/2 ∑ 0<j≤c(N)√N 2Re [ eit j N ]∣∣∣∣∣∣ = ∣∣∣∣∣ 1c(N)N1/2 Re [ eit 1 N − eit c(N) √ N+1 N 1− eit 1N ]∣∣∣∣∣ = ∣∣∣∣∣ 1c(N)N1/2 Re [ e−it 1 2N eit 1 N − eit c(N) √ N+1 N −2i sin( t2N ) ]∣∣∣∣∣ ≤ 1 c(N)N1/2 ∣∣∣∣∣ 22 sin( t2N ) ∣∣∣∣∣ . 141 For 1+ c(N)N1/2 ≤ t2N ≤ pi2 with  > 0 fixed we get as an upper bound 1 c(N)N1/2 ∣∣∣∣∣∣ 1 sin ( 1+ c(N)N1/2 ) ∣∣∣∣∣∣ ≤ 1 1 +  < 1, given N big enough. Finally use that 2 < √ 6 to obtain the claim. Corollary B.0.2. For N ≥ N0, y ∈ N−1Z we have∣∣∣∣NP(Sk = y)− p ( (1 + o(1)) k 3N , y )∣∣∣∣ ≤ C1 {N exp{−kC2}+N1/2k−3/2} , where C1, C2 > 0 are some positive constants. Proof. This result corresponds to Corollary 9 in [3]. The proof works similarly. Instead of the reference given at the beginning of the proof of Corollary 9 in [3], we used Durrett [2], p. 95, Ex. 3.2(ii) and [2], Thm. (3.3). Note in particular that the result of Lemma B.0.1(c) can be extended to t ∈ [ (1 + o(1)) √ N 3 , piN ] if we choose δ > 0 small enough. Indeed, using Lemma B.0.1(b) we obtain |ρ(t)| ≤ e−C t 2 12N ≤ e−C N/312N ≤ (1− δ) as claimed. 142 Bibliography [1] Bhattacharya, R.N. and Ranga Rao, R. Normal approximation and asymptotic expansions. Wiley and Sons, New York-London-Sydney, 1976. MR0436272 [2] Durrett, R. Probability: Theory and Examples, Third edition. Brooks/Cole-Thomsom Learning, Belmont, 2005. [3] Mueller, C. and Tribe, R. Stochastic p.d.e.’s arising from the long range contact and long range voter processes. Probab. Theory Related Fields (1995) 102, 519–545. MR1346264

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