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Stochastic ODEs and PDEs for interacting multi-type populations Kliem, Sandra Martina 2009

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Stochastic ODEs and PDEs for interacting multi-typepopulationsbySandra Martina KliemA THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate Studies(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)September, 2009c Sandra Martina Kliem, 2009iiAbstractThis thesis consists of the manuscripts of three research papers studying stochas-tic ODEs (ordinary di erential equations) and PDEs (partial di erential equa-tions) that arise in biological models of interacting multi-type populations.In the  rst paper I prove uniqueness of the martingale problem for a de-generate SDE (stochastic di erential equation) modelling a catalytic branchingnetwork. This work is an extension of a paper by Dawson and Perkins to ar-bitrary networks. The proof is based upon the semigroup perturbation methodof Stroock and Varadhan. In the proof estimates on the corresponding semi-group are given in terms of weighted H older norms, which are equivalent to asemigroup norm in this generalized setting. An explicit representation of thesemigroup is found and estimates using cluster decomposition techniques arederived.In the second paper I investigate the long-term behaviour of a special class ofthe SDEs considered above, involving catalytic branching and mutation betweentypes. 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-dimensionaldi usion 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. Wecombine critical long-range and  xed 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 modelsin mathematical ecology. Tightness is established via a Kolmogorov tightnesscriterion. Here, estimates on the moments of small increments for the approxi-mate densities are derived via an approximate martingale problem and Green’sfunction representation.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Overview of the Manuscripts . . . . . . . . . . . . . . . . . . . . 21.1.1 Degenerate stochastic di erential equations for catalyticbranching networks . . . . . . . . . . . . . . . . . . . . . 21.1.2 Long-term behaviour of a cyclic catalytic branching sys-tem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.3 Convergence of rescaled competing species processes to aclass of SPDEs . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . 5Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Degenerate Stochastic Di erential Equations for CatalyticBranching Networks . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.1 Catalytic branching networks . . . . . . . . . . . . . . . . 92.1.2 Comparison with Dawson and Perkins [7] . . . . . . . . . 112.1.3 The model . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1.4 Statement of the main result . . . . . . . . . . . . . . . . 132.1.5 Outline of the proof . . . . . . . . . . . . . . . . . . . . . 142.1.6 Weighted H older norms and semigroup norms . . . . . . 182.1.7 Outline of the paper . . . . . . . . . . . . . . . . . . . . . 202.2 Properties of the Semigroup . . . . . . . . . . . . . . . . . . . . 212.2.1 Representation of the semigroup . . . . . . . . . . . . . . 212.2.2 Decomposition techniques . . . . . . . . . . . . . . . . . 262.2.3 Existence and representation of derivatives of the semi-group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.2.4 L1 bounds of certain di erentiation operators applied toPtf and equivalence of norms . . . . . . . . . . . . . . . 32iv2.2.5 Weighted H older bounds of certain di erentiation opera-tors applied to Ptf . . . . . . . . . . . . . . . . . . . . . 402.3 Proof of Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . 52Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 Long-term Behaviour of a Cyclic Catalytic Branching System 583.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.1.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.1.2 Main results and outline of the paper . . . . . . . . . . . 593.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.2.1 Existence and nonnegativity . . . . . . . . . . . . . . . . 603.2.2 The overall sum and uniqueness . . . . . . . . . . . . . . 623.2.3 The normalized processes . . . . . . . . . . . . . . . . . . 683.2.4 Properties of a stationary distribution to the system (3.15)of normalized processes . . . . . . . . . . . . . . . . . . . 713.2.5 Stationary distribution . . . . . . . . . . . . . . . . . . . 733.2.6 Extension to arbitrary networks . . . . . . . . . . . . . . 743.2.7 Complete analysis of the case d = 2 . . . . . . . . . . . . 74Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774 Convergence of Rescaled Competing Species Processes toa Class of SPDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.1.1 The voter model and the Lotka-Volterra model . . . . . . 784.1.2 Spatial versions of the Lotka-Volterra model . . . . . . . 804.1.3 Long-range limits . . . . . . . . . . . . . . . . . . . . . . 804.1.4 Overview of results . . . . . . . . . . . . . . . . . . . . . 824.1.5 Outline of the paper . . . . . . . . . . . . . . . . . . . . . 834.2 Main Results of the Paper . . . . . . . . . . . . . . . . . . . . . 834.2.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . 834.2.2 Main results . . . . . . . . . . . . . . . . . . . . . . . . . 864.2.3 Reformulation . . . . . . . . . . . . . . . . . . . . . . . . 884.3 An Approximate Martingale Problem . . . . . . . . . . . . . . . 944.4 Green’s Function Representation . . . . . . . . . . . . . . . . . . 1014.5 Tightness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.6 Characterizing Limit Points . . . . . . . . . . . . . . . . . . . . 1184.7 Uniqueness in Law . . . . . . . . . . . . . . . . . . . . . . . . . . 120Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1235 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.1 Overview of Results and Future Perspectives of the Manuscripts 1255.1.1 Degenerate stochastic di erential equations for catalyticbranching networks . . . . . . . . . . . . . . . . . . . . . 1255.1.2 Long-term behaviour of a cyclic catalytic branching sys-tem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126v5.1.3 Convergence of rescaled competing species processes to aclass of SPDEs . . . . . . . . . . . . . . . . . . . . . . . . 127Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129AppendicesA Appendix for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . 130A.1 ~a is non-singular . . . . . . . . . . . . . . . . . . . . . . . . . . . 130A.2 Proof of Proposition 3.2.18 . . . . . . . . . . . . . . . . . . . . . 132Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139B Appendix for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . 140Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142viList of Figures2.1 Decomposition from the catalyst’s point of view . . . . . . . . . . 112.2 De nition of NR;NC and  Ri . . . . . . . . . . . . . . . . . . . . 152.3 Decomposition of the system of SDEs . . . . . . . . . . . . . . . 193.1 The de nition of t2n 1 and t2n. . . . . . . . . . . . . . . . . . . . 66viiAcknowledgementsI 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 valuablefeedback after reading my manuscript.I also wish to thank Carl Mueller for a discussion on survival and coexistencequestions related to the limiting SPDEs I obtained in my third manuscript.A special thanks goes to the administration and the computer support ofmy 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 encouragementthroughout my studies.1Chapter 1IntroductionIn the following three Chapters I investigate degenerate stochastic ODEs (or-dinary di erential equations) and SPDEs (stochastic partial di erential equa-tions) that arise in biological models of interacting multi-type populations. Inthe  rst two Chapters I investigate the behaviour of the respective masses of a nite number of interacting populations in a non-spatial setting, and in the lastpaper I study two interacting populations only but add a spatial component.The former will result in the consideration of SDEs (stochastic di erential equa-tions), the latter in the consideration of limiting SPDEs. The former modelscan arise from a network of cooperating branching populations which require thepresence of other types (catalysts) to reproduce, while the latter class includesscaling limits of ecological models for two types competing for resources.When investigating biological models for the evolution of populations overtime, 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 typeof population is Feller’s branching di usion with parameter  and linear drift,i.e. the unique solution to the SDEdxt = bxtdt+p2 xtdBtwith constants b 2 R; 2 R+ and Bt a Brownian motion. Such di usions canbe obtained as the limit of a sequence of rescaled Galton-Watson branchingprocesses at criticality (that is, branching processes with an average numberof descendents approaching one). By adding a spatial component, one obtainssuper-Brownian motion with linear drift instead, which is the unique in lawsolution u(t;x) to the following SPDE@u@t =  u+bu+p u _W;where _W = _W(t;x) is space time white noise. Here,  u models the spatialmotion and dispersion of the population and p u _W models the stochastic  uc-tuations in the population size. ut = u(t; ) can be interpreted as the continuousspatial density of the population at time t.For both models, the degeneracy in the  uctuation term and its lack ofLipschitz continuity leads to di culties in establishing uniqueness. In the above,the additivity properties inherent to both models (for example the sum of two in-dependent (b; )-Feller-di usions starting at x1 respectively x2 is a (b; )-Feller-di usion starting at x1 + x2) can be successfully employed to investigate the2long-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 andintroduce interactions between types by interlinking the equations. Thereby onecan model competition of types for resources but also mutual help between types.As a result, the analysis of the resulting equations becomes more complicatedas additivity properties are not present anymore in this context. In the  rstand third paper of my thesis I shall therefore employ di erent perturbationmethods to derive results on the systems at hand from results of more accessiblemodels. For instance, the  rst paper uses a perturbation method of Stroock andVaradhan to obtain the new system of SDEs as a perturbation of a system ofindependent Feller di usions with constant coe cients. The last paper considersperturbations of the biased voter model for which the long-range limit wasobtained in Mueller and Tribe [12].In what follows, I shall give a short overview of the models, objectives andunderlying literature of each of the three manuscripts of this thesis.1.1 Overview of the Manuscripts1.1.1 Degenerate stochastic di erential equations forcatalytic branching networksIn the  rst paper I investigate weak uniqueness of solutions to the followingsystem of SDEs: For j 2R f1;:::;dg and Cj  f1;:::;dgnfjg:dx(j)t = bj(xt)dt+vuuut2 j(xt)0@Xi2Cjx(i)t1Ax(j)t dBjt (1.1)and for j =2Rdx(j)t = bj(xt)dt+q2 j(xt)x(j)t dBjt: (1.2)Here xt 2 Rd+ and bj; j;j = 1;:::;d are H older-continuous functions on Rd+with  j(x) > 0, and bj(x)  0 if xj = 0. The Bjt;j 2 f1;:::;dg are independentBrownian motions.This system of SDEs models catalytic branching networks, where typesi 2 Cj catalyze the replication of type j;j 2 R (the so-called reactants). Suchsystems 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 2 R in state xt branches at a rate  j(xt)Pi2Cj x(i)tproportional to the sum of masses of types i;i2Cj at time t.The degeneracies in the covariance coe cients of this system and their lackof Lipschitz continuity make the investigation of uniqueness a challenging ques-tion. The former rules out the classic Stroock-Varadhan approach of perturbing3Brownian motion and the latter prevents application of It^o’s pathwise unique-ness arguments. Similar results have been proven in Athreya, Barlow, Bassand Perkins [1] and Bass and Perkins [2] but without the additional singularityPi2Cj x(i)t in the covariance coe cients of the di usion.The question of uniqueness of equations with non-constant coe cients arisesalready 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 systemof SDEs: For j 2R and Cj = fcjg;cj 6= j,dx(j)t = bj(xt)dt+q2 j(xt)x(cj)t x(j)t dBjt;and (1.2) as above. This restriction to at most one catalyst per reactant issu cient for the renormalization analysis for d = 2 types, but for more than 2types one will encounter models where one type may have two catalysts. Thegoal of my  rst paper was to overcome this restriction and to allow considerationof general multi-type branching networks as envisioned in the section on futurechallenges in [6]. In my  rst paper, I extend the techniques of [7] to the settingof general catalytic networks, i.e. to (1.1) and (1.2). My work further includesnatural settings such as competing hypercycles (cf. Eigen and Schuster [8], p.55respectively Hofbauer and Sigmund [10], p.106). This latter work proposed ananalogous system of ODEs as a model for the emergence of long polynucleotidesin prebiotic evolution.1.1.2 Long-term behaviour of a cyclic catalyticbranching systemAs an application of the above I investigate the following special case in mysecond paper, involving cyclic catalytic branching and mutation between types.As I shall point out in this paper, the cyclic setup can easily be extended toarbitrary networks. Questions for survival and coexistence of types in the longtime 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  nitetime survival-extinction but they were not able to determine, if the overall sumdies out in the limit or not. In [10] (p. 86) multi-type branching processes withindependent replication and mutation between types were rejected as a modelsince typically one type would take over, contrary to the observed diversitywhich emerged from the primordial soup.Let the following system of SDEs for d 2 be given:dxit =q2 ixitxi+1t dBit +dXj=1xjtqjidt; i2 f1;:::;dg;4where xd+1t  x1t . I assumed the  i and qji;i6= j to be given positive constantsand the xi0  0;i 2 f1;:::;dg to be given initial conditions. (qji) is a Q-matrix modelling the mutations or migrations from type j to type i. I shallinvestigate in particular the behaviour of the sum of types st =Pdi=1 xit and ofthe normalized processes yit = xit=st in the time-limit. The latter addresses thediversity issue in [10].1.1.3 Convergence of rescaled competing speciesprocesses to a class of SPDEsThe objectives of this paper were threefold. To better understand them, I shall rst introduce the three papers that provide motivation.Firstly, in [12], Mueller and Tribe construct a sequence of rescaled compet-ing species processes  Nt 2 f0;1gZ=N in dimension d = 1 and show that itsapproximate densitiesA( Nt )(x)  1jfy 2 Z=N : 0 <jy xj  1=pNgjXy2Z=N;0<jy xj 1=pN Nt (y); x2N 1Zconverge in distribution to a continuous space time density that solves an SPDE.Here,  Nt 2 f0;1gZ=N denotes the con guration at time t of a \voter process"with bias  =  N . That is, each type (0 or 1) invades a randomly chosen\neighbouring site" with constant rate, where  > 0 would slightly favour 1’s bygiving them a slightly larger invasion rate.  Nt (x) = i if site x2 Z=N is occupiedby type i;i = 0;1 and hence ut can be interpreted as the limiting continuousspace time density of type 1 and 1  ut as the density of type 0. [12]  x   0,i.e. consider the case where the opinion of type 1 is slightly dominant. Theyshow that ut is a solution of the following SPDE, the heat equation with drift,driven by Fisher-Wright noise, namely@u@t = u6 + 2 (1  u)u+p4u(1 u) _W: (1.3)Observe that [12] scale space by 1=N and use fy 2 Z=N : 0 < jy  xj  1=pNg, the set of neighbours of x, to calculate approximate densities. Hence,the number of neighbours of x 2 Z=N is increasing proportionally to 2N1=2.They thus obtain long-range interactions. Finally, they also speed up time toobtain appropriate limits.After rescaling (1.3) appropriately, one obtains the Kolmogorov-Petrovskii-Piscuinov (KPP) equation driven by Fisher-Wright noise. The behaviour of thisSPDE has already been investigated in Mueller and Sowers [11] in detail, wherethe 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 theirmodels to approach 1, the models can also be interpreted as small perturbations5of 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 treatthe low density regime, i.e. where only a  nite number of individuals of type 1is present. Also note that both papers use a di erent 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  xed kernel models in dimensions d  3 and long-range kernelmodels in arbitrary dimension separately. Finally, in Cox and Perkins [5], theresults of [4] for d  3 are used to relate the limiting super-Brownian motionsin the  xed kernel case to questions of coexistence and survival of a rare typein the original Lotka-Volterra model.Thirdly, spatial versions of the Lotka-Volterra model with  nite range wereintroduced and investigated in Neuhauser and Pacala [13]. The model from[13] incorporates a fecundity parameter and models both intra- and interspeci ccompetition. The paper shows that short-range interactions alter the predictionsof the mean- eld model.In my paper I try to extend the approach of [12] for voter models to smallperturbations of voter models similar to the perturbations in [4]. I work atcriticality in the hope to obtain continuous densities in the limit that solve aclass 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] asthe approximating models and to determine their limits. As an additional ex-tension to [12] I shall investigate the weak uniqueness of the limiting class ofSPDEs as weak uniqueness of the solutions to the SPDE will yield in turn weakuniqueness of the limits of the approximating densities.The last objective of this paper was to combine both long-range models atcriticality and  xed kernel models in the perturbations. I investigate if theadditional  xed 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 ableto consider rescaled Lotka-Volterra models with long-range dispersal and short-range competition.1.2 Concluding RemarksTightness of approximating particle systems can be used to prove existence oflimiting points of the approximating particle systems. Often, all limits can beshown to have certain properties in common. For instance, if all limits satisfyan SDE or SPDE as in my third paper for the case of long-range interactionsonly, weak uniqueness for the limiting systems of SDEs or SPDEs then yieldsuniqueness of the limits. Additionally, weak uniqueness of the solutions makesavailable certain tools that are used to investigate the behaviour of the systemsat hand. Therefore, the proof of weak uniqueness in the  rst paper was funda-mental to the analysis of the model for cyclic catalytic branching and mutation6between 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  uctua-tions that stems from catalytic branching (the Fisher-Wright noise term can beseen as an application of a 2-cyclic model). Additionally all three models areparameter-dependent, where the parameters can be used to answer questions ofsurvival and coexistence of types.7Bibliography[1] Athreya, S.R. and Barlow, M.T. and Bass, R.F. and Perkins,E.A. Degenerate stochastic di erential equations and super-Markov chains.Probab. Theory Related Fields (2002) 123, 484{520. MR1921011[2] Bass, R.F. and Perkins, E.A. Degenerate stochastic di erential equa-tions with H older continuous coe cients 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 convergeto 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. Poincar e 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 di usions. 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-brownianmotion. Ann. Probab. (2001) 29, 820{861. MR1849179[10] Hofbauer, J. and Sigmund, K. The Theory of Evolution and DynamicalSystems. 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 KPPEquation with Noise. J. Funct. Anal. (1995) 128, 439{498. MR1319963[12] Mueller, C. and Tribe, R. Stochastic p.d.e.’s arising from the longrange 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 theLotka-Volterra model with interspeci c competition. Ann. Appl. Probab.(1999) 9, 1226{1259. MR17285618[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. MR19154459Chapter 2Degenerate StochasticDi erential Equations forCatalytic BranchingNetworks12.1 Introduction2.1.1 Catalytic branching networksIn this paper we investigate weak uniqueness of solutions to the following systemof stochastic di erential equations (SDEs): For j 2 R  f1;:::;dg and Cj  f1;:::;dgnfjg:dx(j)t = bj(xt)dt+vuuut2 j(xt)0@Xi2Cjx(i)t1Ax(j)t dBjt (2.1)and for j =2Rdx(j)t = bj(xt)dt+q2 j(xt)x(j)t dBjt: (2.2)Here xt 2 Rd+ and bj; j;j = 1;:::;d are H older-continuous functions on Rd+with  j(x) > 0, and bj(x)  0 if xj = 0.The degeneracies in the covariance coe cients of this system make the inves-tigation of uniqueness a challenging question. Similar results have been provenin [1] and [4] but without the additional singularityPi2Cj x(i)t in the covariancecoe cients of the di usion. Other types of singularities, for instance replacingthe additive form by a multiplicative formQi2Cj x(i)t , are possible as well, underadditional assumptions on the structure of the network (cf. Remark 2.1.9 at theend of Subsection 2.1.5).The given system of SDEs can be understood as a stochastic analogue toa system of ODEs for the concentrations yj;j = 1;:::;d of a type Tj. Then1A version of this chapter has been accepted for publication. Kliem, S.M. (2009) DegenerateStochastic Di erential Equations for Catalytic Branching Networks. Ann. Inst. H. Poincar eProbab. Statist.10yj=_yj corresponds to the rate of growth of type Tj and one obtains the followingODEs (see [9]): for independent replication _yj = bjyj, autocatalytic replication_yj =  jy2j and catalytic replication _yj =  j Pi2Cj yi yj. In the catalytic casethe types Ti;i2Cj catalyze the replication of type j, i.e. the growth of type jis proportional to the sum of masses of types i;i2Cj present at time t.An important case of the above system of ODEs is the so-called hypercy-cle,  rstly introduced by Eigen and Schuster (see [8]). It models hypercyclicreplication, i.e. _yj =  jyj 1yj and represents the simplest form of mutual helpbetween di erent 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 branchingrate in the stochastic setting, i.e. type j branches at a rate proportional to thesum of masses of types i;i2Cj at time t.The question of uniqueness of equations with non-constant coe cients arisesalready in the case d = 2 in the renormalization analysis of hierarchically inter-acting two-type branching models treated in [6]. The consideration of successiveblock averages leads to a renormalization transformation on the di usion func-tions of the SDEdx(i)t = c  i  x(i)t dt+p2gi(xt)dBit;i = 1;2with  i  0;i = 1;2  xed. Here g = (g1;g2) with gi(x) = xi i(x) or gi(x) =x1x2 i(x), i = 1;2 for some positive continuous function  i on R2+. The renor-malization transformation acts on the di usion coe cients g and produces anew set of di usion coe cients for the next order block averages. To be ableto iterate the renormalization transformation inde nitely a subclass of di usionfunctions has to be found that is closed under the renormalization transforma-tion. To even de ne the renormalization transformation one needs to show thatthe above SDE has a unique weak solution and to iterate it we need to establishuniqueness under minimal conditions on the coe cients.This paper is an extension of the work done in Dawson and Perkins [7]. Thelatter, motivated by the stochastic analogue to the hypercycle and by [6], provedweak uniqueness in the above mentioned system of SDEs (2.1) and (2.2), where(2.1) is restricted todx(j)t = bj(xt)dt+q2 j(xt)x(cj)t x(j)t dBjt;i.e. Cj = fcjg and (2.2) remains unchanged. This restriction to at most onecatalyst per reactant is su cient for the renormalization analysis for d = 2types, but for more than 2 types one will encounter models where one typemay have two catalysts. The present work overcomes this restriction and allowsconsideration of general multi-type branching networks as envisioned in [6],including further natural settings such as competing hypercycles (cf. [8] page 55resp. [9], p. 106). In particular, the techniques of [7] will be extended to thesetting of general catalytic networks.Intuitively it is reasonable to conjecture uniqueness in the general setting asthere is less degeneracy in the di usion coe cients; x(cj)t changes to Pi2Cj x(i)t ,11Figure 2.1: Decomposition from the catalyst’s point of view: Arrows point fromvertices i 2 NC to vertices j 2 Ri (for the de nition of NC;Ri and N2 seeSubsections 2.1.3 and 2.1.5 to follow). Separate points signify vertices j 2 N2.The dotted arrows signify arrows which are only allowed in the generalizedsetting and thus make a decomposition of the kind used in [7] inaccessible.all coordinates i 2 Cj have to become zero at the same time to result in asingularity.For d = 2 weak uniqueness was proven for a special case of a mutuallycatalytic model ( 1 =  2 = const.) via a duality argument in [10]. Unfortunatelythis 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 signi cant change is the additional dependency among catalysts. In[7] the semigroup of the process under consideration could be decomposed intogroups 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], wherebounds on the semigroup are derived, i.e. Section 2 of [7] (\Properties of thebasic semigroups"), could be reduced to the setting of a single vertex or a singlecatalyst with a  nite number of reactants. In the general setting this strategy isno longer available as one reactant is now allowed to have multiple catalysts (seeagain 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 semigroupare 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 ofreactantsRi). For the general setting it turns out to be more e cient to considerevery catalyst i with the set Ri of its reactants. In particular, the restrictionfrom Ri to  Ri, including only reactants whose catalysts are all zero, turns outto be crucial for later de nitions and calculations. It plays a key role in theextension of the de nition of the weighted H older norms to general networks12(see Subsection 2.1.6).Changes in one catalyst indirectly impact other catalysts now via commonreactants, resulting for instance in new mixed partial derivatives. As a  rst stepa 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 ofsemigroups of each catalyst with its reactants. Now a change in one catalystresp. coordinate of the semigroup impacts in particular the local covarianceof all its reactants. As the other catalysts of this reactant also appear in thiscoe cient, a decomposition becomes impossible. Instead the triangle inequalityhas to be often used to express resulting multi-dimensional coordinate changesof the function G, which is closely related with the semigroup representation(see (2.16)), via one-dimensional ones. As another important tool Lemma 2.2.6was developed in this context.The ideas of the proofs in [7] often had to be extended. Major changes canbe found in the critical Proposition 2.2.25 and its associated Lemmas (especiallyLemma 2.2.29). The careful extension of the weighted H older norms to arbitrarynetworks had direct impact on the proofs of Lemma 2.2.19 and Theorem 2.2.20.2.1.3 The modelLet a branching network be given by a directed graph (V;E) with vertices V =f1;:::;dg and a set of directed edges E = fe1;:::;ekg. The vertices representthe di erent types, whose growth is under investigation, and (i;j) 2 E meansthat type i \catalyzes" the branching of type j. As in [7] we continue to assume:Hypothesis 2.1.1. (i;i) =2 E for all i2V.Let C denote the set of catalysts, i.e. the set of vertices which appear as the1st element of an edge and R denote the set of reactants, i.e. the set of verticesthat appear as the 2nd element of an edge.For j 2R, letCj = fi : (i;j) 2 Egbe the set of catalysts of j and for i2C, letRi = fj : (i;j) 2 Egbe the set of reactants, catalyzed by i. If j =2 R let Cj = ; and if i =2 C, letRi = ;.We shall consider the following system of SDEs:For j 2R:dx(j)t = bj(xt)dt+vuuut2 j(xt)0@Xi2Cjx(i)t1Ax(j)t dBjtand for j =2Rdx(j)t = bj(xt)dt+q2 j(xt)x(j)t dBjt:Our goal will be to show the weak uniqueness of the given system of SDEs.132.1.4 Statement of the main resultIn what follows we shall impose additional regularity conditions on the coe -cients of our di usions, similar to the ones in Hypothesis 2 of [7], which willremain valid unless indicated to the contrary. jxj is the Euclidean length ofx2 Rd and for i2V let ei denote the unit vector in the ith direction.Hypothesis 2.1.2. For i2V, i : Rd+ ! (0;1);bi : Rd+ ! Rare taken to be H older continuous of some positive index on compact subsets ofRd+ such that jbi(x)j  c(1 + jxj) on Rd+, and(bi(x)  0 if xi = 0: In addition,bi(x) > 0 if i2C [R and xi = 0:De nition 2.1.3. If  is a probability on Rd+, a probability P on C(R+;Rd+) issaid to solve the martingale problem MP(A, ) if under P, the law of x0(!) = !0(xt(!) = !(t)) is  and for all f 2 C2b (Rd+),Mf(t) = f(xt)  f(x0)  Z t0Af(xs)dsis a local martingale under P with respect to the canonical right-continuous ltration (Ft).Remark 2.1.4. The weak uniqueness of a system of SDEs is equivalent to theuniqueness of the corresponding martingale problem (see for instance, [12],V.(19.7)).For f 2 C2b (Rd+), the generator corresponding to our system of SDEs isAf(x) = A(b; )f(x)=Xj2R j(x)0@Xi2Cjxi1Axjfjj(x) +Xj =2R j(x)xjfjj(x) +Xj2Vbj(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 useS =8<:x2 Rd+ :Yj2R0@Xi2Cjxi +xj1A> 09=;: (2.3)We  rst note that S is a natural state space for A.14Lemma 2.1.5. If P is a solution to MP(A, ), where  is a probability on Rd+,then xt 2 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 2.1.4provides 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  , on S, there is exactly one solution to MP(A, ).2.1.5 Outline of the proofOur main task in proving Theorem 2.1.6 consists in establishing uniqueness ofsolutions to the martingale problem MP(A, ). Existence can be proven as inTheorem 1.1 of [1]. The main idea in proving uniqueness consists in understand-ing our di usion as a perturbation of a well-behaved di usion and applying theStroock-Varadhan perturbation method (refer to [13]) to it. This approach canbe devided into three steps.Step 1: Reduction of the problem. We can assume w.l.o.g. that  =  x0.Furthermore it is enough to consider uniqueness for families of strong Markovsolutions. Indeed, the  rst reduction follows by a standard conditioning argu-ment (see p. 136 of [3]) and the second reduction follows by using Krylov’sMarkov selection theorem (Theorem 12.2.4 of [13]) together with the proof ofProposition 2.1 in [1].Next we shall use a localization argument of [13] (see e.g. the argument inthe proof of Theorem 1.2 of [4]), which basically states that it is enough if foreach x0 2 S the martingale problem MP( ~A; x0) has a unique solution, wherebi = ~bi and  i = ~ i agree on some B(x0;r0) \ Rd+. Here we used in particularthat a solution never exits S as shown in Lemma 2.1.5.Finally, if the covariance matrix of the di usion is non-degenerate, unique-ness follows by a perturbation argument as in [13] (use e.g. Theorem 6.6.1 andTheorem 7.2.1). Hence consider only singular initial points, i.e. where eithernx(j)0 = 0 orXi2Cjx(i)0 = 0 for some j 2Roornx(j)0 = 0 for some j =2R:oStep 2: Perturbation of the generator. Fix a singular initial point x0 2 Sand set (for an example see Figure 2.2)NR =8<:j 2R :Xi2Cjx0i = 09=;;NC = [j2NRCj;N2 = Vn(NR [NC) ; Ri = Ri \NR;15PSfrag replacements672 : x02 = 05 : x05 = 04 : x04 > 03 : x03 = 0 2 2NC 3 2NC 4 =2NC 5 =2NC 6 2  Ri;i = 1;2;3 x06 > 0 6 2NR 6 =2  R3 6 2R3 x06  0 7 =2NR 1 2NC1 : x01 = 0Figure 2.2: De nition of NR;NC and  Ri. The  ’s are the implications deducedfrom 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) = fi2V : x0i = 0g (if i =2Z, then x0i > 0 and so x(i)s > 0 forsmall s a.s. by continuity). Moreover, if x0 2 S, then NR \Z = ; andNR [NC [N2 = Vis a disjoint union.Notation 2.1.7. In what follows letRA  ff;f : A! Rg resp. RA+  ff;f : A! R+g:for arbitrary A V.Next we shall rewrite our system of SDEs with corresponding generatorA as a perturbation of a well-understood system of SDEs with correspondinggenerator A0, which has a unique solution. The state space of A0 will beS x0 = S0 = fx2 Rd : xi  0 for all i =2NRg.First, we view  x(j) j2NR; x(i) i2NC , i.e. the set of vertices with zerocatalysts together with these catalysts, near its initial point  x0j j2NR; x0i i2NC as a perturbation of the di usion on RNR  RNC+ , which is given by the unique16solution to the following system of SDEs:dx(j)t = b0jdt+vuuut2 0j0@Xi2Cjx(i)t1AdBjt; x(j)0 = x0j; for j 2NRand (2.4)dx(i)t = b0idt+q2 0ix(i)t dBit; x(i)0 = x0i; for i2NC;where for j 2 NR, b0j = bj(x0) 2 R and  0j =  j(x0)x0j > 0 as x0j > 0 ifits catalysts are all zero. Also, b0i = bi(x0) > 0 as x0i = 0 for i 2 NC and 0i =  i(x0)Pk2Ci x0k > 0 if i 2 NC \R as i is a zero catalyst thus having atleast one non-zero catalyst itself, or  0i =  i(x0) > 0 if i2NCnR. Note that thenon-negativity of b0i;i2 NC ensures that solutions starting in fx0i  0g remainthere (also see de nition of S0).Secondly, for j 2N2 we view this coordinate as a perturbation of the Fellerbranching process (with immigration)dx(j)t = b0jdt+q2 0jx(j)t dBjt; x(j)0 = x0j; for j 2N2; (2.5)where b0j = (bj(x0)_0) (at the end of Section 2.3 the general case bj(x0) 2 R isreduced to bj(x0)  0 by a Girsanov transformation),  0j =  j(x0)Pi2Cj x0i > 0if j 2 R by de nition of N2, i.e. at least one of the catalysts being positive,or  0j =  j(x0) > 0 if j =2 R. As for i 2 NC, the non-negativity of b0j;j 2 N2ensures that solutions starting in fx0j  0g remain there (see again de nition ofS0).Therefore we can view A as a perturbation of the generatorA0 =Xj2Vb0j @@xj+Xj2NR 0j0@Xi2Cjxi1A @2@x2j+Xi2NC[N2 0i xi @2@x2i : (2.6)The coe cients b0i; 0i found above for x0 2 S now satisfy8><>: 0j > 0 for all j;b0j  0 if j =2NR;b0j > 0 if j 2 (R[C) \Z;(2.7)whereNR \Z = ;: (2.8)In the remainder of the paper we shall always assume the conditions (2.7) holdwhen dealing with A0 whether or not it arises from a particular x0 2 S asabove. As we shall see in Subsection 2.2.1 the A0 martingale problem is thenwell-posed and the solution is a di usion onS0  S x0 = fx2 Rd : xi  0 for all i2VnNR = NC [N2g: (2.9)17Notation 2.1.8. In the following we shall use the notationNC2  NC [N2:Step 3: A key estimate. SetBf := (A A0)f=Xj2V ~bj(x)  b0j @f@xj +Xj2NR ~ j(x)   0j 0@Xi2Cjxi1A@2f@x2j+Xi2NC2 ~ i(x)   0i xi@2f@x2i ;wherefor j 2V; ~bj(x) = bj(x);for j 2NR; ~ j(x) =  j(x)xj; andfor i2NC2; ~ i(x) = 1fi2Rg i(x)Xk2Cixk + 1fi=2Rg i(x):By using the continuity of the di usion coe cients of A and the localizationargument mentioned in Step 1 we may assume that the coe cients of the op-erator B are arbitrarily small, say less than  in absolute value. The key step(see Theorem 2.3.3) will be to  nd a Banach space of continuous functions withnorm k k, depending on x0, so that for  small enough and  0 > 0 large enough,kBR fk 12 kfk; 8  > 0: (2.10)HereR f =Z 10e  sPsfds (2.11)is the resolvent of the di usion with generator A0 and Pt is its semigroup.The uniqueness of the resolvent of our strong Markov solution will thenfollow 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 branchingnetwork our results carry over to the system of SDEs, where the additive formfor the catalysts is replaced by a multiplicative form as follows. For j 2 R wenow considerdx(j)t = bj(xt)dt+vuuut2 j(xt)0@Yi2Cjx(i)t1Ax(j)t dBjtinstead and for j =2Rdx(j)t = bj(xt)dt+q2 j(xt)x(j)t dBjt18as before. Indeed, if we impose that for all j 2R we have eitherjCjj = 1 orjCjj  2 and for all i1 6= i2;i1;i2 2Cj : i1 2Ci2 or i2 2Ci1;and if we assume that Hypothesis 2.1.2 holds, then we can show a result similarto Theorem 2.1.6.For instance, the following system of SDEs would be included.dx(1)t = b1(xt)dt+q2 1(xt)x(2)t x(3)t x(1)t dB1t;dx(2)t = b2(xt)dt+q2 2(xt)x(3)t x(4)t x(2)t dB2t;dx(3)t = b3(xt)dt+q2 3(xt)x(4)t x(1)t x(3)t dB3t;dx(4)t = b4(xt)dt+q2 4(xt)x(1)t x(2)t x(4)t dB4t:Note in particular, that the additional assumptions on the network ensure thatat most one of either the catalysts in Cj or j itself can become zero, so that weobtain the same generator A0 as in the setting of additive catalysts if we set 0j   j(x0)Qi2fjg[Cj:x0i >0x0i (cf. the derivation of (2.4)).Remark 2.1.10. In [5] the H older condition on the coe cients was successfullyremoved 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 ofa well-understood SDE, one could start extending [5] to arbitrary networks byusing the same generator and semigroup decomposition for the well-understoodSDE as considered in this paper.2.1.6 Weighted H older norms and semigroup normsIn this section we describe the Banach space of functions which will be usedin (2.10). In (2.10) we use the resolvent of the generator A0 with state spaceS0 = S x0 = fx 2 Rd : xi  0 for all i 2 NC2g. Note in particular that thestate space and the realizations of the sets NR;  Ri etc. depend on x0.Next we shall de ne the Banach space of weighted  -H older continuous func-tions on S0, C w(S0)  Cb(S0), in two steps. It will be the Banach space we lookfor and is a modi cation of the space of weighted H older norms used in [4].Let f : S0 ! R be bounded and measurable and  2 (0;1). As a  rst stepde ne the following seminorms for i2NC:jfj ;i = supnjf(x+h)  f(x)j jhj  x =2i _jhj  =2 :jhj> 0;hk = 0 if k =2 fig[  Ri;x;h2 S0o:19a0a0a1a1Figure 2.3: Decomposition of the system of SDEs: un lled circles, resp.  lledcircles, resp. squares are elements of NR, resp. NC, resp. N2. The de nition ofjfj ;i;i 2 NC allows changes in i ( lled circles) and the associated j 2  Ri (un- lled circles), the de nition of jfj ;j;j 2N2 allows changes in j 2N2 (squares).Hence changes in all vertices are possible.For j 2N2 this corresponds to settingjfj ;j = supnjf(x+h)  f(x)j jhj  x =2j _jhj  =2 :hj > 0;hk = 0 if k 6= j;x2 S0o:This de nition is an extension of the de nition in [7], p. 329. In our contextthe de nition of jfj ;i;i2NC had to be extended carefully by replacing the setRi (in [7] equal to the set  Ri) by the set  Ri  Ri. Observe that the seminormsfor i2NC and j 2N2 taken together still allow changes in all coordinates (seeFigure 2.3). The de nition of jfj ;j;j 2 N2 furthermore varies slightly fromthe one in [7]. We use our de nition instead as it enables us to handle thecoordinates i2NC;j 2N2 without distinction.Secondly, set I = NC2. Then letjfjC w = maxj2Ijfj ;j; kfkC w= jfjC w+ kfk1;where kfk1 is the supremum norm of f. kfkC w is the norm we looked for andits corresponding Banach subspace of Cb(S0) isC w(S0) = ff 2 Cb(S0) :kfkC w<1g;the Banach space of weighted  -H older continuous functions on S0. Note thatthe de nition of the seminorms jfj ;j;j 2 I depends on NC;  Ri etc. and henceon x0. Thus kfkC w depends on x0 as well.The seminorms jfj ;i are weaker norms near the spatial degeneracy at xi = 0where we expect to have less smoothing by the resolvent.20Some more background on the choice of the above norms can be found in[4], Section 2. Bass and Perkins ([4]) considerjfj  ;i  supnjf(x+hei)  f(x)jjhj  x =2i : h> 0;x2 Rd+o;jfj   supi djfj  ;i and kfk   jfj  + kfk1instead, where ei denotes the unit vector in the i-th direction in Rd. They showthat if f 2 Cb(Rd+) is uniformly H older of index  2 (0;1], and constant outsideof a bounded set, then f 2 C ; w  ff 2 Cb(Rd+) :kf k  < 1g. On the otherhand, f 2 C ; w implies f is uniformly H older of order  =2.As it will turn out later (see Theorem 2.2.20) our norm kfkC w is equivalentto another norm, the so-called semigroup norm, de ned via the semigroup Ptcorresponding to the generator A0 of our process. As we shall mainly investigateproperties of the semigroup Pt on Cb(S0) in what follows, it is not surprisingthat this equivalence turns out to be useful in later calculations.In general one de nes the semigroup norm (cf. [2]) for a Markov semigroupfPtg on the bounded Borel functions on D where D  Rd and  2 (0;1) viajfj = supt>0kPtf  fk1t =2 ; kfk = jfj + kfk1 : (2.12)The associated Banach space of functions is thenS = ff : D ! R : f Borel ;kfk <1g: (2.13)Convention 2.1.11. Throughout this paper all constants appearing in state-ments of results and their proofs may depend on a  xed parameter  2 (0;1)and fb0j; 0j : j 2Vg as well as on jVj = d. By (2.7)M0 = M0( 0;b0)  maxi2Vn 0i _  0i  1 _  b0i  o_ maxi2(R[C)\Z b0i  1 <1: (2.14)Given  2 (0;1), d and 0 <M <1, we can, and shall, choose the constants tohold uniformly for all coe cients satisfying M0  M.2.1.7 Outline of the paperProofs 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 maybe found on the arXiv at arXiv:0802.0035v2.The outline of the paper is as follows. In Section 2.2 the semigroup Ptcorresponding 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 representationof the semigroup in Subsection 2.2.1. In Subsection 2.2.2 the canonical measureN0 is introduced which is used in Subsection 2.2.3 to prove existence and givea representation of derivatives of the semigroup. In Subsections 2.2.4 and 2.2.521bounds are derived on the L1 norms and on the weighted H older norms ofthose di erentiation operators applied to Ptf, which appear in the de nition ofA0. Furthermore, at the end of Subsection 2.2.4 the equivalence of the weightedH older norm and semigroup norm is shown. Finally, in Section 2.3 bounds onthe resolvent R 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 solutionsto the martingale problem MP(A, ) from this, as in [7].2.2 Properties of the Semigroup2.2.1 Representation of the semigroupIn this subsection we shall  nd an explicit representation of the semigroup Ptcorresponding to the generator A0 (cf. (2.6)) on the state space S0 and fur-ther preliminary results. We assume the coe cients satisfy (2.7) and Conven-tion 2.1.11 holds.Let us have a look at (2.4) and (2.5) again. For i 2 NC or j 2 N2the 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 2 NR becomeindependent time-inhomogeneous Brownian motions (with drift), whose dis-tributions are well understood. Thus if the associated process is denoted byxt =nx(j)toj2NR[NC2=nx(j)toj2V, the semigroup Ptf has the explicit repre-sentationPtf(x) =   i2NC2 Pixi  ZRjNRjf fzjgj2NR ;nx(i)toi2NC2 (2.15) Yj2NRp 0j 2I(j)t zj  xj  b0jt dzj35;where Pixi is the law of the Feller branching immigration process x(i) onC(R+;R+), started at xi with generatorAi0 = b0i @@x + 0i x @2@x2;I(j)t =Z t0Xi2Cjx(i)s ds;and for y 2 (0;1)py(z) := e z22y(2 y)1=2:Remark 2.2.1. This also shows that the A0 martingale problem is well-posed.22For (y;z) = fyjgj2NR ;fzigi2NC2 and xNR  fxjgj2NR, letG(y;z) = Gt;xNR (y;z) = Gt;xNR fyjgj2NR ;fzigi2NC2 (2.16)=ZRjNRjf fujgj2NR ;fzigi2NC2 Yj2NRp 0j 2yj uj  xj  b0jt duj:Notation 2.2.2. In the following we shall use the notationsENC2 =  i2NC2 Pixi ; INRt =nI(j)toj2NR; xNC2t =nx(i)toi2NC2and we shall write E whenever we do not specify w.r.t. which measure weintegrate.Now (2.15) can be rewritten asPtf(x) = ENC2hGt;xNR INRt ;xNC2t i= ENC2hG INRt ;xNC2t i: (2.17)Lemma 2.2.3. Let j 2NR, then(a)ENC224Xi2Cjx(i)t35= Xi2Cj xi +b0it ;ENC224 Xi2Cjx(i)t!235= Xi2Cjxi!2+Xi2Cj0@20@Xk2Cjb0k + 0i1Axi1At+Xi2Cj0@0@Xk2Cjb0k + 0i1Ab0i1At2;ENC224 Xi2Cj x(i)t  xi !235= Xi2Cj2 0i xit+Xi2Cj0@0@Xk2Cjb0k + 0i1Ab0i1At2andENC2hI(j)ti= ENC224Z t0Xi2Cjx(i)s ds35 = Xi2Cj xit+ b0i2 t2 :(b)ENC2  I(j)t  p  c(p)t p mini2Cj (t+xi) p 8p> 0:Note. Observe that the requirement b0i > 0 if i 2 (R [ C) \ Z as in (2.7)is crucial for Lemma 2.2.3(b). As i 2 Cj;j 2 NR implies i 2 C \ Z, (2.7)guarantees b0i > 0. The bound (b) cannot be applied to i 2 N2 in general, as(2.7) only gives b0i  0 in these cases.23Proof of (a). The  rst three results follow from Lemma 7(a) in [7] togetherwith the independence of the Feller-di usions under consideration.Proof of (b). Proceeding as in the proof of Lemma 7(b) in [7] we obtainENC2  I(j)t  p  cpeZ 10ENC2he u 1I(j)tiu p 1du cpemini2Cj Z 10Pixihe u 1I(i)tiu p 1du as I(j)t = Pi2Cj Rt0 x(i)s ds  Pi2Cj I(i)t , where the Feller-di usions under con-sideration are independent. Now we can proceed as in Lemma 7(b) of [7] toobtain the desired result.Lemma 2.2.4. Let Gt;xNR be as in (2.16). Then(a) for j 2NR    @Gt;xNR@xj fyjgj2NR ;fzigi2NC2     =    @Gt;xNR@xj (y;z)     kfk1 ( 0jyj) 1=2;(2.18)and more generally for any k 2 N, there is a constant ck such that     @kGt;xNR@xkj (y;z)      ck kfk1 y k=2j :(b) For j 2NR    @Gt;xNR@yj (y;z)     c1 kfk1 y 1j : (2.19)More generally there are constants ck;k 2 N such that for l1;l2;j1;j2 2NR,     @m1+m2+k1+k2Gt;xNR@xm1l1 @xm2l2 @yk1j1 @yk2j2 (y;z)      cm1+m2+k1+k2 kfk1 y m1=2l1 y m2=2l2 y k1j1 y k2j2for all m1;m2;k1;k2 2 N.(c) Let yNR = fyjgj2NR and zNC2 = fzigi2NC2, then for all zNC2 withzi  0, i 2 NC2 we have that  xNR;yNR ! Gt;xNR yNR;zNC2 is C3 onRjNRj  (0;1)jNRj.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 2Cb(S0) withjPtf(x)  Ptf(x0)j  ckfk1 t 1 jx x0j:24Proof. The outline of the proof is as in the proof of [7], Lemma 12. We shallnevertheless show the proof in detail as it illustrates some basic notational issues,which will appear again in later theorems. Note in particular the frequent useof the triangle inequality resulting in additional sums of the form Pj:j2  Ri0inthe second part of the proof.Using (2.17), we have for x;x0 2 RNR,  Ptf x;xNC2  Ptf x0;xNC2   (2.20)=   ENC2hGt;x INRt ;xNC2t  Gt;x0 INRt ;xNC2t i    kfk1Xj2NRjxj  x0jjq 0jENC2  I(j)t  1=2 (by (2.18)) ckfk1Xj2NRjxj  x0jjq 0jt 1=2 mini2Cjn(t+xi) 1=2o(by Lemma 2.2.3(b)) ckfk1 t 1Xj2NRjxj  x0jj:Next we shall consider x;x0 = x+hei0 2 RNC2 where i0 2NC2 is arbitrarily xed. Assume h> 0 and let xh denote an independent copy of x(i0) starting ath but with b0i0 = 0. Then x(i0) +xh has law Pi0xi0+h(additive property of Fellerbranching processes) and so if Ih(t) =Rt0 xhsds,  Ptf xNR;x0  Ptf xNR;x   =    ENC2 Gt;xNR nI(j)t + 1fi02CjgIh(t)oj2NR; xit + 1fi=i0gxht i2NC2  Gt;xNR nI(j)toj2NR;xNC2t      :For what follows it is important to observe thatfj 2NR : i0 2Cjg = j : j 2  Ri0 ;having made the de nition of  Ri necessary. Next we shall use the triangleinequality to  rst sum up changes in the jth coordinates (where j 2 NR suchthat i0 2 Cj) separately in increasing order, followed by the change in thecoordinate i0. If Th = infft  0 : xht = 0g we thus obtain as a bound for theabove (recall that ek denotes the unit vector in the kth direction):Xj:j2  Ri0ckfk1 ENC2 Ih(t) I(j)t  1 + 2 kfk1 E[Th >t]=Xj:j2  Ri0ckfk1 ENC2[Ih(t)]ENC2  I(j)t  1 + 2 kfk1 E[Th >t]25by (2.19) and as kGk1 kfk1 by the de nition of G. Next we shall use thatE[Th > t]  ht 0i0(for reference see equation (2.26) in Section 2.2.2). Togetherwith Lemma 2.2.3(a), (b) we may bound the above byXj:j2  Ri0ckfk1 htt 1 mini2Cj (t+xi) 1 + 2 kfk1ht 0i0  ckfk1 ht 1:The case x0 = x + hei;i 2 NC2 follows similarly. Note that for i 2 N2 onlythe second term in the above bound is nonzero as the sum is taken over anempty set (  Ri = ; for i2N2). Together with (2.20) (recall that the 1-norm andEuclidean 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 > 0;y 2 Rd+ andI1;I2  f1;:::;dg,jg(y +  Pi12I1 ei1 +  0Pi22I2 ei2)  g(y +  Pi12I1 ei1)(  0) g(y +  0Pi22I2 ei2) +g(y)j(  0) supfy02Qi2f1;:::;dg[yi;yi+ + 0]gXi12I1Xi22I2    @2@yi1@yi2 g(y0)    :Also let f : Rd+ ! R be C3. Then for all  1; 2; 3 > 0;y 2 Rd+ and I1;I2;I3  f1;:::;dg,jf(y +  1Pi12I1 ei1 +  2Pi22I2 ei2 +  3Pi32I3 ei3)( 1 2 3) f(y+  1Pi12I1 ei1 +  3Pi32I3 ei3) +f(y +  2Pi22I2 ei2)( 1 2 3) f(y+  2Pi22I2 ei2 +  3Pi32I3 ei3) +f(y +  3Pi32I3 ei3)( 1 2 3) f(y+  1Pi12I1 ei1 +  2Pi22I2 ei2) +f(y +  1Pi12I1 ei1)  f(y)j( 1 2 3) supfy02Qi2f1;:::;dg[yi;yi+ 1+ 2+ 3]gXi12I1Xi22I2Xi32I3    @3@yi1@yi2@yi3f(y0)    :Proof. This is an extension of [7], Lemma 13, using the triangle inequality tosplit the terms under consideration into sums of di erences in only one coordi-nate at a time.262.2.2 Decomposition techniquesIn 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]. Furtherbackground and motivation on the processes under consideration may be foundin [11], Section II.7.Let fP0x : x  0g denote the laws of the Feller branching process X withno immigration (equivalently, the 0-dimensional squared Bessel process) withgenerator L0f(x) =  xf00(x). Recall that the Feller branching process X canbe constructed as the weak limit of a sequence of rescaled critical Galton-Watsonbranching processes.If ! 2 C(R+;R+) let  (!) = infft> 0 : !(t) = 0g. There is a unique  - nitemeasure N0 onCex = f! 2 C(R+;R+) : !(0) = 0; (!) > 0;!(t) = 0 8t  (!)g (2.21)such that for each h> 0, if  h is a Poisson point process on Cex with intensityhN0, thenX =ZCex  h(d ) has law P0h: (2.22)Citing [11], N0 can be thought of being the time evolution of a cluster giventhat it survives for some positive length of time. The representation (2.22)decomposes X according to the ancestors at time 0.Moreover we also haveN0[  > 0] = (  ) 1 (2.23)and for t> 0 ZCex tdN0( ) = 1: (2.24)For t> 0 let P t denote the probability on Cex de ned byP t [A] = N0[A\f t > 0g]N0[ t > 0]: (2.25)Lemma 2.2.7. For all h> 0P0h[ >t] = P0h[Xt > 0] = 1  e h=(t )  ht : (2.26)Proposition 2.2.8. Let f : C(R+;R+) ! R be bounded and continuous. Thenfor any  > 0,limh#0h 1E0h[f(X)1fX >0g] =ZCexf( )1f  >0gdN0( ):The representation (2.22) leads to the following decompositions of the pro-cesses x(i)t ;i 2 NC2 that will be used below. Recall that x(i)t is the Fellerbranching immigration process with coe cients b0i  0; 0i > 0 starting at xiand with law Pixi. In particular, we can make use of the additive property ofFeller branching processes.27Lemma 2.2.9. Let 0    1.(a) We may assumex(i) = X00 +X1;where X00 is a di usion with generator A00f(x) =  0i xf00(x) + b0if0(x) startingat  xi;X1 is a di usion with generator  0i xf00(x) starting at (1  )xi  0, andX00;X1 are independent. In addition, we may assumeX1(t) =ZCex t (d ) =NtXj=1ej(t); (2.27)where  is a Poisson point process on Cex with intensity (1  )xiN0, fej;j 2 Ngis an iid sequence with common law P t , and Nt is a Poisson random variable(independent of the fejg) with mean (1  )xit 0i.(b) We also haveZ t0X1(s)ds =ZCexZ t0 sds1f t6=0g (d ) +ZCexZ t0 sds1f t=0g (d ) NtXj=1rj(t) +I1(t)and Zt0x(i)s ds =NtXj=1rj(t) +I2(t); (2.28)where rj(t) =Rt0 ej(s)ds;I2(t) = I1(t) +Rt0 X00(s)ds.(c) Let  h be a Poisson point process on Cex with intensity hiN0 (hi > 0),independent of the above processes. Set  x+h =  +  h and Xht = R  t h(d ).ThenXx+ht  x(i)t +Xh(t) =ZCex t x+h(d ) +X00(t) (2.29)is a di usion with generator A00 starting at xi +hi. In additionZCex t x+h(d ) =N0tXj=1ej(t); (2.30)where N0t is a Poisson random variable with mean ((1  )xi +hi)( 0i t) 1, suchthat fejg and (Nt;N0t) are independent.Also Zt0Xx+hs ds =N0tXj=1rj(t) +I2(t) +Ih3 (t); (2.31)where Ih3 (t) =RCexRt0  sds1f t=0g h(d ):282.2.3 Existence and representation of derivatives of thesemigroupLet A0 and Pt be as in Subsection 2.2.1. The  rst and second partial derivativesof Ptf w.r.t. xk;xl;k;l 2 NC2 will be represented in terms of the canonicalmeasure N0.Recall that by (2.17)Ptf(x) = ENC2hG INRt ;xNC2t i;where INRt =nI(j)toj2NRwith I(j)t =Rt0 Pi2Cjx(i)s ds:Notation 2.2.10. If X 2 C R+;RNC2+ ,  ; 0; ; 0 2 Cex (for the de nition ofCex see (2.21)) and k;l2NC2, letG+kt;xNR X;Z t0 sds; t  Gt;xNR  Z t0Xi2CjXisds+ 1fk2CjgZ t0 sds j2NR; Xit + 1fi=kg t i2NC2!andG+k;+lt;xNR X;Z t0 sds; t;Z t0 0sds; 0t  Gt;xNR  Z t0Xi2CjXis + 1fk2Cjg s + 1fl2Cjg 0sds j2NR; Xit + 1fi=kg t + 1fi=lg 0t i2NC2!:Note that if k 2N2 in the above we have 1fk2Cjg = 0 for j 2NR, i.e.G+kt;xNR X;Z t0 sds; t = G+kt;xNR X; 0; t ;G+k;+lt;xNR X;Z t0 sds; t;Z t0 0sds; 0t = G+k;+lt;xNR X; 0; t;Z t0 0sds; 0t (2.32)and for l 2N2G+k;+lt;xNR X;Z t0 sds; t;Z t0 0sds; 0t = G+k;+lt;xNR X;Z t0 sds; t;0; 0t : (2.33)If X 2 C R+;RNC2+ ,  ; 0 2 Cex and k;l2NC2, let G+kt;xNR (X; )  G+kt;xNR X;Z t0 sds; t  G+kt;xNR X; 0;0 29and G+k;+lt;xNR (X; ; 0) (2.34) G+k;+lt;xNR X;Z t0 sds; t;Z t0 0sds; 0t  G+k;+lt;xNR X; 0;0;Z t0 0sds; 0t  G+k;+lt;xNR X;Z t0 sds; t;0;0 +G+k;+lt;xNR X; 0;0;0;0 :Proposition 2.2.11. If f is a bounded Borel function on S0 and t > 0 thenPtf 2 C2b (S0) and for k;l2V = f1;:::;dgk(Ptf)klk1 ckfk1t2 :Moreover if f is bounded and continuous on S0, then for all k;l2NC2(Ptf)k(x) = ENC2 Z G+kt;xNR xNC2;  dN0( ) ; (2.35)(Ptf)kl(x) = ENC2 Z Z G+k;+lt;xNR xNC2; ; 0 dN0( )dN0( 0) : (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 morethan 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 su ces to consider bounded continuous f. In [7], Proposition 14one only proves the existence of (Ptf)kl(x), k;l2NC2 and its representation interms of the canonical measure as in (2.36) based on (2.35). From the methodsused 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 2 NC2.Let 0 <   t. The role of  will be explained at the end of this proof. Inthe  rst case where  0 =  t = 0, use Lemmas 2.2.6 and 2.2.4(b) to see that for30k;l2NC    G+k;+lt;xNR xNC2; ; 0    (2.37)=     G+k;+lt;xNR xNC2;Z t0 sds;0;Z  0 0sds;0! G+k;+lt;xNR xNC2; 0;0;Z  0 0sds;0! G+k;+lt;xNR xNC2;Z t0 sds;0;0;0 +G+k;+lt;xNR xNC2; 0;0;0;0     =    Gt;xNR  Z t0Xi2Cjx(i)s ds+ 1fk2CjgZ t0 sds+ 1fl2CjgZ  0 0sds j2NR;xNC2t  Gt;xNR  Z t0Xi2Cjx(i)s ds+ 1fl2CjgZ  0 0sds j2NR;xNC2t  Gt;xNR  Z t0Xi2Cjx(i)s ds+ 1fk2CjgZ t0 sds j2NR;xNC2t +Gt;xNR  Z t0Xi2Cjx(i)s ds j2NR;xNC2t      Xj1:j12 RkXj2:j22 Rlckfk1 I(j1)t  1 I(j2)t  1Z  0 0sdsZ t0 sds(compare to (49) in [7]).For k or l 2N2 we obtain via (2.32) and (2.33)    G+k;+lt;xNR xNC2; ; 0    = 0:This is consistent with (2.37) if we consider the sum over an empty set to bezero (recall that  Rk = Rk \NR and thus  Rk = ; if k 2 N2). Hence (2.37) is abound for all k;l2NC2.The other cases are proven as in [7] (for the last case use the trivial bound    G+k;+lt;xNR (xNC2; ; 0)    4 kfk1) with the same modi cations as just observed.31Combining all the cases we conclude that    G+k;+lt;xNR (xNC2; ; 0)     1f 0 = t=0g0@ Xj1:j12 RkXj2:j22 Rl I(j1)t  1 I(j2)t  1Z  0 0sdsZ t0 sds1A+ 1f 0 =0; t>0g0@ Xj:j2  Rl I(j)t  1Z  0 0sds1A+ 1f 0 >0; t=0g0@ Xj:j2  Rk I(j)t  1Z t0 sds1A+ 1f 0 >0; t>0g ckfk1  1f 0 = t=0g Z t0x(k)s ds  1 Z t0x(l)s ds  1Z  0 0sdsZ t0 sds+ 1f 0 =0; t>0g Z t0x(l)s ds  1Z  0 0sds+ 1f 0 >0; t=0g Z t0x(k)s ds  1Z t0 sds+ 1f 0 >0; t>0g ckfk1  gt;  xNC2; ; 0 The remainder of the proof works similar to the proof in [7]. Some minorchanges are necessary in the proof of continuity from below in x2 (now to bereplaced by xNC2) following (59) in [7], by considering every coordinate onits own. Also, new mixed partial derivatives appear, which can be treatedsimilarly to the ones already appearing in the proof of Proposition 14 in [7].Other necessary technical changes will reappear in later proofs where they willbe worked out in detail. They are thus omitted at this point.Remark 2.2.12. The necessity for introducing  only becomes clear in thecontext of a complete proof. For instance, the derivation of (2.36) starts byde ning Xh: , independent of x(l) and satisfyingXht = h+Z t0q2 0l XhsdB0s; (h> 0)(i.e. Xh has law P0h) so that x(l) +Xh has law Plxl+h. Therefore (2.35) togetherwith de nition (2.34) implies1h [(Ptf)k(x+hel)  (Ptf)k(x)]= 1hZ Z Z G+k;+lt;xNR  xNC2; ;Xh  1fXh =0g+ 1fXh >0g dN0( )dPNC2dP0h:Now the  rst term can be made arbitrarily small for t  xed and  # 0+. Thesecond term can be further rewritten with the help of Proposition 2.2.8 and will nally yield the representation (2.36) by  rst taking h# 0+ and then  # 0+.322.2.4 L1 bounds of certain di erentiation operatorsapplied to Ptf and equivalence of normsWe continue to work with the semigroup Pt on the state space S0 correspondingto the generator A0. Recall the de nitions of the semigroup norm jfj from(2.12) and of the associated Banach space of functions S from (2.13) in whatfollows.Proposition 2.2.13. If f is a bounded Borel function on S0 then for j 2NR    @@xjPtf(x)     ckfk1ptmaxi2Cjfpt+xig; (2.38)and      maxi2Cjfxig @2@x2j Ptf(x)      ckfk1t : (2.39)If f 2 S , then    @@xj Ptf(x)     cjfj t 2  12maxi2Cjfpt+xig  cjfj t 2  1; (2.40)and      maxi2Cjfxig @2@x2j Ptf(x)      cjfj t 2  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 Ptand @@xj commute.If f 2 S , we proceed as in [2] and write    @@xjP2tf(x)  @@xjPtf(x)    =    @@xjPt(Ptf  f)(x)    :Applying the estimate (2.38) to g = Ptf f and using the de nition of jfj weget    @@xjP2tf(x)  @@xjPtf(x)     ckgk1ptmaxi2Cjfpt+xig  cjfj t =2ptmaxi2Cjfpt+xig:This together with(2:38) ) limt!1    @@xj Ptf(x)    = 033implies that    @@xjPtf(x)     1Xk=0    @@xj (P2ktf  P2(k+1)tf) (x)     jfj 1Xk=0 2kt  2  12 cmaxi2Cjfp2kt+xig jfj t 2  12 cmaxi2Cjfpt+xig:This then immediately yields (2.40). Use (2.39) to derive (2.41) in the sameway as (2.38) was used to prove (2.40).Notation 2.2.14. If w > 0, set pj(w) = wjj! e w. For frj(t)g and fej(t)g as inLemma 2.2.9, let Rk = Rk(t) =Pkj=1 rj(t) and Sk = Sk(t) =Pkj=1 ej(t).Notation 2.2.15. If X 2 C R+;RNC2+ , Y;Y0;Z;Z0 2 C(R+;R+),  ; 0,  ; 0 2Cex and m;n;k;l2NC2, where m6= n letGm;n;+k;+lt;xNR X;Yt;Zt;Y0t;Z0t;Z t0 sds; t;Z t0 0sds; 0t  Gt;xNR  Z t0Xi2Cjnfm;ngXisds+ 1fm2CjgYt + 1fn2CjgY0t+Z t01fk2Cjg s + 1fl2Cjg 0sds j2NR; 1fi=2fm;nggXit + 1fi=mgZt + 1fi=ngZ0t+ 1fi=kg t + 1fi=lg 0t i2NC2!:The notation indicates that the one-dimensional coordinate processesRt0 Xms ds;Xmt resp. Rt0 Xns ds;Xnt will be replaced by the processes Yt;Zt resp.Y0t;Z0t (note that for m 2 N2 this only implies a change from Xmt into Zt).Additionally, we add Rt0  sds; t;Rt0  0sds and  0t as before. The termsGm;+k;+lt;xNR ;Gm;+kt;xNR;Gm;n;+lt;xNR ;Gm;nt;xNR;Gmt;xNR; Gm;+k;+lt;xNR etc. (2.42)will then be de ned in a similar way, where for instance Gmt;xNR only refersto replacing the processes Rt0 Xms ds;Xmt via Yt;Zt but doesn’t involve addingprocesses.34Proposition 2.2.16. If f is a bounded Borel function on S0, then for i2NC2    @@xiPtf(x)     ckfk1ptpt+xi; (2.43)and     xi @2@x2i Ptf(x)     cxi kfk1t(t+xi)  ckfk1t : (2.44)If f 2 S , then    @@xiPtf(x)     cjfj t 2  12pt+xi  cjfj t 2  1;and     xi @2@x2i Ptf(x)     cjfj t 2  1:Proof. The outline of the proof is the same as for [7], Proposition 17. Part ofthe proof will be presented here with its notational modi cations since somecare is needed when working in a multi-dimensional setting and the formulasbecome more involved.As in the proof of Proposition 2.2.11 we assume w.l.o.g. that f is boundedand 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 inProposition 2.2.11 and symmetry allow us to write for k 2NC2 (i.e. l = k)(Ptf)kk(x) = ENC2 Z Z G+k;+kt;xNR  xNC2; ; 0 1f t=0; 0t=0gdN0( )dN0( 0) + 2ENC2 Z Z G+k;+kt;xNR  xNC2; ; 0 1f t=0; 0t>0gdN0( )dN0( 0) +ENC2 Z Z G+k;+kt;xNR  xNC2; ; 0 1f t>0; 0t>0gdN0( )dN0( 0)  E1 + 2E2 +E3:The idea for bounding jE1j;jE2j and jE3j is similar to the one in [7]. In whatfollows we shall illustrate the necessary changes to bound jE3j.Notation 2.2.17. We have N0[ \f t > 0g] = ( t) 1P t [ ] on f t > 0g, where weused (2.25) and (2.23). Whenever we change integration w.r.t. N0 to integrationw.r.t. P t we shall denote this by ( )=.35The decomposition of Lemma 2.2.9 (cf. (2.27) and (2.28)) with  = 0 givesjE3j ( )= ct2    E Z Z nGk;+k;+kt;xNR xNC2;RNt +I2(t);SNt +X00(t); (2.45)Z t0 sds; t;Z t0 0sds; 0t  Gk;+k;+kt;xNR xNC2;RNt +I2(t);SNt +X00(t); 0;0;Z t0 0sds; 0t  Gk;+k;+kt;xNR xNC2;RNt +I2(t);SNt +X00(t);Z t0 sds; t;0;0 + Gk;+k;+kt;xNR xNC2;RNt +I2(t);SNt +X00(t); 0;0;0;0 o dP t ( )dP t ( 0)     ;where for instanceGk;+k;+kt;xNR xNC2;RNt +I2(t);SNt +X00(t);Z t0 sds; t;Z t0 0sds; 0t = Gt;xNR  Z t0Xi2CjnfkgXisds+ 1fk2Cjg (RNt +I2(t))+Z t01fk2Cjg ( s + 0s)ds j2NR; 1fi6=kgXit + 1fi=kg (SNt +X00(t))+ 1fi=kg ( t + 0t) i2NC2!by Notation 2.2.15 and the comment following it.Recall that Rk = Rk(t) = Pkj=1 rj(t) and Sk = Sk(t) = Pkj=1ej(t) withfrj(t)g and fej(t)g as in Lemma 2.2.9. In particular, fej;j 2 Ng is iid withcommon law P t and rj(t) =Rt0 ej(s)ds.We obtain (recall the de nition of Gkt;xNR from (2.42))jE3j = ct2    E Gkt;xNR xNC2;RNt+2 +I2(t);SNt+2 +X00(t)  2Gkt;xNR xNC2;RNt+1 +I2(t);SNt+1 +X00(t) + Gkt;xNR xNC2;RNt +I2(t);SNt +X00(t)      :Observe that in case k 2N2 the above notation Gkt;xNR (xNC2;RNt +I2(t);SNt +X00(t)) only indicates that x(k)t gets changed into SNt +X00(t); for k 2N236the indicated change of Rt0 x(k)s ds into RNt + I2(t) has no impact on the termunder consideration.Let w = xk=( 0kt). The independence of Nt from (fRt0 x(i)s ds;i2Cjnfkg;j 2NRg;x(NC2)nfkgt ;I2(t);X00(t);felg;frlg) yieldsjE3j = ct2   1Xn=0pn(w)EhGkt;xNR xNC2;Rn+2 +I2(t);Sn+2 +X00(t)  2Gkt;xNR xNC2;Rn+1 +I2(t);Sn+1 +X00(t) +Gkt;xNR xNC2;Rn +I2(t);Sn +X00(t) i   :Sum by parts twice and use jGj  kfk1 to bound the above byckfk1 1xkt w(3p0(w) +p1(w)) +1Xn=2wjpn 2(w)  2pn 1(w) +pn(w)j! ckfk1 1xkt wp0(w) +wp1(w) +1Xn=2pn(w)j(w n)2  njw! ckfk1 1xkt 2p1(w) +1Xn=0pn(w)(w n)2 +nw! ckfk1 1xkt:We obtain another bound on jE3j if we use the trivial bound jGj  kfk1 in(2.45). This yields jE3j  ckfk1 t 2 and sojE3j  ckfk1t(t+xk):Combine the bounds on jE1j;jE2j and jE3j to obtain (2.44).The bounds for f 2 S are obtained from the above just as in the proof ofProposition 2.2.13.Recall Convention 2.1.11, as stated in (2.14), for the de nition of M0 inwhat follows.Notation 2.2.18. Set J(j)t =  0j 2I(j)t ;j 2NR.Lemma 2.2.19. For each M  1,  2 (0;1) and d 2 N there is a c =c(M; ;d) > 0 such that if M0  M, thenjfgj  cjfjC w kgk1 + kfk1 jgj (2.46)andkfgk  c kfkC wkgk1 + kfk1 jgj  : (2.47)37Proof. Compared to the proof of [7], Lemma 18, the derivation of a bound forthe second error term E2 below becomes more involved. Again the triangle-inequality has to be used to express multi-dimensional coordinate changes viaone-dimensional ones.Let  xNR;xNC2 2 RjNRj  RjNC2j+ and de ne ~f(y) = f(y)  f(x). Then(2.15) givesjPt(fg)(x)  (fg)(x)j (2.48) jPt( ~fg)(x)j + jf(x)jjPtg(x)  g(x)j kgk1 ENC224ZRjNRj   ~f zNR;xNC2t    Yj2NRpJ(j)t zj  xj  b0jt dzj35+ kfk1 jgj t =2:The above expectation can be bounded by three terms as follows:ENC224Z    ~f zNR;xNC2t    Yj2NRpJ(j)t zj  xj  b0jt dzj35 (2.49) ENC2 Z n   ~f zNR;xNC2t  ~f zNR;xNC2    +  f zNR;xNC2  f xNR +b0NRt;xNC2   +  f xNR +b0NRt;xNC2  f xNR;xNC2     Yj2NRpJ(j)t zj  xj  b0jt dzj  E1 +E2 +E3:For all three terms we shall use the triangle inequality to sum up changes indi erent coordinates separately.The de nition of jfj ;i givesE1  Xi2NC2jfj ;iENC2     x(i)t  xi    x  =2i ^   x(i)t  xi    =2  Xi2NC2jfj ;i  ENC2    x(i)t  xi   2  =2x  =2i!^ENC2    x(i)t  xi   2  =4!:We now proceed as in the derivation of a bound on E1 in the proof of Lemma 18in [7], using Lemma 2.2.3(a) (alternatively compare with estimation of E2 be-low). We  nally obtainE1  cXi2NC2jfj ;it =22 =2  cjfjC wt =22 =2:38Similarly we haveE2  Xk2NRmini:k2  Ri jfj ;iENC2 Z     zk  (xk +b0kt)   x  =2i  ^  zk  (xk +b0kt)   =2 Yj2NRpJ(j)t zj  xj  b0jt dzj   cXk2NRmini:k2  Ri jfj ;iENC2     J(k)t    =2x  =2i ^   J(k)t    =4   cXk2NRmini:k2  Ri jfj ;i  ENC2h   J(k)t   i =2x  =2i ^ENC2h   J(k)t   i =4  as R jzj pJ(z)dz  cJ =2 for  2 (0;1). Next use Lemma 2.2.3(a) which showsthat ENC2hJ(k)ti=  0k2ENC2hI(k)ti Pl2Ck cM2(t2 + xlt). Put this in theabove bound on E2 to see that E2 can be bounded bycXk2NRmini:k2  Ri8<:jfj ;i0@0@ Xl2Ck(t2 +xlt)! =2x  =2i1A^ Xl2Ck(t2 +xlt)! =41A9=;k2NR cjfjC wXk2NR0B@0@Xl2Ckt2 +xltmaxi:k2  Rixi1A =2^ Xl2Ck t2 +t maxi:k2 Rixi ! =41CAk2NR cjfjC wt =2Xk2NR0B@0@ tmaxi:k2 Rixi + 11A =2^ 1 +maxi:k2 Rixit! =41CA cjfjC wt =22 =2:For the third term E3 we  nally haveE3  Xk2NRmini:k2  Rinjfj ;i    b0kt   x  =2i ^   b0kt   =2  o cjfjC wXk2NR  b0kt   =2 cjfjC wt =2:Put the above bounds on E1;E2 and E3 into (2.49) and then in (2.48) toconclude thatjPt(fg)(x)  (fg)(x)j   kgk1 cjfjC w+ kfk1 jgj  t =2and so by de nition of the semigroup normjfgj  cjfjC w kgk1 + kfk1 jgj :This gives (2.46) and (2.47) is then immediate.39Theorem 2.2.20. There exist 0 <c1  c2 such thatc1jfjC w  jfj  c2jfjC w: (2.50)This implies that C w = S and so S contains C1 functions with compact supportin S0.Proof. The idea of the proof was taken from the proof of Theorem 19 in [7]. Thesecond inequality in (2.50) follows immediately by settingg = 1 in Lemma 2.2.19.For the  rst inequality let x;h2 S0, t> 0 and use Propositions 2.2.13 and 2.2.16to see thatjf(x+h)  f(x)j (2.51) jPtf(x+h)  f(x+h)j + jPtf(x)  f(x)j + jPtf(x+h)  Ptf(x)j 2jfj t =2 + jPtf(x+h)  Ptf(x)j 2jfj t =2 +cjfj t 2  120@Xj2NRjhjjmaxl2Cjfpt+xlg +Xi2NC2hipt+xi1A;where we used the triangle inequality together with hl  0;l 2 Cj  NC2 forall j 2NR.By setting t = jhj and bounding  maxl2Cjfpt+xlg  1 and  pt+xi  1 by pt  1 we obtain as a  rst bound on (2.51)cjfj jhj =2: (2.52)Next only consider h2 S0 such that there exists i2NC2 and j 2 fig[  Ri suchthat hj 6= 0 and hk = 0 if k =2 fig[  Ri. (2.51) becomesjf(x+h)  f(x)j 2jfj t =2 +cjfj t 2  120@ Xj:j2  Rijhjjmaxl2Cjfpt+xlg +hipt+xi1A 2jfj t =2 +cjfj t 2  12 1pt+xijhj:In case xi > 0 set t = jhj2xi and bound  pt+xi  1 by  pxi  1 to get as asecond upper boundcjfj x  =2i jhj : (2.53)The  rst inequality in (2.50) is now immediate from (2.52) and (2.53) and theproof is complete.Note. Special care was needed when choosing h 2 S0 in the last part of theproof as it only works for those h which are to be considered in the de nition ofj jC w. Note that this was the main reason to de ne the weighted H older normsfor  Ri instead of Ri.40Remark 2.2.21. The equivalence of the two norms will prove to be crucial laterin Section 2.3, where we show the uniqueness of solutions to the martingaleproblem MP(A, ) as stated in Theorem 2.1.6. All the estimates of Section 2.2are obtained in terms of the semigroup norm. In Section 2.3 we shall furtherneed estimates on the norm of products of certain functions. At this point weshall have to rely on the result of Lemma 2.2.19 for weighted H older norms.The equivalence of norms now yields a similar result in terms of the semigroupnorm.2.2.5 Weighted H older bounds of certain di erentiationoperators applied to PtfThe xj;j 2NR derivatives are much easier.Notation 2.2.22. We shall need the following slight extension of our notationfor ENC2:ENC2 = ENC2xNC2 =  i2NC2 Pixi :Notation 2.2.23. To ease notation letT 12k  t;xNC2  (minl2Ck (t+xl) 1=2 ; k 2NR;(t+xk) 1=2; k 2NC2:Proposition 2.2.24. If f is a bounded Borel function on S0, then for all x;h2S0, j 2NR, i2Cj and arbitrary k 2V,    @@xjPtf(x+hkek)  @@xjPtf(x)     ckfk1t3=2 jhkjT 12k t;xNC2 (2.54)and     (x+hkek)i@2Ptf@x2j (x+hkek)  xi@2Ptf@x2j (x)      ckfk1t3=2 jhkjT 12k t;xNC2 :(2.55)If f 2 S , then    @@xjPtf(x+hkek)  @@xjPtf(x)     cjfj t 2  32jhkjT 12k  t;xNC2 (2.56)and     (x+hkek)i@2Ptf@x2j (x+hkek)  xi@2Ptf@x2j (x)      cjfj t 2  32jhkjT 12k t;xNC2 :(2.57)Proof. The focus will be on proving (2.55) as (2.54) is simpler. Again, it su cesto consider f bounded and continuous. For increments in xk, k 2 NR thestatement follows as in the proof of [7], Proposition 22.41Consider increments in xk, k 2NC2. We start with observing that forhk  0(xi + kihi)@2Ptf@x2j (x+hkek)  xi@2Ptf@x2j (x)=  kihiENC2xNC2+hkek"@2@x2j Gt;xNR INRt ;xNC2t #+xi ENC2xNC2+hkek ENC2xNC2 " @2@x2j Gt;xNR INRt ;xNC2t # E1 +E2;by arguing as in the proof of [7], Proposition 22. The bound on E1 is derivedas 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 fromLemma 2.2.9 with  = 12. Recall the notation Gkt;xNR from (2.42) and thede nition of Rk and Sk as in Notation 2.2.14. ThenjE2j = xi     E"@2@x2j Gkt;xNR xNC2;RN0t +I2(t) +Ih3 (t);SN0t +X00(t)  @2@x2j Gkt;xNR xNC2;RNt +I2(t);SNt +X00(t) #      xi     E"@2@x2j Gkt;xNR xNC2;RN0t +I2(t) +Ih3 (t);SN0t +X00(t)  @2@x2j Gkt;xNR xNC2;RN0t +I2(t);SN0t +X00(t) #     +xi     E"@2@x2j Gkt;xNR xNC2;RN0t +I2(t);SN0t +X00(t)  @2@x2j Gkt;xNR xNC2;RNt +I2(t);SNt +X00(t) #      E2a +E2b:E2a can be bounded as in [7], using Lemmas 2.2.4(b) and 2.2.3(b), and theindependence of xNC2 and Ih3 (t). Next turn to E2b. Recall that Sn = Sn(t) =Pnl=1 el(t), Rn = Rn(t) =Pnl=1rl(t) and pk(w) = e wwk=k!. In the  rst termof E2b we may condition on N0t as it is independent from the other randomvariables and in the second term we do the same for Nt. Thus, if w0 = w+ hk 0kt42and w = xk2 0kt, then by Lemma 2.2.4(a) and Lemma 2.2.3(b),E2b= xi    1Xn=0(pn(w0)  pn(w))E @2@x2j Gkt;xNR xNC2;Rn +I2(t);Sn +X00(t)       cxi1Xn=0     Z w0wp0n(u)du     kfk1 8>><>>:ENC2  I(j)t  1 ; k =2Cj;mini2Cjnfkg ENC2  Rt0 x(i)s ds  1  ^E  Rt0 X00(s)ds  1 ; k 2Cj9>>=>>; ckfk1 xi1Xn=0     Z w0wp0n(u)du     t 1 minl2Cj (t+xl) 1 ;where we used that X00 starts at xk2 and thus by Lemma 2.2.3(b)E" Z t0X00(s)ds  1# ct 1 t+ xk2  1 ct 1(t+xk) 1:We therefore obtain with i2CjE2b  ckfk1 xiZ w0w1Xn=0pn(u)jn uju dut 1(t+xi) 1 ckfk1  Z w0w1pudu!^ Z w0w2du!!t 1;where we used P1n=0pn(u)jn uju = 1uEjN  uj  1upEjN  uj2 = 1pu andP1n=0pn(u)jn uju  P1n=0pn(u) nu + 1 = EjNju + 1 = 2 with N being Poissondistributed with parameter u. HenceE2b  ckfk1 (w0  w) 1pw ^ 2 t 1 = ckfk1 hkt ptpxk^ 2 t 1:As 1pxk ^2pt  c 1pt+xkwe  nally getE2b  ckfk1 t 3=2hk(t+xk) 1=2:The bounds (2.56) and (2.57) can be derived from the  rst two by an argu-ment similar to the one used in the proof of Proposition 2.2.13 (alternativelyrefer to the end of the proof of Proposition 22 in [7]).In what follows recall Notation 2.2.23.43Proposition 2.2.25. If f is a bounded Borel function on S0, then for all x;h2S0, i2NC2 and arbitrary k 2V,    @@xiPtf(x+hkek)  @@xiPtf(x)     ckfk1t3=2 jhkjT 12k t;xNC2 (2.58)and    (x+hkek)i@2Ptf@x2i (x+hkek)  xi@2Ptf@x2i (x)     ckfk1t3=2 jhkjT 12k t;xNC2 :(2.59)If f 2 S , then    @@xiPtf(x+hkek)  @@xiPtf(x)     cjfj t 2  32jhkjT 12k  t;xNC2 and    (x+hkek)i@2Ptf@x2i (x+hkek)  xi@2Ptf@x2i (x)     cjfj t 2  32jhkjT 12k  t;xNC2 :Proof. Proposition 2.2.25 is an extension of Proposition 23 in [7]. The last twoinequalities follow from the  rst two by an argument similar to the one usedin the proof of Proposition 2.2.13 (alternatively refer to the end of the proof ofProposition 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 boundedand continuous.Recall the notation  G+i;+it;xNR (X; ; 0) from (2.34). Proposition 2.2.11 gives(Ptf)ii(x) =4Xn=1ENC2  nGt;xNR xNC2  ; (2.60)where 1Gt;xNR (X)  Z Z G+i;+it;xNR (X; ; 0)1f t= 0t=0gdN0( )dN0( 0); 2Gt;xNR (X)  Z Z G+i;+it;xNR (X; ; 0)1f t>0; 0t=0gdN0( )dN0( 0); 3Gt;xNR (X)  Z Z G+i;+it;xNR (X; ; 0)1f t=0; 0t>0gdN0( )dN0( 0)and 4Gt;xNR (X)  Z Z G+i;+it;xNR (X; ; 0)1f t>0; 0t>0gdN0( )dN0( 0)( )= ct2Z Z G+i;+it;xNR (X; ; 0)1f t>0; 0t>0gdP t ( )dP t ( 0):44Let us consider  rst the increments inxk;k 2NC2. Increments inxk;k 2NRwill follow at the end of this section in Lemma 2.2.30. Let hk  0 and use (2.60)to obtainj(x+hkek)i(Ptf)ii(x+hkek)  xi(Ptf)ii(x)j (2.61) 4Xn=1   xi ENC2xNC2+hkek ENC2xNC2   nGt;xNR xNC2     +hk j(Ptf)kk(x+hkek)j:The last term on the right hand side can be bounded via (2.44) as follows:hk j(Ptf)kk(x+hkek)j  hk ckfk1t(t+xk) ckfk1 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  = 12 to bound the  rst four terms in (2.61).Lemma 2.2.26. For k 2NC2 (and i2NC2) we have   xi ENC2xNC2+hkek ENC2xNC2   1Gt;xNR xNC2      ckfk1t3=2(t+xk)1=2hk:Proof. This Lemma corresponds to Lemma 24 in [7]. In [7] one considered G+i;+i( ) as a second order di erence, 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 termsof the form (t+xi) 1(t+xk) 1. Instead, we shall bound the left hand side byreasoning as for the E2-term in Proposition 22 of [7] (part of the proof can befound in this paper in the proof of Proposition 2.2.24), but with @2@x2jG( );j 2NRreplaced by  G+i;+i( );i2NC2.Lemma 2.2.27. For k 2NC2 (and i2NC2) and n = 2;3 we have   xi ENC2xNC2+hkek ENC2xNC2   nGt;xNR xNC2      ckfk1t3=2(t+xk)1=2hk: (2.62)Proof. By symmetry we only need to consider n = 2. As before let w = xk2 0kt,w0 = w+ hk 0kt, Sn =Pnl=1el(t) and Rn =Pnl=1rl(t). Let Qh be the law of Ih3 (t)as de ned after (2.31). As this random variable is independent of the othersappearing below we may condition on it and use (2.29), (2.30) and (2.31) to45concludexiENC2xNC2+hkek  2Gt;xNR xNC2  = xiE Z Z Z  Gk;+i;+it;xNR xNC2;I2(t) +z +RN0t;X00(t) +SN0t;Z t0 sds; t;Z t0 0sds;0  Gk;+i;+it;xNR xNC2;I2(t) +z +RN0t;X00(t) +SN0t; 0;0;Z t0 0sds;0  Gk;+i;+it;xNR xNC2;I2(t) +z +RN0t;X00(t) +SN0t;Z t0 sds; t;0;0 +Gk;+i;+it;xNR xNC2;I2(t) +z +RN0t;X00(t) +SN0t; 0;0;0;0   1f t>0g1f 0t=0gdN0( )dN0( 0)dQh(z) :When working under ENC2xNC2 there is no Ih3 (t) term. Hence we obtain the sameformula with z replaced by 0 and N0t replaced by Nt. The di erence of theseterms can be bounded by a di erence dealing with the change from z to 0and the change from N0t to Nt separately. For the second term we recall thatpn(u) = e uun=n! and observe that N0t is independent of the other randomvariables. Hence we may condition on its value to see that the l.h.s. of (2.62) isat mostxi    E Z Z Z   Gk;+i;+it;xNR xNC2;I2(t) +z +RN0t;X00(t) +SN0t;Z t0 sds; t;Z t0 0sds;0   Gk;+i;+it;xNR xNC2;I2(t) +RN0t;X00(t) +SN0t;Z t0 sds; t;Z t0 0sds;0   1f t>0g1f 0t=0gdN0( )dN0( 0)dQh(z)     + xi    1Xn=0(pn(w0)  pn(w))E Z Z Gk;+i;+it;xNR xNC2;I2(t) +Rn;X00(t) +Sn;Z t0 sds; t;Z t0 0sds;0  1f t>0g1f 0t=0gdN0( )dN0( 0)      Ea +Eb:The  rst term can be rewritten as the sum of two second order di erences(one in z, one in Rt0  0sds). Together with Lemma 2.2.6, Lemma 2.2.4(b) and46Lemma 2.2.3(b) we therefore obtain (terms including empty sums are againunderstood as being zero)Ea  2xickfk1Xj1:j12 RiXj2:j22 RkE2648><>: I(j1)t  1; k =2Cj1;minm2Cj1nfkg  Rt0 x(m)s ds  1 ^ Rt0 X00(s)ds  1; k 2Cj19>=>;  Z t0X00(s)ds  1#Z Z t0 0sdsdN0( 0)N0[ t > 0]ZzdQh(z) xickfk1 t 2(t+xi) 1(t+xk) 1tt 1hkt ckfk1 hkt 3=2(t+xk) 1=2:Turning to Eb observe that we have the sum of two  rst order di erences(both in Rt0  0sds). Together with the triangle inequality, Lemma 2.2.4(b) andLemma 2.2.3(b) we therefore obtainEb  cxi1Xn=0     Z w0wp0n(u)du     kfk1Xj1:j12 RiE2648><>: I(j1)t  1; k =2Cj1;minm2Cj1nfkg  Rt0 x(m)s ds  1 ^ Rt0 X00(s)ds  1; k 2Cj19>=>;375 Z Z t0 0sdsdN0( 0)N0[ t > 0] cxi1Xn=0     Z w0wp0n(u)du     kfk1 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 getEb  cxit 1=2hk(t+xk) 1=2 kfk1 t 1(t+xi) 1tt 1 ckfk1 hkt 3=2(t+xk) 1=2:The above bounds on Ea and Eb give the required result.Notation 2.2.28. LetGm;n6=mt;xNR (X;Yt;Zt;Y0t;Z0t)  (Gm;nt;xNR (X;Yt;Zt;Y0t;Z0t) if n6= mGmt;xNR (X;Yt;Zt) if n = m:Expressions such as Gm;n6=m;+k;+lt;xNR X;Yt;Zt;Y0t;Z0t;Rt0  sds; t;Rt0  0sds; 0t will be de ned similarly.47Lemma 2.2.29. For k 2NC2 (and i2NC2) we have   xi ENC2xNC2+hkek ENC2xNC2   4Gt;xNR xNC2      ckfk1t3=2(t+xk)1=2hk:Proof. LetE  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 theestimation of the l.h.s. in (2.62), thereby not only decomposing x(k) but alsox(i) (the respective parts of the decomposition of x(k) and x(i) are designatedvia upper indices k resp. i and are independent for k 6= i), we havexiENC2xNC2+hkek  4Gt;xNR xNC2  = xiE Z Z Z Gk;i6=k;+i;+it;xNR xNC2;I(k)2 (t) +z +R(k)N0(k)t;X0(k)0 (t)+S(k)N0(k)t;I(i)2 (t) +R(i)N(i)t;X0(i)0 (t) +S(i)N(i)t;Z t0 sds; t;Z t0 0sds; 0t  1f t>0g1f 0t>0gdN0( )dN0( 0)dQh(z) :Now let for k = i^Gn(z)  EhGkt;xNR xNC2;I(k)2 (t) +z +R(k)n ;X0(k)0 (t) +S(k)n i;respectively for k 6= i,^Gn z;N0(k)t   EhGk;it;xNR xNC2;I(k)2 (t) +z +R(k)N0(k)t;X0(k)0 (t) +S(k)N0(k)t;I(i)2 (t) +R(i)n ;X0(i)0 (t) +S(i)n i:Note that the expectation in the de nition of ^Gn z;N0(k)t excludes the randomvariable N0(k)t . Use w0(k) = xk2 0kt+ hk 0kt(i.e.  = 1=2) to obtain for k = ixkENC2xNC2+hkek  4Gt;xNR xNC2  (2.64)( )= cxkt21Xn=0pn w0(k) Z  ^Gn+2  2 ^Gn+1 + ^Gn (z)dQh(z);and use w(i) = xi2 0i tto obtain for k 6= ixiENC2xNC2+hkek  4Gt;xNR xNC2  (2.65)( )= cxit21Xn=0pn w(i) E Z  ^Gn+2  2 ^Gn+1 + ^Gn  z;N0(k)t  dQh(z) :48A similar argument holds for xiENC2xNC2  4Gt;xNR xNC2  . Indeed, if k = ireplace z by 0 and replace w0(k) by w(k) = xk2 0ktin (2.64). If k 6= i replace z by0 and replace N0(k)t by N(k)t in (2.65).Let us  rst investigate the case k = i. De ne^Hn(z) = ^Gn(z)  ^Gn(0)to get for E as in (2.63),E  cxkt2     1Xn=0pn w0(k) Z  ^Hn+2  2 ^Hn+1 + ^Hn (z)dQh(z)     + cxkt2     1Xn=0 pn w0(k)  pn w(k)   ^Gn+2  2 ^Gn+1 + ^Gn (0)      E1 +E2:We can bound E1 bycxkt21Xn=0   (pn 2  2pn 1 +pn) w0(k)    supn 0    Z^Hn(z)dQh(z)    ;wherepn(w)  0 if n< 0. By using qn(w) = wpn(w) andP1n=0 j(qn 2 2qn 1+qn)(w)j  2 (see [7], (109)) we obtainE1  cxkt2 1w0(k) supn 0    Z^Hn(z)dQh(z)    :Next observe that ^Hn(z) is zero for k 2N2 (recall that for k 2N2 the indicatedchange from Rt0 x(k)s ds into I(k)2 (t) +z+R(k)n resp. I(k)2 (t) +R(k)n has no impacton the terms under consideration) and is a  rst order di erence for k 2NC forwhich we obtain as usual    Z^Hn(z)dQh(z)     ckfk1 t 1(t+xk) 1ZzdQh(z) ckfk1 t 1(t+xk) 1hkt ckfk1 hkt 1=2(t+xk) 1=2:Together with w(k) = xk2 0ktand w0(k) = xk2 0kt+ hk 0ktthis givesE1  ckfk1 hkt 3=2(t+xk) 1=2:49For E2 we obtain with kGk1 kfk1 and Fubini’s theoremE2  ckfk1 xkt21Xn=0   (pn 2  2pn 1 +pn) w0(k) (2.66) (pn 2  2pn 1 +pn) w(k)     ckfk1 xkt2Z w0(k)w(k)1Xn=0   p0n 2  2p0n 1 +p0n (u)  du:As pn(u) = e u unn! we have p0n(u) =  pn(u) + pn 1(u) and thus we obtain incase 0 <u< 1 for the integrand1Xn=0   p0n 2  2p0n 1 +p0n (u)   8:For u 1 we obtain for the integrand as an upper boundp0(u) +p1(u)    31u  1    +1Xn=2pn(u)    n(n 1)(n 2)u3  3n(n 1)u2 + 3nu  1     e u(1 + 3 +u) + 1u31Xn=2pn(u)  (n u)3  3n(n u) + 2n   e u(4 +u) + 1u3 EjNu  uj3 + 3qEN2uE(Nu  u)2 + 2ENu ;where Nu is Poisson with mean u. Note that EjNu  ujm  cmum=2 for m2 Nand u 1. We also have ENu = u and EN2u = u2 +u. This yields as an upperbound for the integrand in (2.66) for u 1cu 32 + 1u3 c3u3=2 + 3p(u2 +u)c2u1 + 2u  cu 32:We thus get for E2E2  ckfk1 xkt2Z w0(k)w(k) u+ 12 0k  3=2du ckfk1 xkt2   w0(k)  w(k)    w(k) + 12 0k  3=2 ckfk1 xkt2 hkt xk +t2 0kt  3=2 ckfk1 hkt 3=2 (t+xk) 1=2:Together with the bound on E1 the assertion now follows for k = i.50Next investigate the case k 6= i. De ne^H1n z;N0(k)t  = ^Gn z;N0(k)t   ^Gn 0;N0(k)t  ;^H2n N0(k)t ;N(k)t  = ^Gn 0;N0(k)t   ^Gn 0;N(k)t  to getE  cxit2    1Xn=0pn w(i) E Z  ^H1n+2  2 ^H1n+1 + ^H1n  z;N0(k)t  dQh(z)     + cxit2    1Xn=0pn w(i) E Z  ^H2n+2  2 ^H2n+1 + ^H2n  N0(k)t ;N(k)t  dQh(z)     :Recall that the expectation in the de nition of ^Gn z;N0(k)t and thus of^H1n z;N0(k)t  excludes the random variable N0(k)t . To bound E we thus takeexpectation w.r.t. N0(k)t , too. Rewriting this yieldsE  cxit21Xn=0   (pn 2  2pn 1 +pn) w(i)     supn 0 E     Z^H1n z;N0(k)t  dQh(z)     +E     Z^H2n N0(k)t ;N(k)t  dQh(z)      and by using qn(w) = wpn(w) and P1n=0 j(qn 2  2qn 1 +qn)(w)j  2 again weobtainE  cxit2 1w(i) supn 0 E     Z^H1n z;N0(k)t  dQh(z)     +E     Z^H2n N0(k)t ;N(k)t  dQh(z)      :Next observe that ^H1n z;N0(k)t is zero for k 2N2 and is a  rst order di erencefor k 2NC for which we obtain    Z^H1n z;N0(k)t  dQh(z)     ckfk1 t 1(t+xk) 1ZzdQh(z) ckfk1 hkt 1=2(t+xk) 1=2:51The other term can be bounded as follows:   ^H2n N0(k)t ;N(k)t     1XN=0   pN w0(k)  pN w(k)        EhGk;it;xNR xNC2;I(k)2 (t) +R(k)N ;X0(k)0 (t) +S(k)N ;I(i)2 (t) +R(i)n ;X0(i)0 (t) +S(i)n i    1XN=0   pN w0(k)  pN w(k)    kGk1;where w(k) = xk2 0ktand w0(k) = xk2 0kt+ hk 0kt. As done before in the proof ofProposition 2.2.24 we use1XN=0     Z w0(k)w(k)p0N(u)du      ct 1=2hk(t+xk) 1=2to  nally get with kGk1 kfk1    Z^H2n N0(k)t ;N(k)t  dQh(z)     ct 1=2hk(t+xk) 1=2 kfk1 :Plugging our results into our estimate for E we getE  cxit2 txickfk1 hkt 1=2(t+xk) 1=2  ckfk1 hkt 3=2(t+xk) 1=2;which proves our assertion.Finally we consider the increments in xk, k 2NR.Lemma 2.2.30. If f is a bounded Borel function on S0, then for all x;h2 S0,i2NC2 and k 2NR    xi@2Ptf@x2i (x+hkek)  xi@2Ptf@x2i (x)     ckfk1t3=2 jhkj minl2Ckn(t+xl) 1=2o:Proof. Except for the necessary adaptations, already used in the proofs of thepreceding assertions, the proof proceeds analogously to Lemma 27 in [7].Continuation of the proof of Proposition 2.2.25. Use Lemmas 2.2.26, 2.2.27and 2.2.29 in (2.61) together with the calculation following (2.61) to obtain thebound for increments in xk;k 2 NC2. Lemma 2.2.30 gives the correspondingbound for increments in xk;k 2NR which completes the proof of (2.59).522.3 Proof of UniquenessAs in Section 3, [7], it is relatively straightforward to use the results from theprevious sections on the semigroup Pt to prove bounds on the resolvent R ofPt.We shall then use these bounds to complete the proof of uniqueness of solu-tions to the martingale problem MP(A, ) satisfying Hypothesis 2.1.1 and 2.1.2,where  is a probability onS =8<:x2 Rd+ :Yj2R0@Xi2Cjxi +xj1A> 09=;(recall (2.3) and Lemma 2.1.5) andAf(x) =Xj2R j(x)0@Xi2Cjxi1Axjfjj(x) +Xj =2R j(x)xjfjj(x) +Xj2Vbj(x)fj(x):(2.67)The proof of uniqueness is identical to the one in [7] except for minor changessuch as the replacement of xcj by Pi2Cj xi at the appropriate places. Note inparticular the change in the de nition of the state space S.In what follows we shall give a sketch of the proofs and indicate wherestatements have to be modi ed. For explicit calculations the reader is referredto [7], Sections 3 and 4.Notation 2.3.1. For i2NC2 let yi = fyjgj2  Ri ;yi ; yi ei =Xj2 Riyjej +yiei and  Ri = Rj Rij  R+; (2.68)where we understand this to be  yi = (yi) in case i 2 N2, i.e.  Ri = ;. Forf 2 C2b (S0) let@f@ xi =  @@xj f j2  Ri; @@xif!;    @f@ xi    = Xj2 Ri    @@xjf    +    @@xif    (2.69)and         @f@ xi        1= sup     @f@ xi (x)    : x2 S0 ; (2.70)where S0 = fx2 Rd : xi  0 for all i2NC2g as de ned in (2.9). Also introduce if =0@(xi @2@x2j f)j2  Ri;xi @2@x2i f1A:De ne j ifj and k ifk1 similarly to (2.69) and (2.70).53With the help of these notations A0 (see (2.6)) can be rewritten toA0f(x) =Xj2Vb0jfj(x) +Xj2NR 0j0@Xi2Cjxi1Afjj(x) + Xi2NC2 0i xifii(x) (2.71)=Xi2NC2Db0 i; @f@ xi(x)E+Xi2NC2D 0 i; if(x)E;where h ; i denotes the standard scalar product in Rk;k 2 N. To prevent over-counting in case  Ri1 \  Ri2 6= ; for i1 6= i2, i1;i2 2NC (see also de nition (2.68))the vector b0i was replaced by b0 i in the above formula, where b0 i has certaincoordinates set to zero so that the above equality holds. The same applies tothe vector  0 i. The details are left to the interested reader.Theorem 2.3.2. There is a constant c such that for all f 2 C w(S0);  1 andk 2NC2,(a)        @R f@ xk        1+ jj kR fjj1  c   =2jfjC w:(b)    @R f@ xk    C w+ j kR fjC w cjfjC w:Note. This result is slightly weaker than the corresponding Theorem 34 in [7]as jfj ;k is replaced by jfjC w in (a).Proof. Firstly we obtain a result similar to Proposition 30 in [7]. This is an easyconsequence of Proposition 2.2.13 and Proposition 2.2.16, using the equivalenceof norms shown in Theorem 2.2.20 and states that there is a constant c suchthat(a) For all f 2 C w(S0);t> 0;x2 S0, and i2NC2,    @Ptf@ xi (x)     cjfjC wt =2 1=2(t+xi) 1=2  cjfjC wt =2 1; (2.72)andk iPtfk1 cjfjC wt =2 1: (2.73)(b) For all f bounded and Borel on S0 and all i2NC2,        @Ptf@ xi        1 ckfk1 t 1:Note in particular that Theorem 2.2.20 gaveC w = S and that every functionin C w(S0) is by de nition bounded.Secondly, an easy consequence of Propositions 2.2.24, 2.2.25 and the triangleinequality, using the equivalence of norms shown in Theorem 2.2.20 and theequivalence of the maximum norm and Euclidean norm of  nite dimensional54vectors, is a result similar to Proposition 32, [7]: There is a constant c such thatfor all f 2 C w(S0);i;k 2NC2 and  hi 2  Ri,(a)    @Ptf@ xk x+  hi ei  @Ptf@ xk (x)     cjfjC wt 3=2+ =2(t+xi) 1=2   hi  ; (2.74)(b)   k(Ptf) x+  hi ei   k(Ptf)(x)   cjfjC wt 3=2+ =2(t+xi) 1=2   hi  : (2.75)Finally recall that R f(x) = R10 e  tPtf(x)dt is the resolvent associatedwith Pt. Now the remainder of the proof works as in the proof of Theorem 34in [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  xed value ~t > 0 and (2.74) resp. (2.75) over thetime interval from ~t to in nity. Appropriate choices for ~t now yield the requiredbounds. Here the choices of ~t are in fact easier due to the replacement of j j ;iin [7] by j jC w.Proof of Theorem 2.1.6. The existence of a solution to the martingale problemfor MP(A; ) follows by standard methods (a result of Skorokhod yields ex-istence of approximating solutions, then use a tightness-argument), e.g. seethe proof of Theorem 1.1 in [1]. Note in particular that Lemma 2.1.5 ensuresthat solutions remain in S  Rd+. The uniform boundedness in M of the termEhPi   XM;iT   ithat appears in the proof of Theorem 1.1 in [1] can easily be re-placed by the uniform boundedness in M of EhPi2V (XM;iT )2ivia a Gronwall-type argument.At the end of this section we shall reduce the proof of uniqueness to thefollowing theorem. The theorem investigates uniqueness of a perturbation of theoperator A0 as de ned in (2.6) (also refer to (2.71)) with coe cients satisfying(2.7) and (2.8). A0 is the generator of a unique di usion on S x0 given by(2.9) with semigroup Pt and resolvent R given by (2.11). For the de nition ofM0 refer to (2.14).In what follows x0 2 S will be arbitrarily  xed.Theorem 2.3.3. Assume that~Af(x) = Xj2NR~ j(x)0@Xi2Cjxi1Afjj(x) (2.76)+Xj2NC2~ j(x)xjfjj(x) +Xj2V~bj(x)fj(x); x2 S x0 ;where ~bk : S x0 ! R and ~ k : S x0 ! (0;1),~ =dXk=1jj~ kjjC w+      ~bk      C w<1:55Let~ 0 =dXk=1    ~ k   0k    1 +      ~bk  b0k      1;where b0k; 0k;k 2V satisfy (2.7). Let Bf = ( ~A A0)f.(a) There exists  1 =  1(M0) > 0 and  1 =  1(M0;~ )  0 such thatif ~ 0   1 and    1 then BR : C w ! C w is a bounded operator withkBR k 1=2.(b) If we assume additionally that ~ k and ~bk are H older continuous of index 2 (0;1), constant outside a compact set and ~bkjfxk=0g  0 for all k 2VnNR,then the martingale problem MP( ~A; ) has a unique solution for each probability on S x0 .Proof. Let ~R be the associated resolvent operator of the perturbation operator~A. Using the de nition B = ~A A0 and recalling (2.71) we get for f 2 C w thatkBR fkC w Xi2NC2          ~b(x)  b0 i;@R f@ xi(x)         C w+Xi2NC2          ~ (x)   0 i; iR f(x)         C w:Using (2.46) (recall in particular the discussion on the reasons for using twodi erent norms from Remark 2.2.21) we obtain for instance for arbitraryi2NCand j 2  Ri     ~bj(x)  b0j @R f@xj (x)    C w c       ~bj(x)  b0j      C w        @R f@xj (x)        1+      ~bj(x)  b0j      1    @R f@xj (x)       ch ~ +M0    =2jfjC w + ~ 0jfjC wiby Theorem 2.3.2, (2.50) and the assumptions of this theorem. By arguingsimilarly for the other terms we get indeed kBR f kC w 12 kf kC w for  bigenough thus  nishing the proof of part (a).For part (b) we proceed as in the proof of [7], Theorem 37. The proof ofTheorem 37 in [7] involves the proof of Lemma 38 in [7], where one shows thatfor f 2 C w~R f = R f + ~R BR f: (2.77)Note that the proof of Lemma 38 relies amongst others on an estimate, derivedin Corollary 33 of [7], which we now obtain for free in Proposition 2.2.11 as wetreated all vertices in one step only.The proof of Theorem 37 now concludes as follows. Iteration of (2.77) yields~R f(x) =1Xn=0R ((BR )nf)(x):56Using kBR kC w 1=2 from part (a) and kfk1 kfkC w we get kR ((BR )nf)k1 k(BR )nfk1 k(BR )nfkC w 2 n kfkC w :Thus the series converges uniformly and the error term approaches zero. Theuniqueness of MP( ~A; ) now follows from the uniqueness of its resolvents ~R .Continuation of the proof of Theorem 2.1.6. Recall \Step 1: Reduction ofthe problem", in Subsection 2.1.5. The remainder of the proof of uniquenessof MP(A; x0) works analogously to [7] (compare the proof of Theorem 4 onpp. 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. theargument in the proof of Theorem 1.2 of [4]), which basically states that it isenough if for each x0 2 S the martingale problem MP( ~A; x0) has a uniquesolution, where bi = ~bi and  i = ~ i agree on some neighbourhood of x0. Bycomparing the de nition of A (see (2.67)) and ~A (see (2.76)) one chooses~bk(x) = bk(x) for all k 2V;~ j(x) = xj j(x) for j 2NR;~ j(x) =0@Xi2Cjxi1A j(x) for j 2RnNR~ j(x) =  j(x) for j =2R:By settingb0k  ~bk(x0) and  0k  ~ k(x0)and choosing ~bk and ~ k in appropriate ways, the assumptions of Theorem 2.3.3(a),(b) will be satis ed in case b0k  0 for all k 2N2 (and hence by Hypothesis 2.1.2for allk 2NC2). In particular the boundedness and continuity of the coe cientsof ~A will allow us to choose ~ 0 arbitrarily small. In case there exists k 2 N2such that b0k < 0 a Girsanov argument as in the proof of Theorem 1.2 of [4]allows the reduction of the latter case to the former case.57Bibliography[1] Athreya, S.R. and Barlow, M.T. and Bass, R.F. and Perkins,E.A. Degenerate stochastic di erential equations and super-Markov chains.Probab. Theory Related Fields (2002) 123, 484{520. MR1921011[2] Athreya, S.R. and Bass, R.F. and Perkins, E.A. H older norm esti-mates for elliptic operators on  nite and in nite-dimensional spaces. Trans.Amer. Math. Soc. (2005) 357, 5001{5029 (electronic). MR2165395[3] Bass, R.F. Di usions and Elliptic Operators. Springer, New York, 1998.MR1483890[4] Bass, R.F. and Perkins, E.A. Degenerate stochastic di erential equa-tions with H older continuous coe cients and super-Markov chains. Trans.Amer. Math. Soc. (2003) 355, 373{405 (electronic). MR1928092[5] Bass, R.F. and Perkins, E.A. Degenerate stochastic di erential equa-tions arising from catalytic branching networks. Electron. J. Probab. (2008)13, 1808{1885. MR2448130[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. Poincar e 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 di usions. 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] Hofbauer, J. and Sigmund, K. The Theory of Evolution and DynamicalSystems. London Math. Soc. Stud. Texts, vol. 7, Cambridge Univ. Press,Cambridge, 1988. MR1071180[10] Mytnik, L. Uniqueness for a mutually catalytic branching model. Probab.Theory Related Fields (1998) 112, 245{253. MR1653845[11] 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[12] Rogers, L.C.G. and Williams, D. Di usions, Markov Processes, andMartingales, vol. 2, Reprint of the second (1994) edition. Cambridge Math-ematical Univ. Press, Cambridge, 2000. MR1780932[13] Stroock, D.W. and Varadhan, S.R.S. Multidimensional Di usion Pro-cesses. Grundlehren Math. Wiss., vol. 233, Springer, Berlin-New York,1979. MR53249858Chapter 3Long-term Behaviour of aCyclic Catalytic BranchingSystem13.1 Introduction3.1.1 BasicsIn this paper we investigate the long-term behaviour of the following system ofstochastic di erential equations (SDEs) for d 2:dXit =q2 iXitXi+1t dBit +dXj=1Xjtqjidt; i2 f1;:::;dg; (3.1)where Xd+1t  X1t . We shall assume the  i and qji;i 6= j to be given positiveconstants and the Xi0  0;i 2 f1;:::;dg to be given initial conditions. (qji) isa Q-matrix modelling mutations from type j to type i.This system involves both cyclic catalytic branching and mutation betweentypes. 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 solutionsshall be shown by standard methods. To show weak uniqueness we shall employthe results of Dawson and Perkins [3] once we show that a solution does not hit0 2 Rd in  nite time.The given system of SDEs can be understood as a stochastic analogue to asystem of ODEs for the concentrations yj;j = 1;:::;d of a type Tj. Then yj=_yjcorresponds to the rate of growth of type Tj and one obtains the following ODEs(see Hofbauer and Sigmund [6]): for independent replication _yj = bjyj, auto-catalytic replication _yj =  jy2j and catalytic replication _yj =  j Pi2Cj yi yj.In the cyclic catalytic case type Tj+1 catalyzes the replication of type j, i.e. thegrowth 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 betweendi erent types. It was  rstly introduced by Eigen and Schuster (see Eigen andSchuster [4]).1A version of this chapter will be submitted for publication. Kliem, S.M. (2009) Long-termBehaviour of a Cyclic Catalytic Branching System.59The 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 branchingrate in the stochastic setting, i.e. type j branches at a rate proportional to themass of type j + 1 at time t.Results on weak uniqueness for catalytic branching networks can be foundfor instance in [3] and Kliem [9]. The former proved weak uniqueness for cat-alytic replication under the restriction to networks with at most one catalystper reactant, which includes the hypercyclic case. The latter removed this re-striction. Both papers allow more general di usion- and drift- coe cients undersome H older-continuity conditions. These conditions were weakened in Bass andPerkins [1] to continuity only.Our main interest shall be the long-time behaviour of the above system. Inparticular, we shall investigate survival and coexistence of types. Such questionsnaturally arise in biological competition models. For instance, Fleischmannand Xiong [5] investigated a cyclically catalytic super-Brownian motion. Theyshowed global segregation (noncoexistence) of neighbouring types in the limitand other results on the  nite time survival-extinction but they were not ableto 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 convergesto zero but does not hit zero in  nite time. To further analyze the relativebehaviour of types while they approach zero, we turned our attention to thenormalized processes Y it  Xit=Pj Xjt - note that Xit=Xjt = Y it =Yjt - andshowed weak convergence to a unique stationary distribution that does notcharge the set where at least one of the coordinates is zero.3.1.2 Main results and outline of the paperAs a  rst step we shall show existence and nonnegativity of solutions Xit;i 2f1;:::;dg to the above SDE by standard methods in Subsection 3.2.1. As anext step we shall prove in Subsection 3.2.2 that the sum of all coordinates,i.e. St  Pdi=1Xit, converges to zero but does not hit zero in  nite time a.s.We then establish the weak uniqueness of the system by Theorem 4 of [3] orTheorem 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 = Xit=St to get some insight on the relative behaviourof types. Existence of solutions follows again by standard methods and the weakuniqueness of solutions in [0;1]d follows by establishing a connection betweenthe system at hand and the original system of SDEs. In Subsection 3.2.4 weshow that any stationary distribution for Yt does not charge the set where atleast one of the coordinate processes becomes extinct. We shall use this result inSubsection 3.2.5 to prove weak convergence to a unique stationary distributionby adapting the proof of Theorem 2.3 of Dawson, Greven, den Hollander, Sunand Swart [2] to our setup.Finally, in Subsection 3.2.7 we shall give a complete analysis of the cased = 2 by using methods of speed and scale.603.2 Main Results3.2.1 Existence and nonnegativityLet ( ;F;(F)t;P) be a  ltered probability space that satis es the usual condi-tions (cf. Rogers and Williams [10], Introduction to Chapter IV). Consider thefollowing system of SDEs for d 2:dXit =q2 iXitXi+1t dBit +dXj=1Xjtqjidt; i2 f1;:::;dg; (3.2)where Xd+1t  X1t . We shall assume the  i and qji;i 6= j to be given strictlypositive constants and the Xi0  0;i2 f1;:::;dg to be given initial conditions.As the qji model mutations from type j to type i we imposeqii =  Xj:j6=iqij ()Xjqij = 0: (3.3)Letqmax = max1 i;j djqijj; qmin = min1 i;j djqijj> 0 and  max = max1 i d i > 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  rst quadrant (herewe replaced the terms under the square root with their absolute values to beable to consider solutions on all of Rd). For this purpose we shall  rst show thatthe local time of the coordinate processes at zero is zero.Corollary 3.2.2. Let i2 f1;:::;dg be arbitrarily  xed. Then the local time l0tat zero of the process Xit is zero.Proof. The proof proceeds along the lines of standard techniques for local times.Let i 2 f1;:::;dg be arbitrarily  xed. By [10], IV.(45.3) (\occupation densityformula") we have for ’(x) = 1[0; ](x),  > 0,Z t0’(Xis)d<Xi >s=Z t01f0 Xis  g2 i  XisXi+1s   ds =Z  0latda =ZRlat’(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 Z  0latda lim #0+1 Z t01f0<Xis  g2 i   Xi+1s   ds = 0;the last by dominated convergence, proving the assumption.61Notation 3.2.3. In what follows we shall denote the martingale part of Xit byMit  Z t0q2 iXisXi+1s dBis; i2 f1;:::;dg:Lemma 3.2.4. The processes Xit, i 2 f1;:::;dg are nonnegative if we startat Xi0  0;8i 2 f1;:::;dg. We also obtain that the Mit;i 2 f1;:::;dg aremartingales.Proof. To the purpose of proving this Lemma we shall use that Xit  =Z t0 1fXis 0gdXis + 12l0t =Z t0 1fXis 0gdXis; (3.4)the last by Corollary 3.2.2Before we continue, observe that Mit  Rt0q2 iXisXi+1s dBis is a martingale.Indeed, Mi is a continuous local martingale and so it su ces to show thatE <Mi >t <1 for all t> 0.To show this de nes a sequence of stopping timesTn  infft 0 : maxdi=1 jXitj  ng. As Cauchy Schwarz’ inequality yieldsE <Mi >t^Tn  CZ t0qE (Xis^Tn)2 E (Xi+1s^Tn)2 ds<1; (3.5)Mi ^Tn is a continuous martingale. In particular, Eh Mit^Tn 2i= E <Mi ^Tn >t and we obtain thatE (Xit^Tn)2  C8><>:(Xi0)2 + Eh Mit^Tn 2i+ E2640@Z t^Tn0dXj=1Xjsqjids1A23759>=>;(3:5) C 1 +Z t0dmaxi=1 E (Xis^Tn)2 ds :Hence Gronwall’s lemma gives maxdi=1 E (Xit^Tn)2  CeCt. As Tn ! 1 forn ! 1 we can now apply the monotone convergence theorem in (3.5) to getE <Mi >t <1 for all t> 0. Thus Mi is indeed a continuous martingale.Taking expectations in (3.4) this impliesEh Xit  i E24Z t0dXj=11Xjs 0( Xjs)qjids35:Sum both sides over i and use (3.3) to obtain 0  Pi Eh Xit  i 0. ThusXit  0 a.s. for all i2 f1;:::;dg and t 0.623.2.2 The overall sum and uniquenessIn what follows we shall investigate the behaviour of our system for t! 1. Weshall show that the sum of all coordinates converges to zero but does not hitzero at a  nite time. At the end of this Subsection we shall use this result toestablish the weak uniqueness of solutions to (3.2).Notation 3.2.5. Let St  Pdi=1 Xit.Corollary 3.2.6. St converges a.s. for t! 1.Proof. First note that by Lemma 3.2.4, St is a nonnegative process. Using (3.2)we obtain for StdSt =dXi=1q2 iXitXi+1t dBit +dXi=1dXj=1Xjtqjidt (3:3)=dXi=1q2 iXitXi+1t dBit:Using that the Mit are martingales as shown in Lemma 3.2.4, we obtain that Stis 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=Z t0dXi=12 iXisXi+1s ds 2 maxZ t0S2sds:Next we shall use a time-change to be able to take advantage of this inequality.LetIs  dXi=12 iXisXi+1s and Ct  Z t0Is2 maxS2s ds for t  ;where   infft 0 : 9 > 0 such that Is = 0 8s2 [t;t+ ]g.Note that if St0 = 0 for some t0 > 0, then St = 0 for all t  t0 by theoptional sampling theorem as St is a continuous nonnegative martingale. AsIs  2 maxS2s we therefore have Ss > 0 for all s< .Also note that Ct < 1 for all t <  as 0  Is and Is  2 maxS2s whichyields0  Is2 maxS2s 1 for all 0  s< : (3.6)In particular the de nition of  implies that Ct is a continuous strictly increas-ing function de ning a homeomorphism between [0; ] and [0; ] for  < 1respectively [0;1) if  = 1 (let us also denote this by [0; ]), where   C .Let D : [0; ] ! [0; ] be the continuous strictly increasing inverse to Ct.Under this time-change we now obtain for St,Zt  SDt;t  ;63where<Z >t=<S >Dt=Z Dt0Isds =Z t0IDs 2 maxS2DsIDs ds = 2 maxZ t0Z2sds;i.e.d<Z >t= 2 maxZ2tdt; 8t  :Thus Z is a geometric Brownian motion (see for instance Karatzas and Shreve[8], Exercise 5.5.31). In particular Zt > 0 8t   if  < 1 respectively Zt > 08t< if  = 1 follows, given that Z0 = S0 > 0. We therefore obtain8<:SDt > 0 8t  )St > 0 8t  ; if  <1 respectivelySDt > 0 8t< )St > 0 8t< ; if  = 1 (3:6))  = 1 : (3.7)In what follows let  = infft  0 : St = 0g. To  nish our proof it remainsto show that  = 1 a.s. By (3.7) it can easily be seen that we have    and by continuity of St the in mum in the de nition of  is attained given < 1. Finally note that ST = 0 implies St = 0 8t  T as St is a nonnegativemartingale. Indeed, T is a stopping time and so this follows from the optionalsampling theorem (see for instance [7], II.2.13).Let us suppose by contradiction that  <1. We are left with two cases.1. case.  (!) =  (!) <1.This yields a contradiction as S = S = 0 by de nition of  but S > 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 = Rt0q2 iXisXi+1s dBis that   Mi t  Mit  !0 for  t t! 1 a.s.Proof. Mit is a continuous martingale with d < Mi >t= 2 iXitXi+1t dt andMi0 = 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 probabilityspace. Now0  <Mi > t  <Mi >t=Z  tt2 iXisXi+1s ds Z  ttdXi=12 iXisXi+1s ds =<S > t  <S >t! 0 for  t t! 1as St converges a.s. and thus limt!1 <S >t<1 a.s. Hence we obtain  Mi t  Mit  =  B<Mi> t  B<Mi>t  ! 0 for  t t! 1 a.s. (3.8)as required.64Continuation of the proof of Lemma 3.2.7.2. case.  (!) < (!) <1.Suppose  < <1. This implies that there exists  > 0 such that Ij[ ; + ) = 0and Sj[ ; + ) > 0. By de nition of It, St and the continuity of the processesunder consideration this requires that there exists  =  (!) > 0 and i = i(!) 2f1;:::;dg such thatXitj[ ; + ) = 0 and Xi+1t j[ ; + ) > : (3.9)Next consider increments in the ith coordinate to see that for all 0  t< 0 = Xi +t  Xi = Mi +t  Mi  +Z  +t dXj=1;j6=i Xjsqji  Xisqij ds  Mi +t  Mi   (d 1) maxi6=jqijZ  +t Xisds+Z  +t Xi+1s qi+1;ids  Mi +t  Mi  +tqi+1;i :As Ij[ ; + ) = 0 and d < S >s= Isds we get d<S>sds   [ ; + )= 0. Similarlyto the proof of Corollary 3.2.8 this implies d<Mi>sds   [ ; + )= 0. RewritingMit = B<Mi>t as done in the Corollary we get Mij[ ; + )  const. and thusMi +t Mi = 0. Plugging this in the above inequality we obtain 0  tqi+1;i >0, a contradiction.Taking both cases together we have shown that  = 1 a.s., i.e. infft  0 :St = 0g = 1 a.s.Remark 3.2.9. We have actually shown more. As  <1 was not used in theproof of case 2 of Lemma 3.2.7, we have moreover shown that  =  = 1.Also observe that the proof of Lemma 3.2.7 only uses qij  0;j 6= i andqi+1;i > 0 for all i2 f1;:::;dg and so in particular holds for nearest neighbourrandom 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 ! 1. As the processes Xit  St, i 2 f1;:::;dg 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 S1 was shown in Corollary 3.2.6. Suppose by contradictionthat S1 = S1(!) > 0 for ! element of a set of positive measure. Choose0 <  < S1 arbitrarily small. By (3.8) there exists T = T( ;!)  0 such thatfor all  t>t T  Mi t  Mit    ;i2 f1;:::;dg; and jSt  S1j   : (3.10)65Now observe that for  t>t  Xi t  Xit      Mi t  Mit  +Z  ttdXj=1;j6=i Xjsqji  Xisqij ds    Mi t  Mit  +qminZ  ttdXj=1;j6=iXjsds (d 1)qmaxZ  ttXisds    Mi t  Mit  +qminZ  tt(Ss  Xis)ds (d 1)qmaxZ  ttXisds=    Mi t  Mit  +qminZ  ttSsds [(d 1)qmax +qmin]Z  ttXisds:Hence we have for  t>t T  Xi t  Xit     +qmin( t t) (S1   )  [(d 1)qmax +qmin] ( t t) sups2[t; t]Xis:This is equivalent tosups2[t; t]Xis    +qmin( t t) (S1   )    Xi t  Xit  [(d 1)qmax +qmin] ( t t) : (3.11)In what follows  x i2 f1;:::;dg arbitrary and use the following notation0  I  lim inft!1 Xit  lim supt!1Xit  S  S1 <1:We shall prove that S1 > 0 implies I > 0, which will provide us with thedesired contradiction in the end.1. case: S1 > 0 and I <S.As Xit is a continuous process, there exists an increasing sequence ftngn2N,tn ! 1 for n! 1, independent of the choice of  , such thatXitn = I + (S I);n2 N;where 0 <  < 1  xed but arbitrarily small (see Figure 3.1). Without loss ofgenerality letsups2[t2n 1;t2n]Xis = Xit2n 1 = Xit2n = I + (S I);n2 N:Applying (3.11) with t = t2n 1; t = t2n (choose n2 N such that t T) givesI + (S I)    +qmin(t2n  t2n 1) (S1   )[(d 1)qmax +qmin] (t2n  t2n 1): (3.12)66PSfrag replacementsIt1 t2 t3 t4SI + (S I)Figure 3.1: The de nition of t2n 1 and t2n.As the choice of n may depend on  we have to  nd an estimate for the term    t2n t2n 1   before considering  ! 0+. First observe that for 0  u < v to bespeci ed later  Xiv  Xiu     Miv  Miu  +Z vudXj=1  Xjsqji  ds   Miv  Miu  + (v u)qmax sup0 s<1Ssand thus as St converges a.s.v u   Xiv  Xiu     Miv  Miu  qmax sup0 s<1Ss :Now (3.8) yields that for all  > 0 small there exists T0 = T0( ;!) (note thatT0 is independent of the choice of  ) such that for all v>u T0v u   Xiv  Xiu    qmax sup0 s<1Ss: (3.13)Moreover, choose T0 such that for all t2n 1  T0 we haveinfs2[t2n 1;t2n]Xis <I +  2(S I);which is possible by de nition of I;S and  . We get9u;v 2 [t2n 1;t2n] s.t. jXiv  Xiuj>  2(S I) (3:13)) v u  2(S I)   qmax sup0 s<1Ss:By choosing  su ciently small we obtain that there exists T0 = T0( ;!; ) suchthat jt2n  t2n 1j  v u> const(!; ) > 0 for all t2n 1  T0.Let us return to (3.12). Letting  ! 0+ yieldsI + (S I)  qminS1[(d 1)qmax +qmin]:67Now let  ! 0+ to  nally obtain I > 0.2. case: S1 > 0 and I = S.As I = S is equivalent to Xit being convergent, we can choose T as de ned in(3.10) to additionally satisfyjXis  Ij   ; 8s T:Now (3.11) gives with  t>t = T,  t arbitraryI +  sups2[T; t]Xis    +qmin( t T) (S1   )    Xi t  XiT  [(d 1)qmax +qmin] ( t T)   +qmin( t T) (S1   )  2 [(d 1)qmax +qmin] ( t T):Taking  ! 0+ yields I > 0 once more.Taking both cases together we have shown lim inft!1 Xit > 0 for alli2 f1;:::;dg,given S1 > 0, as i was chosen arbitrary in the calculations. As the Xit are con-tinuous processes this gives a contradiction tolimt!1 <S >t= R10 Pdi=1 2 iXisXi+1s ds< 1 a.s., as this requires that the limitinferior of the nonnegative integrand becomes zero for t ! 1. S1 > 0 thusgives a contradiction, i.e. we have shown S1 = 0 a.s.Lemma 3.2.11. The solution to the given SDE (3.2) is unique in law for allX0 2 Rd s.t. Xi0  0;i2 f1;:::;dg and Qdi=1 Xi0 +Xi+10  > 0.Proof. We shall apply Theorem 4 in [3] (or alternatively Theorem 1.6 of [9]). TheHypotheses under which this Theorem is stated hold, except for condition (3) inHypothesis 2. Here we have that bi(x) > 0 if xi = 0, except for the case wherex = 0 2 Rd (here bi(0) = 0). Thus we have to consider this case separately.By modifying the drift coe cients in a small open neighbourhood of 0, sayin B (0) with  > 0 arbitrarily small, we can achieve that the drift coe cientssatisfy all conditions such as H older continuity on compact subsets of Rd+ andjbi(x)j  c(1 +jxj) and additionally are strictly positive on B (0). For instance,let~bi(x)  bi(x) + (  jxj) _ 0:By solving the system with these modi ed coe cients we obtain existence anduniqueness of the modi ed solution. Finally take > 0 so small that the startingpoint x0 =2B (0). As the di usion and drift coe cients of the modi ed SDE areidentical 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 showedthat every solution to (3.2) does not hit 0 in  nite time (see Lemma 3.2.7) weobtain the assertion.683.2.3 The normalized processesCorollary 3.2.12. The corresponding SDEs for the normalized processesY it  XitSt (3.14)with 0  Y it  1 are as follows.dYit =  Y itXj6=iq2 jY jt Yj+1t dBjt + (1  Y it )q2 iYit Y i+1t dBit (3.15)+Y itXj6=i2 jY jt Yj+1t dt+ (Y it  1)2 iYit Yi+1t dt+dXj=1Yjt qjidt:Idea of the proof. The proof is an easy application of It^o’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 Xit and ask for existence and uniquenessof solutions. As we have not shown nonnegativity of the Y it yet, we replace theterms under the square root with their absolute values.Proposition 3.2.13. The SDE (3.15) started at Y0 2 [0;1]dn@[0;1]d withPi Yi0 = 1 has a unique in law solution. Moreover the solution satis es Yt 2[0;1]d and Pi Yit = 1 for all t 0 a.s.Proof. Existence follows immediately from the existence of solutionsXt to (3.2).Indeed, let X0  Y0, then Yit  Xit=Pj Xjt solves (3.15) by Corollary 3.2.12with Yi0 = Xi0=Pj Xj0 = Yi0=Pj Yj0 = Y i0 .As in Corollary 3.2.2 one can show that the local times at zero of the pro-cesses Yit are zero. The nonnegativity of the processes Y it can be shown asfollows.Lemma 3.2.14. The processes Y it , i2 f1;:::;dg are nonnegative.Proof. We have for all 1  i d andt TM = TM(!)  infft 0 : maxi jY it j  Mg, M 2 N  xed, Yit  =Z t0 1fY is 0gdYis= mart. +Z t0 1fY is 0g0@Y is Xj6=i2 jY js Yj+1s + (Y is  1)2 iYisYi+1s+dXj=1Y js qji1Ads:69We can bound the above bymart. +Z t0C(M) Y is   1fY is 0gXj6=iYjs qjids mart. +Z t0C(M) Yis  +CXj6=i Yjs  ds:Taking expectations on both sides and summing over all coordinates we get forall t 0,XiEh Y it^TM  i Z t0C(M)XiEh Y is^TM  i+CXiXj6=iE  Y js^TM   ds C(M)Z t0XiEh Y is^TM  ids:An application of Gronwall’s lemma yields Eh Y it^TM  i= 0. Take M ! 1and use Fatou’s lemma to obtain the claim.As we shall show in the following Corollary Pi Yi0 = 1 implies PiY it = 1for all t 0.Corollary 3.2.15. Every solution (Y it )1 i d to (3.15) with Pi Yi0 = 1 has tosatisfy Pi Yit = 1 for all t 0 a.s.Proof. We have from (3.15) and (3.3) thatd dXi=1Y it!= 1 dXi=1Y it! dXj=1 q2 jYjt Y j+1t dBjt  2 jY jt Yj+1t dt = 1 dXi=1Y it!(dNt  d<N >t);where Nt  Rt0dPj=1q2 jY js Yj+1s dBjs and Nt^TM is a martingale starting at 0.Setting Dt  dPi=1Y it and applying It^o’s formula we obtain (D0  1 = 0)(Dt^TM  1)2= mart. +Z t^TM0 2(Ds  1)( 1 +Ds) + 122(1  Ds)2 d<N >sgivingE(Dt^TM  1)2  3C(M)Z t0E(Ds^TM  1)2ds:Here we used that d<N >s C(M)ds for s t^TM. Now Gronwall’s lemmayields Dt  1  0 for all t TM a.s. Take M ! 1 to obtain the claim.70Continuation 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 2 [0;1]dn@[0;1]d andsuch that Pi Yi0 = 1. The following Lemma gives the existence of processes Xitsuch that Yit = Xit=Pi Xit. This will enable us later to derive uniqueness ofsolutions to (3.15) from uniqueness of solutions to (3.2).Lemma 3.2.16. Given a process (Y it )1 i d that satis es (3.15) with Y0 2[0;1]dn@[0;1]d and with Pi Yi0 = 1, we can  nd a system of processes (Xit)1 i dthat satis es (3.2) and (3.14) with St  PiXit.Proof. We start with a motivation for the proof. Let (Y it )1 i d be given by(3.14). De nition (3.14) and St =PiXit implied in the former setting thatdSt =dXi=1dXit =dXi=1q2 iXitXi+1t dBit = StdXi=1q2 iY it Y i+1t dBit;i.e.dSt  StdMt (3.16)being solved bySt = S0 exp Mt  12 <M>t :In the given setting the above calculation may be taken as a motivation tode neRt  R0 exp Mt  12 <M>t ; where Mt  Z t0dXi=1q2 iYisY i+1s dBis;such that (3.16) holds with St replaced by Rt and R0 > 0 to be chosen arbi-trarily. Let TM  infft 0 : maxiY it  Mg. For t TM we have <M>t<1a.s. This  nally leads to the de nition of (Xit)1 i d, for t TM, byXit  Y it Rt = Y it R0 exp Mt  12 <M>t ; (3.17)where Xi0  Yi0R0.Let us check that (Xit)1 i d as de ned in (3.17) satis es (3.2). Indeed, wehavedXit (3:17)= RtdYit +Y it dRt+ <Y i;R>t(3:15);(3:16)= Rtq2 iYit Y i+1t dBit +RtdXj=1Yjt qjidt(3:17)= q2 iXitXi+1t dBit +dXj=1Xjtqjidt;71which proves our claim after taking M ! 1 and observing that by Corol-lary 3.2.15, Rt =PiY it Rt =Pi Xit.Conclusion of the proof of Proposition 3.2.13.The uniqueness of the solution to (3.15) follows from the uniqueness of thesolution (Xit)1 i d to (3.2). Indeed, observe that the uniqueness of Xt yieldsthe uniqueness of Rt =Pi Yit Rt =PiXit and thus of Y it = Xit=Rt.3.2.4 Properties of a stationary distribution to thesystem (3.15) of normalized processesRecall that we are given the system of SDEs (3.15) and look for solutions sat-isfying PiY it = 1, where 0  Y it  1, 1  i d.In what follows we shall look for stationary distributions to this system. ByProposition IV.9.2 of [7], a measure is a stationary distribution for our process,that is, it is a stationary distribution for A, where A denotes the generator ofour system of SDEs, if and only if R Agd = 0 for all g 2 C2. Hence we shallinvestigate necessary properties of a measure  satisfying R[0;1]d Agd = 0 forall g 2 C2. In particular we want to show the following Proposition.Proposition 3.2.17. If  is stationary then it does not put mass on the setN  fy 2 [0;1]d : 9i2 f1;:::;dg : yi = 0g;i.e. on the set where at least one of the coordinate processes becomes extinct.Proof. The generator A of our di usion Y can be determined to beAg(x) =dXi=1@ig(x)bi(x) + 12dXi;j=1@ijg(x)aij(x); (3.18)wherebi(x) = xiXj6=i2 jxjxj+1 + (xi  1)2 ixixi+1 +dXj=1xjqji; (3.19)aii(x) = (xi)2Xj6=i2 jxjxj+1 + (1  xi)2 2 ixixi+1 and for i6= jaij(x) = xixjXk=2fi;jg2 kxkxk+1  (1  xi)xj2 ixixi+1  xi(1  xj)2 jxjxj+1(see [10], V.(1.7) for the de nition of A).In what follows we shall try to  nd a function g 2 C2 which leads to acontradiction toR Agd = 0 in case  puts mass on the set N. Thereby we shalltake advantage of the observation that for xi = 0 for i 2 f1;:::;dg arbitrarily xed we haveaii(x) = 0 and bi(x)  Xj6=ixjqmin = (1  xi)qmin = qmin > 0; (3.20)72where we set qmin  mini6=jqij.To make things easier we shall  x i2 f1;:::;dg arbitrarily and only look forfunctions g(x) = g(xi);g 2 C2. Thus we obtain for (3.18) with f(xi) = f(x)  @ig(x) = @ig(xi);f 2 C1 that R[0;1]d Agd = 0 is equivalent toZ[0;1]df(xi)24xiXj6=i2 jxjxj+1 + (xi  1)2 ixixi+1 +dXj=1xjqji35d (x)=  Z[0;1]d12@if(xi)24(xi)2Xj6=i2 jxjxj+1 + (1  xi)2 2 ixixi+135d (x):Using that x2 [0;1]d we getZ[0;1]d8<: jf(xi)jC1xi +f(xi)Xj6=ixjqji9=;d (x)  C2Z[0;1]dj@if(xi)jxid (x);where all constants under consideration are nonnegative. Assuming that f isnonnegative and as Pj6=ixjqji  (1  xi)qmin (see (3.20)) we  nally getZ[0;1]df(xi) [ C1xi + (1  xi)qmin]d (x)  C2Z[0;1]dj@if(xi)jxid (x): (3.21)In what follows we shall try to  nd a nonnegative function f 2 C1 whichgives a contradiction to the assumption that  puts mass on the set N.As we want to investigate the behaviour of  on the set N, we shall de nea function f 2 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 givenf 2 C1+ with support in [0;1] or ( 1;1]and let f (x)  f x  thenf 2 C1+ with support in [0; ] or ( 1; ] and f0 (x) = f0 x  1 :Choose for instance f(x) = exp  11 x 1(x 1). Plugging this into (3.21) andabbreviating A  [0; ] [0;1]d 1 we obtain for arbitrary 0 < < 1ZA f x  [ C1 + (1   )qmin]d (x;z)  ZA    f0 x     C2x d (x;z);73where we assumed without loss of generality that i = 1. This yieldsZA exp  11 x  ([ C1 + (1   )qmin]  1 1  x  2x C2)d (x;z) (3.22) I1( ) +I2( )  0for all  > 0.For the  rst part I1 of the integral observe that the absolute value of theintegrand is bounded for 0 <    0 via C1 0 +qmin. Hence we can apply thedominated convergence theorem to the  rst integral to obtainlim #0+I1( ) = e 1qmin  f0g [0;1]d 1 : (3.23)For the second part of the integral I2 note that for x 2 [0; ] the absolutevalue of the integrand is bounded by 4e 2 x C2  4e 2C2. As this result isuniform in  we can apply the dominated convergence theorem again to obtainlim #0+I2( ) = 0.Plugging this and (3.23) into (3.22) we get with qmin > 0e 1qmin  f0g [0;1]d 1  0 )  f0g [0;1]d 1 = 0;i.e.  does not put mass on N as stated above.3.2.5 Stationary distributionRecall that we are given the system of SDEs (3.15) which we rewrite asdYt =  (Yt)dBt +b(Yt)dtand look for solutions satisfying PiY it = 1, where 0  Y it  1, 1  i d.Proposition 3.2.18. The above system of SDEs has a unique stationary dis-tribution  supported by S   [0;1]dn@[0;1]d \ fy : Piyi = 1g. Moreover,L(YtjY0 = y) ) holds for all y 2S.Proof. Let Yt be the unique strong Markov solution to (3.15). Recall that wealready showed that every equilibrium distribution for A doesn’t put mass onN = fy : 9i : yi = 0g in Proposition 3.2.17 and that PiY it = 1 for all t  0in Proposition 3.2.13. Hence, if Yt has any equilibrium distributions, they areconcentrated onS   [0;1]dn@[0;1]d \(y :Xiyi = 1):In what follows we shall consider the process ~Yt  (Y 1t ;:::;Y d 1t ) 2 [0;1]d 1instead. The martingale problem for the resulting SDE for ~Y is consequently74well-posed as the corresponding martingale problem for Y is well-posed. For~x2 ~S   [0;1]d 1 \(d 1Xi=1~xi  1)!n(~x : 9i : ~xi = 0 ord 1Xi=1~xi = 1)= [0;1]d 1n@[0;1]d 1 \(~x : 0 <Xi~xi < 1);~aij(~x) is non-singular by Corollary A.1.1 of the Appendix. Also observe that~S is an open subset of [0;1]d 1 compact. Now the reasoning of [2], Section 3.1can be applied to show that the system of SDEs for ~Y has a unique stationarydistribution ~ supported by ~S and that L(~Ytj~Y0 = ~y) ) ~ holds for all ~y 2 ~S.Note that, as in [2], the non-singularity of ~aij(~x) on ~S is crucial.The claim now follows from ~Yt  (Y 1t ;:::;Y d 1t ) and Ydt = 1  Pd 1i=1 Y it .A complete proof is given in the Appendix, Subsection A.2.3.2.6 Extension to arbitrary networksInstead of (3.1) we can considerdXit =s2 iXitXj2CiXjtdBit +dXj=1Xjtqjidt; i2 f1;:::;dg; (3.24)where Ci  f1;:::;dg with i =2 Ci and jCij  1. We can think of Ci as theset of catalysts of i. The cyclic case corresponds to Ci = fi + 1g. We shallassume as above that  i and qji;i 6= j are given positive constants and theXi0  0;i 2 f1;:::;dg are given initial conditions. (qji) is again a Q-matrixmodelling mutations from type j to type i.For this setup the above proofs directly carry over. Observe in particularthat (3.9) changes to the requirement that there exists  =  (!) > 0 andi = i(!);j = j(!) 2 f1;:::;dg such thatXitj[ ; + ) = 0 and Xjt j[ ; + ) > :Also note that the restriction on the state space for the initial condition inLemma 3.2.11 changes to Qdi=1 Xi0 +Pj2Ci Xj0 > 0 as we now use Theo-rem 1.6 of [9].3.2.7 Complete analysis of the case d = 2Remark 3.2.19. We shall denote  i by  i in what follows, as for instance  2might easily be misunderstood.Recall that the normalized processes Y it = XitSt for our given SDE satisfy(3.15).75Corollary 3.2.20. For d = 2 we obtain the following SDE for Yt  Y 1t (notethat Y 2t = 1  Y 1t )dYt =p2(1  Yt)Yt ((Yt)2 2 + (Yt  1)2 1)dBt (3.25)+ 2(1  Yt)Yt (Yt 2 + (Yt  1) 1)dt+Yt (q11  q21)dt+q21dt:Idea of the proof. We can calculate the SDE for Y 1t from (3.15), using ourcyclic de nition Y 2+1t = Y1t . We get in particular thatd<Y 1 >t= 2(1  Y 1t )Y1t  (Y1t )2 2 + (Y1t  1)2 1 dt:Hence we can rewrite Y 1t on a possibly enlarged probability space in terms of aBrownian motion Bt as above (cf. [10], Theorem IV.(34.11)).In what follows we shall prove existence and pathwise uniqueness of solutionsto the SDE (3.25) under the constraint Yt 2 [0;1]. First observe that (3.25)implies thatdYt =  (Yt)dBt +b(Yt)dt; (3.26)where Yt 2I  [0;1] with (x) =p2(1  x)x(x2 2 + (x 1)2 1) (3.27)andb(x) = 2(1  x)x(x 2 + (x 1) 1) +x(q11  q21) +q21:Lemma 3.2.21. If Y0 2 [0;1], the SDE (3.26) has a pathwise unique solution,taking values in [0;1].Proof. Replace  (x) and b(x) in (3.26) by continuous functions ~ (x);~b(x) withcompact support such that they coincide with (x);b(x) on [0;1]. Then existenceof solutions to the SDE with modi ed coe cients follows by Theorem V.3.10 in[7].By reasoning as in Lemma 3.2.14 ( rst consider (Yt) , then (1  Yt) ) wecan further show that any solution to the modi ed SDE satis es 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. di usions (see V.(40.1) in [10] and replacethe term under the square root in (3.27) by its absolute value).As every one-dimensional di usion can be uniquely characterized by its scalefunction and speed measure, we shall calculate the scale function as a  rst steptowards investigating the long-time behaviour of the given one-dimensional SDEfor Yt = Y 1t .Lemma 3.2.22. The scale function of Y is given by (up to increasing a netransformations)s0(x) =  x2 2 + (x 1)2 1    1+ q112 2  q212 1 j1  xjq11 2 jxj q21 1 exp   q11 +q21p 1 2 arctan ( 1 + 2)x  1p 1 2  :76This yields in particular that s0(0) = s0(1) = 1. We obtain moreovers(0) =(const. > 1; q21 1 < 1; 1; o.w. and s(1) =(const. <1;  q11 2 < 1;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 is8>><>>:recurrentnever hitrecurrentnever hit9>>=>>;and 1 is8>><>>:recurrentrecurrentnever hitnever hit9>>=>>;for8>><>>: q11 < 2;q21 < 1; q11 < 2;q21   1; q11   2;q21 < 1; q11   2;q21   1;9>>=>>;;where \0 is recurrent" should mean P(9n such that Yt 6= 0 for all t >n) = 0 and \1 never hit" that P(9t 0 : Yt = 1) = 0.(ii) In all cases R10 1f0;1g(Yt)dt = 0.(iii) The scale function of Y is given (up to increasing a ne transformations)as in Lemma 3.2.22.(iv) The speed measure of the di usion Z  s(Y) in natural scale on [s(0);s(1)](read this as [s(0);s(1)] = ( 1;s(1)] for  1 = s(0) <s(1) <1 etc.) ism(dz) = e2R s 1(z)122b(u) (u) 2du (s 1(z)) 21fz2(s(0);s(1))gdz= 1(s0 )2  s 1(z)1fz2(s(0);s(1))gdz: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 fPtg be the transition function of our di usion in natural scale onR. Then for each x:limt!1 k  Pt(x; )k= 0;where (dz)  m(dz)m(R) :Here k k denotes the total variation norm of a measure.Idea of the proof. This follows easily from [10], Theorem V.54.5.77Bibliography[1] Bass, R.F. and Perkins, E.A. Degenerate stochastic di erential equa-tions arising from catalytic branching networks. Electron. J. Probab. (2008)13, 1808{1885. MR2448130[2] 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. Poincar e Probab. Statist. (2008) 44, 1038{1077.MR2469334[3] Dawson, D.A. and Perkins, E.A. On the uniqueness problem for cat-alytic branching networks and other singular di usions. Illinois J. Math.(2006) 50, 323{383 (electronic). MR2247832[4] Eigen, M. and Schuster, P. The Hypercycle: a principle of natural self-organization. Springer, Berlin, 1979.[5] Fleischmann, K. and Xiong, J. 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MR178093278Chapter 4Convergence of RescaledCompeting SpeciesProcesses to a Class ofSPDEs14.1 IntroductionWe investigate convergence of certain rescaled models that have their applica-tions in biology. Such convergence results can for instance be used to relate thelimits to questions of coexistence and survival of types in the original models.We start by introducing the underlying models and concepts for our laterde nitions in Subsections 4.1.1, 4.1.2 and 4.1.3. In Subsection 4.1.4 an overviewof the results of this paper follows. Finally, in Subsection 4.1.5 we outline theremaining parts of the paper.4.1.1 The voter model and the Lotka-Volterra modelAn 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  t : Z ! f0;1g withthe following interpretation. x 2 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 t(x) = 0, at time t, the coordinate x is occupied by an individual of population0. As we shall consider approximate densities later on, this interpretation willsuit our purpose better in what follows.The evolution of the process in time is given via in nitesimal rates. Followingthe notation in [7], let c(x; ) denote the rate at which the coordinate  (x)  ipsfrom 0 to 1 or from 1 to 0 when the system is in state  . Then the process  twill satisfyP( t(x) 6=  0(x)) = c(x; 0)t+o(t) for t# 0+:For the voter model, the rates can for instance be given by a random walk kernel1A 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.79on Z, i.e. 0  p(x)  1 and Px2Zp(x) = 1 such that0 ! 1 at rate c(x; ) =Xyp(x y) (y);1 ! 0 at rate c(x; ) =Xyp(x y)(1   (y)):Under certain conditions on the rate or kernel, it can now be shown that thegiven rates determine indeed a unique, f0;1gZ-valued Markov process  t.A possible interpretation of the kernel p( ) is that at exponential times withrate 1, the individual x 2 Z selects a site at random according to the kernelp(x  ) and in case this site has opposite opinion, changes its opinion to theopinion of the selected site. The exponential times and choices according to therandom kernel are independent for all x2 Z.Finally observe that a special case of this model is the case where we  xa  nite set N  Z of neighbours of 0. If we choose the random walk kernelp(x) = 1jNj1(x 2 N), then a neighbour gets chosen with equal probability.Moreover,fi(x; ) = 1jNjXy2N1( (y) = i) =Xy2Zp(x y)1( (y) = i); i = 0;1 (4.1)can be understood as the frequency of type i in the neighbourhood x+ N of xin con guration  .In general, we can set fi(x; ) =Py2Zp(x y)1( (y) = i);i = 0;1 to rewritethe rates from above to0 ! 1 at rate c(x; ) = f1(x; ); (4.2)1 ! 0 at rate c(x; ) = f0(x; ):For the Lotka-Volterra model we consider rate-changes0 ! 1 at rate (4.3)c(x; ) = f1(x; ) (f0(x; ) + 01f1(x; )) = f1(x; ) + ( 01  1)f21(x; );1 ! 0 at ratec(x; ) = f0(x; ) (f1(x; ) + 10f0(x; )) = f0(x; ) + ( 10  1)f20(x; )instead, where we used that f0(x; ) +f1(x; ) = 1 by de nition. The de nitionwill become clear in the Subsection to follow (choose  = 1). Observe in partic-ular that if we choose  01; 10 close to 1, the Lotka-Volterra model can be seenas a small perturbation of the voter model.Finally, we can consider biased voter models by multiplying the rate c(x; )of the change 0 ! 1 by a factor of (1 + ), i.e. (4.2) becomes0 ! 1 at rate c(x; ) = (1 + )f1(x; ); (4.4)1 ! 0 at rate c(x; ) = f0(x; )80instead. For  > 0 small we thus have a slight favour for type 1 and for  < 0small we have a slight favour for type 0. The biased Lotka-Volterra model isconstructed analogously.4.1.2 Spatial versions of the Lotka-Volterra modelAs a further example consider spatial versions of the Lotka-Volterra model with nite range as introduced in [10] (they considered (x) 2 f1;2ginstead of f0;1g).They use rates0 ! 1 at rate c(x; ) =  f1(x; ) f1(x; ) +f0(x; )(f0(x; ) + 01f1(x; )); (4.5)1 ! 0 at rate c(x; ) = f0(x; ) f1(x; ) +f0(x; )(f1(x; ) + 10f0(x; ));where  01; 10  0; > 0. Here fi is as in (4.1) and N = fy : 0 < jyj  Rgwith R 1.We can think of R as the  nite interaction range of the model. [10] usethis model to obtain results on the parameter regions for coexistence, foundercontrol and spatial segregation of types 0 and 1 in the context of a model thatincorporates short-range interactions and dispersal. As a conclusion they obtainthat the short-range interactions alter the predictions of the mean- eld 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 rst 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 describesthe e ect of intraspeci c competition,  01f1 the e ect of interspeci c competi-tion. [10] assume that the intraspeci c competition is the same for both species.The replacement of a particle of opposite type is regulated by the fecundity pa-rameter  . The  rst factors of both rate-changes added together yield 1. Thusthey can be seen as weighted densities of the two species. If  > 1, species 1has a higher fecundity than species 0.4.1.3 Long-range limitsIn [9], Mueller and Tribe show that the approximate densities of type 1 ofrescaled biased voter processes, de ned as in (4.1) and (4.4) with  =  N , con-verge to continuous space time densities which solve the heat equation withdrift, driven by Fisher-Wright noise, namely@u@t = u6 + 2 (1  u)u+p4u(1 u) _W: (4.6)Observe that [9] scale space by 1=N and consider N = f0 < jyj  N 1=2g.Hence, the number of neighbours of x 2 Z=N is increasing, namely jNj =2c(N)N1=2 with c(N) N!1! 1, and we thus obtain long-range interactions. Fi-nally, they also rescale time by speeding up the rates of changec(x; ) as follows.81Let p(N)(x) = 1jNj1(x2 N), then in the Nth-model we have0 ! 1 at rate cN x; N = N1=2 + N 1=2 jNjXy2Z=Np(N)(x y) N(y)= 2c(N)(N + )f1 x; N ;1 ! 0 at rate cN x; N = N1=2jNjXy2Z=Np(N)(x y)(1   N(y))= 2c(N)Nf0 x; N :They  x   0, i.e. consider the case where the opinion of type 1 is slightlydominant. See the introduction and Theorem 2 of [9] for more details.In [3] it was shown that stochastic spatial Lotka-Volterra models, suitablyrescaled in space and time, converge weakly to super-Brownian motion withlinear drift. As they choose the parameters  01; 10 (recall (4.3)) such thatN  (N)i(1 i)  1 ! i 2 R; i = 0;1 (4.7)(see (H3) in [3]) their models can also be interpreted as small perturbations ofthe 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 thelow density regime, i.e. where only a  nite number of individuals of type 1 ispresent. Instead of investigating limits for approximate densities, both papersde ne measure-valued processes XNt byXNt = 1N0Xx2Z=(MNpN) Nt (x) x;i.e. they assign mass 1=N0, N0 = N0(N) to each individual of type 1 andconsider weak limits in the space of  nite Borel measures on R. In particular,they establish the tightness of the sequence of measures and the uniqueness ofthe martingale problem, solved by any limit point.Note that both papers use a di erent scaling in comparison to [9]. Usingthe notation in [2], for d = 1 they take N0 = N and the space is scaled byMNpN with MN=pN ! 1 (see for instance Theorem 1.1 of [2] for d = 1) inthe long-range setup. According to this notation, [9] used MN = pN, which isat the threshold of the results in [2], but not included. By letting MN = pN inour setup several non-linear terms will arise in our limiting SPDE below. Alsonote the brief discussion of the case where MN=pN ! 0 in d = 1 before (H3)in [2].Additionally, [2] and [3] consider  xed kernel models in dimensions d  2respectively d  3 with MN = 1 and a  xed random walk kernel q satisfyingsome additional conditions such that p(x) = q(pNx) on x2 Z=(MNpN).Finally, in Cox and Perkins [4], the results of [3] for d 3 are used to relatethe limiting super-Brownian motions to questions of coexistence and survival ofa rare type in the original Lotka-Volterra model.824.1.4 Overview of resultsIn the present paper we  rst prove tightness of the local densities for scalinglimits of more general particle systems. The generalization includes two features.Firstly, we shall extend the model in [9] to limits of small perturbations ofthe 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 powerseries (for extensions to polynomials of degree 2 recall (4.3)), thereby includingcertain analytic functions. Recall in particular from (4.7) that we shall allowthe coe cients of the power series to depend on N.Secondly, we shall combine both long-range interaction and  xed kernel in-teraction for the perturbations. As we shall see, the tightness results will carryover. As a special case we shall be able to consider rescaled Lotka-Volterramodels with long-range dispersal and short-range competition, i.e. where (4.3)gets generalized to0 ! 1 at rate c(x; ) = f1(x; ) (g0(x; ) + 01g1(x; ));1 ! 0 at rate c(x; ) = f0(x; ) (g1(x; ) + 10g0(x; )):Here fi(x; );i = 0;1 is the density corresponding to a long-range kernel pL andgi(x; );i = 0;1 is the density corresponding to a  xed kernel pF (also recall theinterpretation of both multiplicative factors in Subsection 4.1.2).Finally, in the case of long-range interactions only we show the limit pointsare solutions of a SPDE similar to (4.6) but with a drift depending on the choiceof our perturbation and small changes in constants due to simple di erences inscale factors. Hence, we obtain a class of SPDEs that can be characterizedas the limit of perturbations of the long-range voter model. If the limitinginitial condition u0 satis es R u0(x)dx < 1, we can show the weak uniquenessof solutions to the limiting SPDE and therefore show weak convergence of therescaled particle densities to this unique law.When there exists a  xed kernel, the question of uniqueness of all limit pointsand of identifying the limit remains an open problem. Also, when we considerlong-range interactions only with R u0(x)dx = 1 the proof of weak uniquenessof solutions to the limiting SPDE remains open.The proof of our results generalizes the work done in [9]. In [9], limits areconsidered for both the long-range contact process and the long-range voterprocess. Full details are given for the contact process. For the voter process,once the approximate martingale problem is derived, almost all of the remainingsteps are left to the reader. Many arguments of our proof are similar to [9] butas additions and adaptations are needed due to our broader setup and as theydid not provide details for the long-range voter model we shall sometimes bemore detailed.834.1.5 Outline of the paperIn Section 4.2 to follow we shall  rst set up our model and give the mainresults. Then we shall reformulate the model so that it can be approachedby the methods used in [9]. A statement of the main results in the reformulatedsetting follows.In Section 4.3 we shall introduce a graphical construction for each approxi-mating model  N . This allows us to write out the time-evolution of our models.By integrating it against a test function and summing over x2 Z=N we  nallyobtain an approximate martingale problem for the Nth-process. We de ne theapproximate density A( Nt )(x) as the average density of particles of type 1 onZ=N in an interval centered at x of length 2=pN (see (4.10) below). By choos-ing a speci c test function, the properties of which are under investigation atthe beginning of Section 4.4, an approximate Green’s function representationfor the approximate densities A( Nt )( ) is derived towards the end of Section 4.4and bounds on error-terms appearing in it are given. Making use of the Green’sfunction representation, tightness of A( Nt )( ) is proven in Section 4.5. Here themain part of the proof consists in  nding estimates on pth-moment di erences.In Section 4.6 the tightness of the approximate densities is used to showtightness of the measure corresponding to the sequence of con gurations  Nt .Finally, in the special case with no  xed kernel, every limit is shown to solvea certain SPDE. In Section 4.7 we prove that this SPDE has a unique weaksolution if < u0;1 >< 1. In this case, weak uniqueness of the limits of thesequence of approximate densities follows.4.2 Main Results of the Paper4.2.1 The modelWe de ne a sequence of rescaled competing species models  Nt in dimensiond = 1, which can be described as perturbations of voter models. In the Nth-model the sites are indexed by x 2 N 1Z. We label the state of site x at timet by  Nt (x) where  Nt (x) = 0 if the site is occupied at time t by type 0 and Nt (x) = 1 if it is occupied by type 1.In what follows we shall write x y if and only if 0 <jx yj  N 1=2, i.e.if and only if x is a neighbour of y. Observe that each x has2c(N)N1=2; c(N) N!1! 1neighbours.The rates of change incorporate both long-range models and  xed kernelmodels with  nite range. The long-range interaction takes into account thedensities of the neighbours of x at long-range, i.e.f(N)i (x; )  12c(N)pNX0<jy xj 1=pN;y2Z=N1( N(y) = i); i = 0;184and the  xed kernel interaction considersg(N)i (x; )  Xy2Z=Np(N(x y))1( N(y) = i); i = 0;1;where p(x) is a random walk kernel on Z of  nite range, i.e. 0  p(x)  1,Px2Zp(x) = 1 and p(x) = 0 for all jxj  Cp. In what follows we shall oftenabbreviate f(N)i (x; ) by f(N)i and g(N)i (x; ) by g(N)i if the context is clear.Now de ne the rates of change of our con gurations. At site x in con gura-tion  N 2 f0;1gZ=N the coordinate  N(x) makes transitions0 ! 1 at rate Nf(N)1 +f(N)1ng(N)0 G(N)0 f(N)1 +g(N)1 H(N)0 f(N)1 o; (4.8)1 ! 0 at rate Nf(N)0 +f(N)0ng(N)0 G(N)1 f(N)0 +g(N)1 H(N)1 f(N)0 o;where G(N)i ;H(N)i ;i = 0;1 are power series as follows.Hypothesis 4.2.1. We assume thatG(N)i (x)  1Xm=0 (m+1;N)i xm and H(N)i (x)  1Xm=0 (m+1;N)i xm; x2 [0;1]with i = 0;1,  (m+1;N)i ; (m+1;N)i 2 R, m 0 and that there exists N0 2 N suchthatsupN N0Xi=0;11Xm=0n (m+1;N)i _ 0 + (m+1;N)i _ 0 +m    (m+1;N)i ^ 0   +m    (m+1;N)i ^ 0   o<1:Remark 4.2.2. The above rates determine indeed a unique, f0;1gZ=N-valuedMarkov process  Nt for N  N0 with N0 as in Hypothesis 4.2.1 as we nowshow. See for instance Theorem B3, p.3 in Liggett [8] and note the uniformboundedness assumption on the rates from p.1 of [8]. Following the notationin [8], let c(x; N) denote the rate at which the coordinate  N(x)  ips from 0to 1 or from 1 to 0 when the system is in state  N. Then using (4.8), 0  f(N)i ;g(N)i  1;i = 0;1 and Hypothesis 4.2.1 yieldssupx2Z=Nsup N2f0;1gZ=Nc x; N  N + 1Xm=0    (m+1;N)0   +    (m+1;N)0   !_ 1Xm=0    (m+1;N)1   +    (m+1;N)1   ! N +C0(N) <185andsupx2Z=NXu2Z=Nsup N2f0;1gZ=N  c x; N  c x; Nu    supx2Z=NXu xsup N2f0;1gZ=N  c x; N  c x; Nu   + supx2Z=NXu2Z=Nsup N2f0;1gZ=NXi=0;1   g(N)i  x; N  g(N)i  x; Nu     C0(N) 2c(N)N1=22(N +C0(N)) + supx2Z=NXu2Z=N2p(N(x u))C0(N) 2c(N)N1=22(N +C0(N)) + 2C0(N)<1;where Nu (v) =( N(v); v 6= u;1   N(v); v = u:Following [8], the two conditions are su cient to ensure the above.Additionally, the closure in the space of continuous functions on f0;1gZ=N ofthe operator  f  N = Pxc x; N  f( Nx )  f( N) , which is de ned on thespace of  nite cylinder functions on f0;1gZ=N, is the Markov generator of theprocess  Nt .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  xed kernel can be obtained by choosing G(N)i  H(N)i ;i = 0;1 and we get0 ! 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 con gurations Nt 2 f0;1gZ=N we de ne approximate densitiesA( Nt )viaA( Nt )(x) = 12c(N)N1=2Xy x Nt (y); x2N 1Z (4.10)and note that A( Nt )(x) = f(N)1  x; Nt  . By linearly interpolating between siteswe obtain approximate densities A( Nt )(x) for all x2 R.Notation 4.2.4. Set C1  ff : R ! [0;1] cont.g and let C1 be equipped withthe topology of uniform convergence on compact sets.We obtain that t7!A( Nt ) is cadlag C1-valued, where we used that0  A( Nt )(x)  1 for all x2N 1Z:Hence, we can consider the law of A( N) on the space of cadlag C1-valued pathswith the Skorokhod topology.864.2.2 Main resultsBefore stating our main results we need some more notation.Notation 4.2.5. For f;g : N 1Z ! R, we set<f;g>= 1NXxf(x)g(x):Let  be a measure on N 1Z. Then we set< ;f >=Zfd :Remark 4.2.6. We can rewrite every con guration  Nt in terms of its corre-sponding measure. Let Nt  1NXx x1( Nt (x) = 1);then< Nt ; >=< Nt ; >:De nition 4.2.7. Let S be a Polish space and let D(S) denote the space ofcadlag paths from R+ to S with the Skorokhod topology. Following the  rstde nition on p.148 of Perkins [11], we shall say that a collection of processeswith paths in D(S) is C-tight if and only if it is tight in D(S) and all weak limitpoints are a.s. continuous. Recall that for Polish spaces, tightness and weakrelative compactness are equivalent.Remark 4.2.8. In what follows we shall investigate tightness of fA( N ) : N  N0g in D(C1) and tightness of f Nt : N  N0g in D(M(R)), where M(R) isthe space of Radon measures equipped with the vague topology (M(R) is indeedPolish, see Dawson [5], Section 3.1.3).Theorem 4.2.9. Suppose that A( N0 ) ! u0 in C1. Let the transition rates of N(x) be as in (4.8) with G(N)i ;H(N)i ;i = 0;1 satisfying Hypothesis 4.2.1. Then A( Nt ) : t 0 are C-tight as cadlag C1-valued processes and the   Nt : t 0 are C-tight as cadlag Radon measure valued processes with the vague topology.If A  Nkt ; Nkt t 0converges to (ut; t)t 0, then  t(dx) = ut(x)dx for allt 0.Remark 4.2.10. The above applies in particular to models where G(N)i ;H(N)iare  nite sums (see Hypothesis 4.2.1).Hypothesis 4.2.11. Let us consider the special case with no  xed kernel (seeRemark 4.2.3). Additionally to Hypothesis 4.2.1 we assume that (m+1;N)i N!1!  (m+1)i for all i = 0;1 and m 087with (take sgn(0) = 0)sgn  (m+1;N)i  0 for all N  N0 or sgn  (m+1;N)i  0 for all N  N0for all i = 0;1;m 0 and thatlimN!1Xi=0;11Xm=0n (m+1;N)i _ 0 +m    (m+1;N)i ^ 0   o(4.11)=Xi=0;11Xm=0n (m+1)i _ 0 +m    (m+1)i ^ 0   o:Remark 4.2.12. The additional conditions of Hypothesis 4.2.11 are necessaryto transform the given rates into rates with positive coe cients in a uniformway in Subsection 4.2.3 and to later characterize limit points of the approximatedensities by taking limits in N ! 1 inside in nte sums in (4.55).De nition 4.2.13. Under the assumptions of Theorem 4.2.9 and Hypothe-sis 4.2.11 we let for x2 [0;1],Gi(x)  limN!1G(N)i (x) = limN!11Xm=0 (m+1;N)i xm =1Xm=0 (m+1)i xm;i = 0;1:This is well-de ned by (4.11) and Royden [12], Proposition 11.18.Theorem 4.2.14. We obtain under the assumptions of Theorem 4.2.9 andHypothesis 4.2.11 for the special case with no  xed kernel that the limit pointsof A( Nt ) are continuous C1-valued processes ut which solve@u@t = u6 + (1  u)ufG0(u)  G1(1  u)g +p2u(1 u) _W (4.12)with initial condition u0. If we assume additionally < u0;1 >< 1, then ut isthe unique (in law) [0;1]-valued solution to the above SPDE.Remark 4.2.15. As an example consider spatial versions of the Lotka-Volterramodel as introduced in Subsection 4.1.2. In what follows we shall choose thecompetition 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" (N)f(N)1 (N)f(N)1 +f(N)0 f(N)0 + (N)01 f(N)1 #;1 ! 0 at rate N"f(N)0 (N)f(N)1 +f(N)0 f(N)1 + (N)10 f(N)0 #:We suppose that (N)  1 +  0N;  (N)01  1 + 01N ;  (N)10  1 + 10N :88Using f(N)0 +f(N)1 = 1 we can therefore rewrite the rates as0 ! 1 at rate (N + 0)f(N)1 1 +  01N f(N)1 Xn 0   0Nf(N)1 n; (4.13)1 ! 0 at rate Nf(N)0 1 +  10N f(N)0 Xn 0   0Nf(N)1 n= Nf(N)0 1 +  10N f(N)0 Xk 0 f(N)0 k  0N k Xn k nk    0N n k:Here we used that   f(N)i    1;i = 0;1 and that    0N   ! 0 for N ! 1. We canuse the explicit calculations for a geometric series, in particular that we havePn 0njqjn < 1 and Pn k jqjn k nk = 1(1 jqj)k+1 for jqj < 1 to check thatHypothesis 4.2.1 and Hypothesis 4.2.11 are satis ed.Using Theorem 4.2.14 we further obtain that the limit points of A( Nt ) arecontinuous C1-valued processes ut which solve@u@t = u6 + (1  u)uf( 0 +u( 01   0))  (  0 + (1  u) ( 10 + 0))g+p2u(1  u) _W= u6 + (1  u)uf 0   10 +u( 01 + 10)g +p2u(1  u) _Wby rewriting the above rates (4.13) in the form (4.9) and taking the limit forN ! 1. For <u0;1 ><1, ut is the unique weak [0;1]-valued solution to theabove SPDE.4.2.3 ReformulationWe proceed as in [9]. Recall from Hypothesis 4.2.1 and (4.8) that the rates ofchange are given by0 ! 1 at rate Nf(N)1 +g(N)01Xm=1 (m;N)0 f(N)1 m+g(N)11Xm=1 (m;N)0 f(N)1 m;(4.14)1 ! 0 at rate Nf(N)0 +g(N)01Xm=1 (m;N)1 f(N)0 m+g(N)11Xm=1 (m;N)1 f(N)0 m;where  (m;N)j ; (m;N)j 2 R for all j = 0;1;m2 N.Following [9] we shall model each term in the rate-changes via independentfamilies of i.i.d. Poisson processes. For instance, if  (m;N)0 is non-negative, theterm g(N)0  (m;N)0 f(N)1 mof the rate-change 0 ! 1 in (4.14) can be modeledvia i.i.d. Poisson processes Qt(x;y1;:::;ym;z) : x;y1;:::;ym;z 2N 1Z 89of rate (m;N)0(2c(N))mNm=2p(N(x z)):At a jump of Qt(x;y1;:::;ym;z) the voter at x adopts the opinion 1 providedthat all of y1;:::;ym have opinion 1 and z has opinion 0.As we want to allow the  (m;N)i ; (m;N)i to be negative, too, we  rst 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 allresulting coe cients are non-negative.Corollary 4.2.16. We can rewrite our transitions as follows.0 ! 1 at rate (4.15) N   (N) f(N)1 +f(N)18<:Xi=0;1a(N)i g(N)i +Xm 2;i;j=0;1q(0;m;N)ij g(N)i f(N)j f(N)1 m 29=;;1 ! 0 at rate N   (N) f(N)0 +f(N)08<:Xi=0;1b(N)i g(N)i +Xm 2;i;j=0;1q(1;m;N)ij g(N)i f(N)j f(N)0 m 29=;;with corresponding  (N);a(N)i ;b(N)i ;q(k;m;N)ij 2 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 tosimplify notation.Suppose for instance  (m;N)0 < 0 for some m 1 in (4.14). Using that xm = (1  x)m 1Xk=1xk  xand recalling that 1  f1 = f0 we obtaing0 (2k+1;N)0 f2k+11 = g0(   (2k+1;N)0 f02kXl=1fl1 + (2k+1;N)0 f1):Finally, we can use g0 = 1  g1 to obtaing0 (2k+1;N)0 f2k+11 = g0   (2k+1;N)0 f02kXl=1fl1+g1   (2k+1;N)0 f1+ (2k+1;N)0 f1:All terms on the r.h.s. but the last can be accommodated into an existingrepresentation (4.15) as follows:q(0;m;N)00 !q(0;m;N)00 +   (2k+1;N)0 for 2  m 2k + 1;a(N)1 !a(N)1 +   (2k+1;N)0 :90Finally, we can assimilate the last term into the  rst part of the rate 0 ! 1,i.e. we replace (N) ! (N)   (2k+1;N)0 :As we use the representation (4.15), a change in the  rst part of the rate0 ! 1 also impacts the rate 1 ! 0 in its  rst term. Therefore we have to x the rate 1 ! 0 by adding a term of (  (2k+1;N)0 )f0 = g0f0(  (2k+1;N)0 ) +g1f0(  (2k+1;N)0 ) to the second and third term of the rate, i.e. by replacingb(N)0 !b(N)0 +   (2k+1;N)0 ;b(N)1 !b(N)1 +   (2k+1;N)0 :The case m = 2k;k  1 follows similarly and the general case with multiplenegative  0s and/or  0s now follows inductively.Remark 4.2.17. The above construction yields the following non-negative co-e cients: (N)  Xj=0;11Xn=1   (n;N)j 1  (n;N)j < 0 +   (n;N)j 1  (n;N)j < 0 ;q(0;m;N)00  1Xn=m   (n;N)0 1  (n;N)0 < 0 ;a(N)0   (1;N)0 1  (1;N)0  0 +( 1Xn=1   (n;N)0 1  (n;N)0 < 0 +1Xn=1   (n;N)1 1  (n;N)1 < 0 +1Xn=1   (n;N)1 1  (n;N)1 < 0 );q(0;m;N)01   (m;N)0 1  (m;N)0  0 ; q(0;m;N)10  1Xn=m   (n;N)0 1  (n;N)0 < 0 ;a(N)1   (1;N)0 1  (1;N)0  0 +( 1Xn=1   (n;N)0 1  (n;N)0 < 0 +1Xn=1   (n;N)1 1  (n;N)1 < 0 +1Xn=1   (n;N)1 1  (n;N)1 < 0 );q(0;m;N)11   (m;N)0 1  (m;N)0  0 ; q(1;m;N)01  1Xn=m   (n;N)1 1  (n;N)1 < 0 ;b(N)0   (1;N)1 1  (1;N)1  0 +( 1Xn=1   (n;N)1 1  (n;N)1 < 0 +1Xn=1   (n;N)0 1  (n;N)0 < 0 +1Xn=1   (n;N)0 1  (n;N)0 < 0 );91q(1;m;N)00   (m;N)1 1  (m;N)1  0 ; q(1;m;N)11  1Xn=m   (n;N)1 1  (n;N)1 < 0 ;b(N)1   (1;N)1 1  (1;N)1  0 +( 1Xn=1   (n;N)1 1  (n;N)1 < 0 +1Xn=1   (n;N)0 1  (n;N)0 < 0 +1Xn=1   (n;N)0 1  (n;N)0 < 0 );q(1;m;N)10   (m;N)1 1  (m;N)1  0 :By Hypothesis 4.2.1 this implies in particular that there exists N0 2 N such thatsupN N0Xi;j;k=0;1Xm 2q(k;m;N)ij <1: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   (N) f(N)1 +f(N)1Xm 2;i;j=0;1q(0;m;N)ij g(N)i f(N)j f(N)1 m 2;1 ! 0 at rate N   (N) f(N)0 +f(N)0Xm 2;i;j=0;1q(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 instancea(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 0with 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 coe cients 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 particularthat they will be assumed to be non-negative.Hypothesis 4.2.19. Assume that there exists N0 2 N such thatsupN N0Xi;j;k=0;1Xm 2q(k;m;N)ij <1for non-negative q(k;m;N)ij , i;j;k = 0;1 and m  2. We can use this conditionas in Remark 4.2.2 to show that the rewritten rates can be used to determine af0;1gZ=N-valued Markov process  Nt for N  N0.92Hypothesis 4.2.20. In the special case with no  xed kernel, i.e. whereq(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 (N) N!1!  ;q(k;m;N)0j N!1! q(k;m)0j for all j;k = 0;1 and m 2andlimN!1Xj;k=0;1Xm 2q(k;m;N)0j =Xj;k=0;1Xm 2q(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 2were obtained from  (m;N)j ;j = 0;1;m 1 as described earlier in Remark 4.2.17and 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 a2 R we letFk(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 usingCorollary 4.2.24 below.Theorem 4.2.23. Suppose that A( N0 ) ! u0 in C1. Let the transition ratesof  N(x) be as in (4.16) and q(k;m;N)ij satisfying Hypothesis 4.2.19. Then the A( Nt ) : t 0 are C-tight as cadlag C1-valued processes and the   Nt : t 0 are C-tight as cadlag Radon measure valued processes with the vague topology.If A  Nkt ; Nkt t 0converges to (ut; t)t 0, then  t(dx) = ut(x)dx for allt 0.For the special case with no  xed kernel we further obtain that if Hypothe-sis 4.2.20 holds, then the limit points of A( Nt ) are continuous C1-valued pro-cesses ut which solve@u@t = u6 +Xk=0;1(1 2k)Xm 2;j=0;1q(k;m)0j Fj(u) (F1 k(u))m 1Fk(u)+p2u(1  u) _W(4.18)with initial condition u0. If we assume additionally < u0;1 >< 1, then ut isthe unique (in law) [0;1]-valued solution to the above SPDE.In the next Corollary we assume there is no  xed kernel and the q(k;m)0j ;j;k =0;1;m  2 are de ned from the  (m)j ;j = 0;1;m  1 as in Remark 4.2.17 andRemark 4.2.18 without the N’s.93Corollary 4.2.24. Under the assumption above, the SPDE (4.18) may berewritten asut =  u6 +(1 u)u1Xm=0 (m+1)0 um u(1 u)1Xm=0 (m+1)1 (1 u)m+p2u(1 u) _W:Proof. Indeed  rst use the de nition 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 followswe shall considere (x) = exp( jxj)for  2 R and we letC = ff : R ! [0;1) cont. with jf(x)e (x)j ! 0 as jxj ! 1 for all  < 0gbe the set of non-negative continuous functions with slower than exponentialgrowth. De nekfk = supxjf(x)e (x)jand give C the topology generated by the norms (k k :  < 0).Remark 4.2.25. We work on the space C instead of C1 because in Section 4.4 weshall introduce functions 0   zt (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 di erences of ^A( t)(z)  A( t)(z) <  0; zt >, where A( 0) ! u0 in C to  nally establish the tightnessclaim for the sequence of approximate densities A( N)(x).Notation 4.2.26. For x2N 1Z;f : N 1Z ! R and  > 0 we shall useD(f; )(x) = supfjf(y)  f(x)j : jy xj   ;y 2N 1Zg; (4.19) (f)(x) = N   (N)2c(N)N1=2Xy x(f(y)  f(x));where we suppress the dependence on N.944.3 An Approximate Martingale ProblemWe shall now derive a graphical construction and evolution in time of our ap-proximating processes Nt . The graphical construction uses independent familiesof i.i.d. Poisson processes: Pt(x;y) : x;y 2N 1Z i.i.d. P.p. of rate N   (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 2N 1Z i.i.d. P.p. of rate q(k;m;N)ij(2c(N))mNm=2p(N(x z)):Note that we suppress the dependence on N in the family of Poisson processesPt(x;y) and Qm;i;j;kt (x;y1;:::;ym;z).At a jump of Pt(x;y) the voter at x adopts the opinion of the voter at yprovided 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 kprovided that y1 has opinion j, all of y2:::;ym have opinion 1  k and z hasopinion i.This yields the following SDE to describe the evolution in time of our ap-proximating processes  Nt : Nt (x) = N0 (x) +Xy xZ t0  0  Ns (x)  1  Ns (y)   1  Ns (x)  0  Ns (y)  (4.21) dPs(x;y)+Xk=0;1(1  2k)Xm 2;i;j=0;1Xy1;:::;ym xXzZ t0 k  Ns (x)   j  Ns (y1) mYl=2 1 k  Ns (yl)  i  Ns (z) dQm;i;j;ks (x;y1;:::;ym;z)for all x2N 1Z.We now explain why the above system (4.21) has a unique solution. Theproblem with (4.21) is that although there is a  rst  ip time for  Nt (x) (thejump rate there is bounded as can be shown using Hypothesis 4.2.19 and as thesum of Poisson processes is a Poisson process again, we have at most one  ipat a time), what to do at this time depends on the states of the  nite numberof \communicating sites" y;z 2 Z=N;y  x;jz xj  Cp=N. The equations foreach of these sites will in turn involve its communicating sites. If we now try togo backwards in time to determine the con guration at x2 Z=N, starting witht> 0, we may encounter an accumulation of jumps before time zero.95To see that the described problem cannot occur, we shall use the uniformboundedness of the  ip-rates together with the  nite interaction range R  dN 1(pN _Cp)e, where dxe denotes the next largest integer.Remark 4.3.1. We have avoided random walk kernels p with in nite range forthe  xed kernel interactions to simplify the analysis of the above jump equations(4.21).We show that up to time t, the evolution of  N can be divided up into nite random \islands" that do not communicate with each other. Indeed, tworegions of Z=N do not interact with each other up to time t, if we can  ndan intermediate region of length 2R where no  ips occur in [0;t]. We can nowpartition Z=N into such regions. The sums of  ips for each region up to timet are independent and can be bounded by i.i.d. Poisson random variables asfollows. The  ips in the region centered at Z2R;Z 2 Z can be bounded byPZt  Xx2Z=N;Z2R R<x Z2R+RPt;x;wherePt;x  Xy xPt(x;y) +Xm 2;i;j;k=0;1Xy1;:::;ym x;jz xj Cp=NQm;i;j;kt (x;y1;:::;ym;z):Using (4.20) we obtain that each PZt has meant2RN 2c(N)N1=2 N   (N)2c(N)N1=2+Xm 2;i;j;k=0;1 2c(N)N1=2 m q(k;m;N)ij(2c(N))mNm=21A Xjz xj Cp=Np(N(x z))= t2RN0@N   (N) + Xm 2;i;j;k=0;1q(k;m;N)ij1A:Thus by Hypothesis 4.2.19, (PZt )Z2Z is a sequence of i.i.d. Poisson randomvariables of  nite mean. Let XZ be a sequence of random variables with XZ = 1if PZt > 0 and XZ = 0 if PZt = 0. Then (XZ)Z2Z is an i.i.d. sequence ofBernoulli variables with P(X0 = 0) > 0. In particular we can show that withprobability one there exists a random sequence:::;Z 2;Z 1;Z0;Z1;Z2;:::suchthat 1  jZi  Zi 1j<1 for all i2 Z and XZi = 0. Hence, up until time t, wecan partitionZ=N = [i2Z(Zi2R;Zi+12R]\ Z=Ninto  nite regions that do not communicate with each other up to time t. Forall x of each region we can now uniquely solve (4.21) on [0;t]. As the region has nite length, we only need to consider a  nite number of sites. To see this, note96that as Pt;x is a Poisson random variable of  nite mean for each x in the region,we can have at most a  nite number of  ips in each region up until time t. Nowiterate on successive intervals of length t to uniquely solve the entire system forall 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-communicatingislands 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 morestraightforward calculation given above was possible.Having solved the equation (4.21) it remains to ensure that the solution isthe spin- ip system with rates c(x; N) 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  N constructed from the given rates is the uniquein law solution to the martingale problem for  , where f  N =Xxc x; N  f( Nx )  f( N) with f in the space of  nite cylinder functions on f0;1gZ=N.Hence it remains to show that for the Markov process  N constructed in(4.21),f( Nt )  f( N0 )  Z t0 f( Ns )ds= f( Nt )  f( N0 )  Z t0Xxc(x; Ns ) f(( Ns )x)  f( Ns ) dsis a martingale for all f in the space of  nite cylinder functions on f0;1gZ=N.Since f depends on  nitely many coordinates, this is an exercise in stochasticcalculus for jump processes, see Remark 4.3.4 below.In what follows we shall often drop the superscripts w.r.t. N to simplifynotation.We now derive the approximate martingale problem. We take a test function : [0;1) N 1Z ! R with t7! t(x) continuously di erentiable and satisfyingZ T0<j sj + 2s + j@s sj;1 >ds<1 (4.22)(this condition ensures that the following integration and summation are well-de ned). We apply integration by parts to  t(x) t(x), sum over x and multiply97by 1N , to obtain for t T (recall that by Remark 4.2.6 < t; >= < t; >)< t; t > (4.23)=< 0; 0 > +Z t0< s;@s s >ds+ 1NXxXy xZ t0 s (y) ( s(x)   s(y))dPs(x;y) (4.24)+ 1NXxXy xZ t0 s (x) s(x) (dPs(y;x)  dPs(x;y)) (4.25)+Xk=0;1(1  2k)Xm 2;i;j=0;11NXxXy1;:::;ym xXzZ t0 k( s (x))  j( s (y1))mYl=2 1 k( s (yl)) i( s (z)) s(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 oncewe analyze term (4.26) in detail. The latter is the only term where calculationschanged seriously compared to [9]. Hence, we shall only summarize the resultsfor terms (4.24) and (4.25) in what follows.We break term (4.24) into two parts, an average term and a  uctuation termand after proceeding as for term (3.1) in [9] we obtain(4:24) =Z t0< s ; ( s) >ds+E(1)t ( );whereE(1)t ( )  1NXxXy xZ t0 s (y) ( s(x)   s(y)) (dPs(x;y)  dhP(x;y)is):We have suppressed the dependence on N in E(1)t ( ). E(1)t ( ) is a martingale(recall that if N  Pois( ), then Nt  t is a martingale with quadratic variationhNit =  t) with predictable brackets process given byd E(1)( ) t          D  t; 1pN         2 < 1;e 2 >dt: (4.27)Alternatively we also obtain the boundd E(1)( ) t  4 k tk0<j tj;1 >dt (4.28)with k tk0= supx j t(x)j.The second term (4.25) is a martingale which we shall denote by M(N)t ( )(in what follows we shall drop the superscripts w.r.t. N and write Mt( )). It98can be analyzed as the martingale Zt( ) of (3.3) in [9]. We obtain in particularthathM( )it = 2N   (N)N Z t0< s ; 2s >ds Z t0<A( s  s); s  s >ds :(4.29)Using thatjA( s  s)(x)j       12c(N)N1=2Xy x s (y) s(y)      12c(N)N1=2Xy xj s(y)j  supy xj s(y)j:we can further dominate hM( )it byhM( )it  C( )Z t0 k sk2 < 1;e 2 > ^ (k sk0< s ;j sj>)ds: (4.30)We break the third term (4.26) into two parts, an average term and a  uctua-tion term. Recall Notation 4.2.22 and observe that if we only consider a2 f0;1gwe have Fk(a) =  k(a).(4:26) =Xk=0;1(1  2k)Xm 2;i;j=0;11NXxXy1;:::;ym xXzZ t0 k( s (x)) j( s (y1)) mYl=2 1 k( s (yl)) i( s (z)) s(x) q(k;m;N)ij(2c(N))mNm=2p(N(x z))ds+E(3)t ( )=Xk=0;1(1  2k)Xm 2;i;j=0;1q(k;m;N)ijZ t01NXx 12c(N)N1=2Xy1 x j( s (y1))! mYl=2 12c(N)N1=2Xyl x 1 k( s (yl))! Xzp(N(x z)) i( s (z))!  k( s (x)) s(x)ds+E(3)t ( )=Xk=0;1(1  2k)Xm 2;i;j=0;1q(k;m;N)ijZ t01NXxFj(A( s )(x)) (F1 k(A( s )(x)))m 1Fi((pN   s )(x)) k( s (x)) s(x)ds+E(3)t ( )=Xk=0;1(1  2k)Xm 2;i;j=0;1q(k;m;N)ijZ t0< (Fj  A( s )) (F1 k  A( s ))m 1 Fi  (pN   s ) ( k   s ); s >ds+E(3)t ( );99where for x2 Z=N we set pN  f (x)  Xz2Z=Np(N(x z))f(z) (4.31)andE(3)t ( )  Xk=0;1(1  2k)Xm 2;i;j=0;11NXxXy1;:::;ym xXzZ t0 k( s (x))  j( s (y1))mYl=2 1 k( s (yl)) i( s (z)) s(x)  dQm;i;j;ks (x;y1;:::;ym;z)  q(k;m;N)ij(2c(N))mNm=2p(N(x z))ds!:We have suppressed the dependence on N in E(3)t ( ). Here, E(3)t ( ) is a mar-tingale with predictable brackets process given by E(3)( ) t  Xm 2;i;j;k=0;1q(k;m;N)ij 1N2XxmYl=0 Xyl x12c(N)N1=2!(4.32)  Xzp(N(x z))!Z t0 2s(x)ds Xm 2;i;j;k=0;1q(k;m;N)ij 1NZ t0< 2s;1 >ds 1NXm 2;i;j;k=0;1q(k;m;N)ijZ t0k sk2 <e 2 ;1 >ds:Taking the above together we obtain the following approximate semimartin-gale decomposition from (4.23).< t; t >= < 0; 0 > +Z t0< s;@s s >ds (4.33)+Z t0< s ; ( s) >ds+E(1)t ( ) +Mt( )+Xk=0;1(1  2k)Xm 2;i;j=0;1q(k;m;N)ijZ t0< (Fj  A( s )) (F1 k  A( s ))m 1 Fi  (pN   s ) ( k   s ); s >ds+E(3)t ( ):100Remark 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  xed kernel. Indeed, uniqueness of the limit ut of A( Nt )will be derived by proving that ut solves the martingale problem associated withthe SPDE (4.18).Remark 4.3.4. For all f in the space of  nite cylinder functions on f0;1gZ=Nthe Markov process  (=  N) constructed in (4.21) yields a martingalef( t) f( 0) Z t0 f( s)ds = f( t) f( 0) Z t0Xxc(x; s) (f(( s)x)  f( s))ds:(4.34)Indeed, every  nite cylinder function on f0;1gZ=N, f( ) = f( (x1);:::; (xn)),n 2 N; (xi) 2 f0;1g;xi 2 Z=N can be rewritten as a linear combinationof functions of the form g( )   (xi1)    (xim), where m 2 N;m  n andfi1;:::;img  f1;:::;ng. By linearity we only need to consider functions ofthis form.Now rewrite (4.21) as t(x) =  0(x) +Z t0c(x; s ) (1  2 s (x))ds+ mart. (4.35)by breaking both integrals in (4.21) into an average term and a  uctuation term.Observe here that we can rewrite the sum of both average terms as in (4.35) byusing for example 0( s (x))Xy x 1( s (y)) N   (N)2c(N)N1=2 =  0( s (x)) N   (N) f1(x; s )and Xz i( s (z))p(N(x z)) = gi(x; s ):Now use the representation of the rates c(x; ) from (4.16). Both  uctuationterms turn out to be martingales that only depend on the Poisson processesPt(x;y) and Qm;i;j;kt (x;y1;:::;ym;z). Hence, for x 6= x0 the martingale termsare orthogonal.For m = 2, that is for g( ) =  (x) (x0) with x 6= x0, we can now use theintegration by parts formula (cf. Theorem VI.(38.3) in Rogers and Williams[13]), the orthogonality of the martingale terms of  t(x) and  t(x0) andg( x)  g( ) = (1   (x)) (x0)   (x) (x0) = (1  2 (x)) (x0)to obtain that (4.34) is a martingale for f = g. Now iterate the above reasoningto obtain the claim for all m2 N.1014.4 Green’s Function RepresentationAnalogous to [9], de ne a test function zt (x)  0 for t 0;x;z 2N 1Zas the unique solution, satisfying (4.22) and s.t.@@t zt =   zt; z0(x) = N1=22c(N)1(x z)with  zt (x) = N   (N)2c(N)N1=2Xy x( zt (y)   zt (x)) (4.36)as in (4.19). Note that  z0 was chosen s.t. <  t; z0 >= A( t)(z) and that wesuppress the dependence on N.Next observe that  is the generator of a simple random walk Xt 2N 1Z,jumping at rate N  (N)2c(N)N1=2  2c(N)N1=2 = N   (N) = (1 + o(1))N with sym-metric steps of variance 1N  13 +o(1) , where we used that c(N) N!1! 1. Hereo(1) denotes some function that satis es o(1) ! 0 for N ! 1. De ne  zt (x) = NP(Xt = xjX0 = z)then< z0;   xt > = 1NXy z0(y)   xt (y) = 1NXyN1=22c(N)1(y  z)NP(Xt = yjX0 = x)(4.37)= N1=22c(N)Xy zP(Xt = yjX0 = x) = Ex[ z0(Xt)] =  zt (x):As we shall see later in Lemma 4.5.2(b), when linearly interpolated, thefunctions  zt (x) and   zt (x) converge to p t3;z x (the proof follows), wherept(x) = 1p2 te x22t is the Brownian transition density:The next Lemma gives some information on the test functions  and   fromabove. Later on, this will provide us with estimates necessary for establishingtightness.102Lemma 4.4.1. There exists N0 <1 s.t. for N  N0;T  0;z 2N 1Z;  0,(a) < zt;1 >=<   zt;1 >= 1 and k zt k0 CN1=2 for all t 0.(b) <e ; zt +   zt > C( ;T)e (z) for all t T,(c) k zt k  C( ;T) N1=2 ^t 2=3 e (z) for all t T,(d) <    zt    zs  ;1 > 2Njt sj for all s;t 0.If we further restrict ourselves to N  N0, N 3=4  s < t  T;y;z 2 N 1Z,jy zj 1, we get(e)k zt   yt k  C( ;T)e (z) jz yj1=2t 1 +N 1=2t 3=2 ;(f)k zt   zs k  C( ;T)e (z) jt sj1=2s 3=2 +N 1=2s 3=2 ;(g)       D  zt;N 1=2 ( )        C( ;T)e (z)N 1=4t 1:Proof. First we shall derive an explicit description for the test functions  zt and  zt . 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 <jjj  pN).Set (t) = E eitY1 and Sk =kXi=1Yi: (4.38)Note that E[Y 21 ] = 1+o(1)3N , where o(1) ! 0 for N ! 1. Similarly, E[Y 41 ] =1+o(1)5N2 , where o(1) may change from line to line.The relation between the test functions  zt;   zt and Sk is as follows. zt (x) = Ex[ z0(Xt)] =1Xk=0((N   (N))t)kk! e ((N  (N))t)NP(Sk+1 = x z);(4.39)  zt (x) = NP(Xt = xjX0 = z) =1Xk=0((N   (N))t)kk! e ((N  (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 x2N 1Z;k  1.(b) follows as in the proof of Lemma 3(b), [9], where we shall use the boundEhe jY1ji exp 5 2 1N 103for all   0 to obtain the claim.(c) Following the proof of Lemma 3(c) in [9], we  rst show that we have, fork 2 N and jxj  1, P(Sk = x)  1N P(Sk  jxj 1), which we can use to obtainP(Sk = x)  1Ne  (jxj 1) exp 5k 2 1N .Substituting this bound into (4.39) gives for any   0 zt (x)  C( ;T) expf  jx zjg (4.40)for all t T and jx zj 1.From (4.39) we further have for N big enough zt (x)  E p (1 +o(1))(Pt + 1)3N ;x z  +E(N;t;x z);where Pt  Pois((N   (N))t). Using Corollary B.0.2 we get as in the proof of[9], Lemma 3(c),jE(N;t;x)j  C 1N 1 +t 3=2 for N 3=4  t:Here we used that for P  Pois(r);r > 0 we haveE[(P + 1)a]  C(a)ra for all a< 0:(This is obviously true for 0 <r < 1. For r  1  xed, prove the claim  rst forall a2 Z. Then extend this result to general a< 0 by an application of H older’sinequality.)Using the trivial bound p(t;x)  Ct 1=2 we get from the above zt (x)  C(T)t 2=3 for N 3=4  t T:Finally, we obtaink zt k =supxnj zt (x)je jxjo(4:40) supfx:jx zj 1gnC( ;T)e  jx zje jxjo_ supfx:jx zj<1;N 3=4 t TgnC(T)t 2=3e jxjo_ supfx:jx zj<1;0 t N 3=4gnj zt (x)je jxjo(a) nC( ;T)e jzjo_n1 N 3=4  t T C(T)t 2=3e1e jzjo_n1 0  t N 3=4 CN1=2e1e jzjo C( ;T) N1=2 ^t 2=3 e (z) for all t T:104This proves part (c).(d) follows along the lines of the proof of [9], Lemma 3(d).(e) For the remaining parts (e)-(g) we  x N 3=4  s<t T;y;z 2 N 1Z,jy zj 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 nd as in [9] thatk zt   yt k0 C(T) jz yjt 1 +N 1t 3=2 : (4.41)Now recall (4.40) with  = 2 to get  zt (x) + yt (x)  C( ;T) expf 2 jx zjgfor jx zj  1;jx yj  1;jy  zj  1 and thus in particular for jx zj  2;jy zj  1. This yieldsk zt   yt k  supfx:jx zj<2gk zt   yt k0 e (x)+ supfx:jx zj 2gnC( ;T) k zt   yt k1=20 e  jx zje (x)o C( ;T)e (z) k zt   yt k0 + k zt   yt k1=20  C( ;T)e (z)  jz yjt 1 +N 1t 3=2 + jz yjt 1 +N 1t 3=2 1=2  C( ;T)e (z) jz yj1=2t 1 +N 1=2t 3=2 :This proves (e).(f) The proof of part (f) follows analogously to the proof of part (e), withchanges as suggested in the proof of [9], Lemma 3(f).(g) Finally, to prove part (g), use part (e),  zt (y) =  yt (z) (see (4.39)) andthe de nition ofD  zt;N 1=2 (x) = supnj zt (y)   zt (x)j : jx yj  N 1=2;y 2N 1Zoto getkD  zt;N 1=2 ( )k  supfx:jx zj<2g(supy:jx yj N 1=2fj zt (y)   zt (x)jge jxj)+ supfx:jx zj 2g(supy:jx yj N 1=2fj zt (y)   zt (x)jge jxj)(4:40) C( ) supfx:jx zj<2g(supy:jx yj N 1=2fj zt (y)   zt (x)jge jzj)+C( ;T) supfx:jx zj 2g(supy:jx yj N 1=2nj zt (y)   zt (x)j1=2oe  jx zje jxj):105Next use that  at (b) =  bt (a) to get as a further upper boundC( ) supfx:jx zj<2g(supy:jx yj N 1=2fj yt (z)   xt (z)jge jzj)+C( ;T) supfx:jx zj 2g(supy:jx yj N 1=2nj yt (z)   xt (z)j1=2oe jzj)(4:41) C( ;T)e (z) supx;y:jx yj N 1=2n jx yjt 1 +N 1t 3=2 + jx yjt 1 +N 1t 3=2 1=2  C( ;T)e (z)N 1=4t 1;where we used N 3=4 <t T. This  nishes the proof of (g) and it also  nishesthe proof of the Lemma.The following Corollary uses the results of Lemma 4.4.1 to obtain estimatesthat we shall need later.Corollary 4.4.2. There exists N0 < 1 s.t. for N  N0, 0    u  t  Tand y;z 2N 1Z;  0, we have(a) Rtu k zt sk ds C( ;T)(t u)1=3e (z)and Rt0 k zt sk2 ds C( ;T)N1=4e2 (z).(b) For jy zj  1 and   t N 3=4 we further havesup0 s  k zt s   yt sk  C( ;T)e (z) jz yj1=2(t  ) 1 +N 1=2(t  ) 3=2 .(c) We also have Rt k zt s   yt sk ds C( ;T) (e (z) +e (y)) (t  )1=3.(d) For N 3=4  u  we havesup0 s  k zt s   zu sk  C( ;T)e (z) (t u)1=2(u  ) 3=2 +N 1=2(u  ) 3=2 .(e) Finally, we have Ru k zt s   zu sk ds C( ;T)e (z)(u  )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)Z tu     zt s    n ds C( ;T)Z tuNn=2 ^ (t s) 2n=3dsen (z):For n = 1 further bound the integrand by (t s) 2=3, for n = 2 and u = 0 usethe 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)Z t k zt s   yt sk ds C( ;T) (e (z) +e (y))Z t (t s) 2=3ds:(d) follows from Lemma 4.4.1(f).(e) Using Lemma 4.4.1(c) once more, we getZ u k zt s   zu sk ds C( ;T)e (z)Z u (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;1) with <f;1 ><1; 2 R we have(a) < s; zt s >=<A( s);   zt s >,(b) j< t;f > <A( t);f >j C( ) kD(f;N 1=2)k .Proof. (a) follows easily from< s; zt s > =< s; zt s >= 1NXx s(x) zt s(x) = 1NXx s(x) xt s(z)(4:37)= 1NXx s(x) < x0;   zt s >= 1NXx s(x) 1NXyN1=22c(N)1(y  x)  zt s(y)= 1NXy(Xx12c(N)N1=2 1(y  x) s(x))  zt s(y)= 1NXyA( s)(y)   zt s(y) =<A( s);   zt s >:Part (b) follows as in the proof of Lemma 5(b) in [9]. Observe in particularthat < t;e  > C( ) as will be shown before and in (4.44) below.Taken all together this  nishes the proof.Next use the test function s   xt s for s tin the semimartingale decomposition (4.33) and observe that  satis es (4.22).Here the initial condition is chosen so that < t; t >=< t; x0 >= A( t)(x).The test function chosen in [9] at the beginning of page 526, namely  s =e c(t s) xt s was chosen so that the drift term <  s; c s > ds of the semi-martingale decomposition (2.9) in [9] would cancel out with the drift term107< s;@s s >ds. As we have multiple coe cients, this is not possible. Also, itturned out that the calculations become easier once we consider time di erencesin Section 4.5 to follow.With the above choice we obtain, for a  xed value of t, an approximateGreen’s function representation for A( t), namelyA( t)(x) = < 0; xt > +Z t0< s;    xt s >ds (4.42)+Z t0< s ;   xt s >ds+E(1)t   xt   +Mt  xt   +Xk=0;1(1  2k)Xm 2;i;j=0;1q(k;m;N)ijZ t0< (Fj  A( s )) (F1 k  A( s ))m 1 Fi  (pN   s ) ( k   s ); xt s >ds+E(3)t ( xt  ):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 signi cant statement will bepart (b).Lemma 4.4.4. Suppose that the initial conditions satisfy A( 0) ! u0 in C asN ! 1. Then for T  0;p 2; > 0,(a) E supt T < t;e  >p  C( ;p).(b) We further haveEh   E(1)t   zt      p_   E(3)t   zt      pi C( ;p;T) 1 +Cp=2Q N p=16e p(z)for all t T and N big enough, where we setCQ  supN N0Xm 2;i;j;k=0;1q(k;m;N)ij : (4.43)(c) Finally, kE[A( t)]k  p 1 for all t T.Proof. First observe that we have  t 2 f0;1gZ=N and 0  A( t)  1. Therefore,parts (a) and (c) follow immediately. Indeed, for (a) we only need to observethat0  < t;e  >=< t;e  j j >= 1NXx t(x)e  jxj  1NXx:x2Z=Ne  jxj 2N1Xj=0e  j=N = 2N 11  e  =N N!1! 2 :108Note in particular that we showed that<e  ;1 > C for all  > 0; N = N( ) big enough; (4.44)which will prove useful later.For (c) we further havekE[A( t)]k  p= supxjE[A( t)(x)]je  pjxj  supxe  pjxj  1:It only remains to show that (b) holds.(b) First observe that CQ <1 by Hypothesis 4.2.19.We shall apply a Burkholder-Davis-Gundy inequality in the formE sups tjXsjp  C(p)E hXip=2t + sups tjXs  Xs jp (4.45)for a cadlag martingale X with X0 = 0 (this inequality may be derived from itsdiscrete 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 martingalesE(1)t ( zt  ) respectively E(3)t ( zt  ) are bounded a.s. by CN 1=2. Indeed,E(1)t ( zt  ) = 1NXxXy xZ t0 s (y)  zt s(x)   zt s(y) (dPs(x;y)  dhP(x;y)is)and thus, using Lemma 4.4.1(a), the maximal jump size is bounded by1N2 supt T k zt k0 CN1=2 (4.46)(the maximal number of jumps at a  xed time is 1). The bound on the maximaljump size of E(3)t ( zt  ) follows analogously.Now choose t  T. We shall start with E(3)t ( zt  ). By (4.45), (4.46) and(4.32) we haveEh   E(3)t ( zt  )   pi C(p)0@ Xm 2;i;j;k=0;1q(k;m;N)ij 1NZ t0     zt s    2 <e 2 ;1 >ds1Ap=2+C(p)N p=2(4:44) C( ;p)Cp=2Q N p=2( Z t0     zt s    2 ds p=2+ 1):By Corollary 4.4.2(a) this is bounded from above byC( ;p;T)Cp=2Q N p=2nNp=8e p(z) + 1o= C( ;p;T)Cp=2Q N 3p=8e p(z):109It remains to investigate E(1)t ( zt  ). Here (4.45), (4.46), (4.27) and (4.28)yieldEh   E(1)t ( zt  )   pi C(p) Z t0 k zt sk0< zt s;1 > ^"        D  zt s; 1pN         2 < 1;e 2 >#ds!p=2+C(p)N p=2:This in turn is bounded from above byC(p) Z t0hC(T)(t s) 2=3i^"        D  zt s; 1pN         2 C( )#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) tothe second part of the integrand, we need to ensure that N 3=4  t s. AsN 3=4  N 3=8 we get as a further upper boundC(p) Z N 3=80C(T)s 2=3ds+Z tN 3=8^t C( ;T)e (z)N 1=4s 1 2C( )ds!p=2+C(p)N p=2 C( ;p;T)e p(z)(  N 3=8 1=3+N 1=2 N 3=8  1 p=2+N p=2) C( ;p;T)N p=16e p(z):This  nishes the proof.4.5 TightnessIn what follows we shall derive estimates on pth-moment di erences of^A( t)(z)  A( t)(z) < 0; zt >:Recall the assumptionA( 0) !u0 in Cfrom Theorem 4.2.9 resp. Theorem 4.2.23 from the beginning. Also note thatLemma 4.5.2(b) to come will yield that  zt (x) converges to p t3;z x . Theestimates of Lemma 4.5.1 and the convergence of  zt taken together will besu cient to show C-tightness of the approximate densities A( t)(z) at the endof this Section.110Lemma 4.5.1. For 0  s  t  T;y;z 2 N 1Z;jt sj  1;jy zj  1; > 0and p 2 we haveEh   ^A( t)(z)  ^A( s)(y)   pi C( ;p;T) 1 +CpQ e p(z) jt sjp=24 + jz yjp=24 +N p=24 :Proof. Fix s;t;T;y;z; ;p as in the statement. We decompose the increment^A( t)(z)  ^A( s)(y) into a space increment ^A( t)(z)  ^A( t)(y) and a time in-crement ^A( t)(y)  ^A( s)(y).We consider  rst the space di erences. 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( ) and E(1)t ( );E(3)t ( ) in  , we getEh   ^A( t)(z)  ^A( t)(y)   pi C( ;p;T) 1 +Cp=2Q N p=16e p(z) + E   Mt  zt    yt     p + Eh   Xk=0;1(1  2k)Xm 2;i;j=0;1q(k;m;N)ijZ t0< (Fj  A( s )) (F1 k  A( s ))m 1 Fi  (pN   s ) ( k   s );  zt s   yt s >ds   pi:Recall de nition (4.31) and observe that 0   pN   s  (x)  1 follows from s 2 f0;1gZ=N. Use this and 0  A( s )(x)  1 together with the de nitionof Fk from Notation 4.2.22 to getEh   ^A( t)(z)  ^A( t)(y)   pi(4.47) C( ;p;T) 1 +Cp=2Q N p=16e p(z) + E   Mt  zt    yt     p + Eh Xm 2;i;j;k=0;1q(k;m;N)ij Z t0< (F1 k  A( s )) ( k   s );   zt s   yt s  >ds pi C( ;p;T) 1 +Cp=2Q N p=16e p(z) + E   Mt  zt    yt     p +CpQEh Z t0<A( s ) + s ;   zt s   yt s  >ds pi:Note that this is the main step to see why the  xed kernel interaction doesnot impact our results on tightness.In what follows, we shall employ a similar strategy to the proof of Lemma 6in [9] to obtain estimates on the above. We nevertheless give full calculationsas we proceeded in a di erent logical order to highlight the ideas for obtainingbounds. Minor changes in the exponents of our bounds ensued, both due to thedi erent logical order and the di erent setup.111Let us  rst derive a bound on E   Mt  zt    yt     p . Using the Burkholder-Davis-Gundy inequality (4.45) from above and observing that the jumps of themartingales Mt( xt  ) are bounded a.s. by CN 1=2 we have for any 0    tE   Mt  zt    yt     p (4.48)(4:30) C( ;p)E" Z  0k zt s   yt sk2 < 1;e 2 >ds+Z t k zt s   yt sk0< s ;   zt s   yt s  >ds p=2#+C(p)N p=2(4:44) C( ;p)E  T sup0 s  k zt s   yt sk2 1 +Z t k zt s   yt sk0< s ;   zt s   yt s  >ds p=2#+C(p)N p=2:Now observe that by Lemma 4.4.3(a) and Lemma 4.4.1(a),< s ;   zt s   yt s  > < s ; zt s + yt s > (4.49)=<A( s );   zt s +   yt s > < 1;   zt s +   yt s >= 2:We can therefore apply the estimates from Corollary 4.4.2(b) to the  rst term in(4.48) and Corollary 4.4.2(c) to the second term, assuming    t N 3=4 _0and using jy zj 1 to obtainE   Mt  zt    yt     p  C( ;p;T)e p(z)njz yjp=2(t  ) p +N p=2(t  ) 3p=2 + (t  )p=6o+C(p)N p=2:Now set = t   jz yj1=4 _N 1=4 ^t and observe that    t N 3=4 _ 0 follows. We obtaint  = jz yj1=4 _N 1=4 ^t andjz yj1=4  N 1=4 ) jz yjp=2(t  ) p +N p=2(t  ) 3p=2 + (t  )p=6(4.50) jz yjp=2jz yj p=4 +N p=2N3p=8 +N p=24= jz yjp=4 +N p=8 +N p=24;jz yj1=4 >N 1=4 ) jz yjp=2(t  ) p +N p=2(t  ) 3p=2 + (t  )p=6 jz yjp=2jz yj p=4 +N p=2N3p=8 + jz yjp=24= jz yjp=4 +N p=8 + jz yjp=24:112Plugging this back in the above estimate we  nally haveE   Mt  zt    yt     p  C( ;p;T)e p(z)njz yjp=24 +N p=24o:Next we shall get a bound on the last term of (4.47). Recall that < t; >=< t; >. We getEh Z t0<A( s ) + s ;   zt s   yt s  >ds pi C(p)(E" Z  0<A( s ) + s ;e  >ds sup0 s  k zt s   yt sk !p#+E" Z t <A( s ) + s ;e  >k zt s   yt sk ds p#):Now use that<A( s ) + s ;e  >=<A( s ) + s ;e  > < 2;e  >(4:44) C( ) (4.51)to obtain that the above is bounded byC(p)( TC( ) sup0 s  k zt s   yt sk  p+ Z t C( ) k zt s   yt sk ds p) C( ;p;T)e p(z)njz yjp=2(t  ) p +N p=2(t  ) 3p=2+(t  )p=3o;where we used Corollary 4.4.2(b),(c) and jy  zj  1. Here we assumed    t N 3=4 _ 0 when we applied Corollary 4.4.2(b). Now choose = t   jz yj1=4 _N 1=4 ^t   t N 3=4 _0 as before. Reasoning as in(4.50), we getC( ;p;T)e p(z) N p=8 + jz yjp=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 jz yj  1)Eh   ^A( t)(z)  ^A( t)(y)   pi C( ;p;T) 1 +Cp=2Q N p=16e p(z)+C( ;p;T) 1 +CpQ e p(z)njz yjp=24 +N p=24o C( ;p;T) 1 +Cp=2Q +CpQ e p(z) jz yjp=24 +N p=24 :113Next we derive a similar bound on the time di erences. We start by sub-tracting the two Green’s function representations again, this time for the timedi erences, using (4.42) and Lemma 4.4.4(b) for the error terms.Eh   ^A( t)(z)  ^A( u)(z)   pi(4.52) C( ;p;T) 1 +Cp=2Q N p=16e p(z) + E   Mt  zt    Mu  zu     p + Eh   Xk=0;1(1  2k)Xm 2;i;j=0;1q(k;m;N)ij  Z t0< (Fj  A( s )) (F1 k  A( s ))m 1 Fi  (pN   s )  ( k   s ); zt s >ds Z u0< (Fj  A( s )) (F1 k  A( s ))m 1 Fi  (pN   s )  ( k   s ); zu s >ds    pi C( ;p;T) 1 +Cp=2Q N p=16e p(z) + E   Mt  zt    Mu  zu     p + Eh Xm 2;i;j;k=0;1q(k;m;N)ij  Z tu< (Fj  A( s )) (F1 k  A( s ))m 1 Fi  (pN   s )  ( k   s ); zt s >ds+Z u0< (Fj  A( s )) (F1 k  A( s ))m 1 Fi  (pN   s )  ( k   s );   zt s   zu s  >ds  pi C( ;p;T) 1 +Cp=2Q N p=16e p(z) + E   Mt  zt    Mu  zu     p + Eh Xm 2;i;j;k=0;1q(k;m;N)ij Z tu< (F1 k  A( s )) ( k   s ); zt s >ds+Z u0< (F1 k  A( s )) ( k   s );   zt s   zu s  >ds  pi C( ;p;T) 1 +Cp=2Q N p=16e p(z) + E   Mt  zt    Mu  zu     p +CpQEh Z tu<A( s ) + s ; zt s >ds+Z u0<A( s ) + s ;   zt s   zu s  >ds pi:For the martingale term we further get via the Burkholder-Davis-Gundy in-114equality (4.45)E   Mt  zt    Mu  zt     p  C(p) E   Mt  zt    Mu  zt     p + E   Mu  zt    Mu  zu     p   C(p)Eh   M   zt    t   M   zt    u  p=2i+C(p)Eh   M   zt    zu    u  p=2i+C(p)N p=2 C( ;p)E" Z tuk zt sk0< s ; zt s >ds p=2#+C( ;p) Z  ^u0k zt s   zu sk2 < 1;e 2 >ds!p=2+C( ;p)E" Z u ^uk zt s   zu sk0< s ;   zt s   zu s  >ds p=2#+C(p)N p=2;where we used equation (4.30) to bound the  rst and second term. Using (4.44)and reasoning as in (4.49) the above can further be bounded byE   Mt  zt    Mu  zu     p  C( ;p) Z tuk zt sk0 ds p=2+C( ;p;T) sup0 s  ^uk zt s   zu skp +C( ;p) Z u ^uk zt s   zu sk0 ds p=2+C(p)N p=2:Under the assumption N 3=4 ^ u  u  ( ^ u) we obtain from Corol-lary 4.4.2(a), (d), (e) thatE   Mt  zt    Mu  zu     p (4.53) C( ;p;T)e p(z)n(t u)p=6 + jt ujp=2 +N p=2 (u ( ^u)) 3p=2+ (u ( ^u))p=6 +N p=2o:Finally observe that with = u   jt uj1=4 _N 1=4 ^u we get N 3=4 ^u u  and by proceeding as in (4.50) we obtainE   Mt  zt    Mu  zu     p  C( ;p;T)e p(z)n(t u)p=6 + jt ujp=24 +N p=24 +N p=2o:115Finally, we can bound the last expectation of the last line of (4.52) by using<A( t s) + s ; zt s >  < 1 + 1; zt s >= 2:Here the last equality followed from Lemma 4.4.1(a). We thus obtain as anupper bound on the last expectation of the last line of (4.52),C(p) jt ujp + E  Z u0<A( s ) + s ;   zt s   zu s  >ds p  :We further have for the second termEh Z u0<A( s ) + s ;   zt s   zu s  >ds pi C(p)(E" Z  ^u0<A( s ) + s ;e  >ds sup0 s  k zt s   zu sk !p#+E  Z u ^u<A( s ) + s ;e  >k zt s   zu sk ds p  (4:51) C( ;p;T)  sup0 s  ^uk zt s   zu sk  p+ Z u ^uk zt s   zu sk ds p  C( ;p;T)e p(z)n(t u)p=2(u ( ^u)) 3p=2 +N p=2(u ( ^u)) 3p=2+(u ( ^u))p=3o;where we assumed N 3=4 ^u u ( ^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) onto obtainC( ;p;T)e p(z)njt ujp=24 +N p=24oas an upper bound.Taking all bounds together we have for the time di erences from (4.52)Eh   ^A( t)(z)  ^A( u)(z)   pi C( ;p;T) 1 +Cp=2Q N p=16e p(z)+C( ;p;T)e p(z)n(t u)p=6 + jt ujp=24 +N p=24 +N p=2o+C(p)CpQnjt ujp +C( ;p;T)e p(z) jt ujp=24 +N p=24 o C( ;p;T) 1 +Cp=2Q +CpQ e p(z)njt ujp=24 +N p=24o:The bounds on the space di erence and the time di erence taken togethercomplete the proof.116We now show that these moment estimates imply C-tightness of the approx-imate densities. We shall start including dependence on N again to clarify thetightness argument. First de ne~A( Nt )(z) = ^A( Nt )(z) on the grid z 2N 1Z;t2N 2N0:Linearly interpolate  rst in z and then in t to obtain a continuous C-valuedprocess. Note in particular that we can use Lemma 4.5.1 to show that for0  s t T;jt sj  1 and y;z 2 R;jy zj 1,Eh   ~A( Nt )(z)  ~A( Ns )(y)   pi C( ;p;T) 1 +CpQ e p(z) jt sjp=48 + jz yjp=24 for  > 0;p 2 arbitrarily  xed.The next Lemma shows that ~A( Nt ) and ^A( Nt ) remain close. The advantageof using ~A( Nt ) is that it is continuous.Using Kolmogorov’s continuity theorem (see for instance Corollary 1.2 inWalsh [15]) on compacts R(i1;i2)1  f(t;x) 2 R+  R : (t;x) 2 (i1;i2) + [0;1]2gfor i1 2 N0;i2 2 Z we obtain tightness of ~A( Nt )(x) in the space of contin-uous functions onn(t;x) : (t;x) 2R(i1;i2)1o. Indeed, we can use the Arzel a-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( Nt )(x) together withLemma 4.5.2(b) below. Now use a diagonalization argument to obtain tight-ness of ( ~A( Nt )(x) : t 2 R+;x 2 R)N2N in the space of continuous functionsfrom 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 continuousfunctions from R+ to the space of continuous functions from R to R, bothequipped with the topology of uniform convergence on compact sets, tightnessof ( ~A( Nt )(x) : t2 R+;x2 R)N2N in the former space is equivalent to tightnessof ( ~A( Nt )( ) : t2 R+)N2N in the latter.Finally, tightness of (A( Nt ) : t 2 R+)N2N as cadlag C1-valued processes(recall that 0  A( Nt )(x)  1 by construction) and also the continuity of allweak limit points follow from the next Lemma.Lemma 4.5.2. For any  > 0;T <1 we have(a) P supt T k ~A( Nt )  ^A( Nt )k   7N 1=4 ! 0 as N ! 1.(b) supt T k< N0 ;  t > Pt=3u0k  ! 0 as N ! 1.Proof. The proof is very similar to the proof of Lemma 7 in [9]. We shall onlygive some additional steps for part (a) to complement the proof of the givenreference.117(a) For 0  s t we havek< N0 ;  t > < N0 ;  s >k  = supz  <A( N0 );   zt    zs >  e  jzj supz< 1;    zt    zs  > 2Njt sj:Here we used Lemma 4.4.3(a) in the  rst line, 0  A( N0 )  1 in the second lineand Lemma 4.4.1(d) in the last. Hence, this only changes by O(N 1) betweenthe (time-)grid points in N 2N0. We obtain thatP supt Tk ~A( Nt )  ^A( Nt )k   7N 1=4 = P 9t2 [0;T]\N 2N0;s2 [0;T];js tj  N 2 s.t.k ^A( Nt )  ^A( Ns )k   7N 1=4  P 9t2 [0;T]\N 2N0;s2 [0;T];js tj  N 2 s.t.kA( Nt )  A( Ns )k  +    < N0 ;  t    s >       7N 1=4  P 9t2 [0;T]\N 2N0;s2 [0;T];js tj  N 2 s.t.kA( Nt )  A( Ns )k   6N 1=4 for N big enough.Next note that the value of A( Nt )(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 forsome y1;:::;ym  x and arbitrary z 2 N 1Z and that each jump of A( Nt ) isby de nition of A( Nt ) bounded by N 1=2. Then, writing P(a) for a Poissonvariable with mean a, we get as a further bound on the aboveXl2ZP 9z 2N 1Z \ (l;l+ 1];9t2 [0;T]\N 2N0;9s2 [t;t+N 2] with   A( Nt )(z)  A( Ns )(z)  ^  A( Nt+N 2(z)  A( Ns )(z)    N 1=4e (jlj 1)  Xl2ZN(N2T)P CN 1=2 Xy 0PN 2(0;y)+Xi;j;k=0;1;m 2Xy1;:::;ym 0XuQm;i;j;kN 2 (0;y1;:::;ym;u)1A N 1=4e (jlj 1)1A Xl2ZC(T)N3P CN 1=2P N 2  N   (N) +CQ   N 1=4e (jlj 1)  Xl2ZC(T)N3P  P N 2 (N +CQ)  p  CNp=4e p(jlj 1) 118for some p > 0. Now apply Chebyshev’s inequality. Choose p > 0 such that3  p=4 < 0. Then the resulting sum is  nite and goes to zero for N ! 1.(b) The proof of part (b) follows as the proof of Lemma 7(b) of [9].4.6 Characterizing Limit PointsTo conclude the proof of Theorem 4.2.9, Theorem 4.2.14 and Theorem 4.2.23we can proceed as in Section 4 in [9], except for the proof of weak uniquenessof (4.12) respectively (4.18).We shall give a short overview in what follows. The interested reader isreferred to [9] for complete explanations.In short, Lemma 4.4.3(b) implies for all  2 Cc thatsupt  < Nt ; > <A( Nt ); >   C( ) kD( ;N 1=2)k N!1! 0: (4.54)The C-tightness of (A( Nt ) : t 0) in C1 follows from the results of Section 4.5.This in turn implies the C-tightness of ( Nt : t 0) as cadlag Radon measurevalued processes with the vague topology. Indeed, let ’k;k 2 N be a sequence ofsmooth functions from R to [0;1] such that ’k(x) is 1 for jxj  k and 0 for jxj  k + 1. Then a diagonalization argument shows that C-tightness of ( Nt : t 0)as cadlag Radon measure valued processes with the vague topology holds if andonly if C-tightness of (’kd Nt : t  0) as cadlag MF ([ (k + 1);k + 1])-valuedprocesses with the weak topology holds. Here, MF ([ (k + 1);k + 1]) denotesthe space of  nite measures on [ (k + 1);k + 1]. Now use Theorem II.4.1 in[11] to obtain C-tightness of (’kd Nt : t 0) in D(MF ([ (k+ 1);k+ 1])). Thecompact containment condition (i) in [11] is obvious. The second condition (ii)in [11] follows from (4.54) and the C-tightness of (A( Nt ) : t 0) in C1 togetherwith Lemma 4.4.4(a).Observe in particular, that (4.54) implies the existence of a subsequence A( Nkt ); Nkt that converges to (ut; t). Hence, we can de ne variables withthe same distributions on a di erent probability space such that with probabilityone, for all T <1; > 0; 2 Cc,supt T      A( Nkt )  ut        ! 0 as k ! 1;supt T   < ; Nkt > < ; t >   ! 0 as k ! 1;where we used 0  A( Nkt )  1 and thus 0  ut(x)  1 a.s. for the  rst limit.We obtain in particular t(dx) = ut(x)dx for all t 0:It remains to investigate ut in the special case, i.e. with no  xed kernel, i.e.whereq(k;m;N)0j = q(k;m;N)1j ;j = 0;1:119Take  t   2 C3c in (4.33). We getM(N)t ( ) =< Nt ; > < N0 ; > Z t0< Ns ; ( ) >ds E(1)t ( ) (4.55) Xk=0;1(1  2k)Xm 2;j=0;1q(k;m;N)0jZ t0< Fj  A  Ns     F1 k  A  Ns   m 1  k   Ns  ; >ds E(3)t ( ):From (4.27) and (4.32) and the Burkholder-Davis-Gundy inequality (4.45) weobtain 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) ( ) (xN) = N   (N)2c(N)N1=2Xy xN( (y)   (xN))= N   (N)c(N)NpN2Xy xN( (y)   (xN))!   6 (x) as xN !x and N ! 1on the support of  .Using this in (4.55) we can show that M(N)t ( ) converges to a continuousmartingale Mt( ) satisfyingMt( ) =Z (x)ut(x)dx Z (x)u0(x)dx Z t0Z   (x)6 us(x)dxds (4.56) Xk=0;1(1  2k)Xm 2;j=0;1q(k;m)0jZ t0ZFj(us(x)) (F1 k(us(x)))m 1Fk(us(x)) (x)dxds:To exchange the limit in N ! 1 with the in nite 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( ) is indeed a martingale weused in particular (4.30) to see that hM(N)( )it  C( )t k k2 < 1;e 2 > isuniformly bounded. Therefore, (M(N)t ( ) : N  N0) and all its moments areuniformly 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) forN ! 1 together with the uniform integrability of ((M(N)t ( ))2 : N  N0).Use our results for  2 C3c, note that C3c is dense in C2c with respect to thenorm kfk kfk1 + kf0k1 + k fk1, and use (4.56) to see that ut solves themartingale problem associated with the SPDE (4.18). It is now straightforwardto 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).1204.7 Uniqueness in LawTo show uniqueness of all limit points of Section 4.6 in the case with no  xedkernel and with < u0;1 >< 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, as0  A( Nt )(x)  1 by de nition, any limit point has to satisfy ut(x) 2 [0;1].We shall choose to prove weak uniqueness of (4.18), i.e. of@u@t = u6 +Xk=0;1(1  2k)Xm 2;j=0;1q(k;m)0j Fj(u) (F1 k(u))m 1Fk(u) (4.57)+p2u(1  u) _W= u6 +u(1  u)Xk=0;1(1  2k)Xm 2;j=0;1q(k;m)0j Fj(u) (F1 k(u))m 2+p2u(1  u) _W  u6 +u(1  u)Q(u) +p2u(1 u) _Wwith initial condition u0 in what follows. Observe that jQ(us(x))j  CQ withCQ as in (4.43) because 0  us(x)  1.To check uniqueness in law of [0;1]-valued solutions we shall apply a versionof 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 theunique law of the [0;1]-valued solution to the SPDE@v@t = v6 +p2v(1  v) _W (4.58)with v0 = u0. Reasons for existence and uniqueness of a [0;1]-valued solutionto the latter can be found in Shiga [14], Example 5.2, p. 428. Note in particularthat the solution vt takes values in C1.To prove weak uniqueness, we shall follow the reasoning of the proof ofTheorem IV.1.6(a),(b) in [11] in a univariate setup. To follow the reasoningfrom [11], we need to show the following Lemma  rst.Lemma 4.7.1. Given u0 = v0 satisfying <u0;1 ><1, we have Pu-a.s.Rt0 < us;1 > ds < 1 for all t  0 and Pv-a.s. Rt0 < vs;1 > ds < 1 for allt 0.Proof. We shall prove the claim for Pu. The other claim then follows by consid-ering the special case Q 0. As a  rst step we shall use a generalization of theweak form of (4.57) to functions in two variables. In the proof of Theorem 2.1121on p. 430 of [14] it is shown that for every  2 D2rap(T) and 0 <t<T we have<ut; t >= <u0; 0 > +Z t0<us; @@s + 6  s >ds (4.59)+Z t0<us(1  us)Q(us); s >ds+Z t0Z p2us(x)(1  us(x)) s(x)dW(x;s):Here we have for T > 0,C(R) =ff : R ! R continuousg;Crap = f 2 C(R) s.t. supxe jxjjf(x)j<1 for all  > 0 ;C2rap =  2 C2(R) s.t.  ; 0; 00 2 Crap ;D2rap(T) =  2 C1;2([0;T)  R) s.t.  (t; ) is C2rap-valued continuous and@ @t (t; ) is Crap-valued continuous in 0  t<T :Also observe that the condition (2.2) of [14] is satis ed as we have 0  us(x)  1and therefore jQ(us(x))j  CQ.Now recall that the Brownian transition density is ps(x) = 1p2 se x22s . Let(Ps )(x) =R ps3 (y x) (y)dy with  2 C1c ;  0 and let (s;x) =  s(x) = eCQ(T s) (PT s )(x) and thus  2 D2rap(T):Note that @@s (PT s )(x) =   6 (PT s )(x), where we used that @@sps(x) =12 ps(x). We obtain for the drift term in (4.59) that<us; @@s + 6  s > + <us(1  us)Q(us); s >=<us; CQ s   6  s +  6  s > + <us(1  us)Q(us); s > 0using that  (s;x)  0 for   0. Additionally, the local martingale in (4.59) isa true martingale asDZ  0Z p2us(x)(1  us(x)) s(x)dW(x;s)Et=Z t0< 2us(1  us); 2s >ds 2e2CQTZ t0< 1;(PT s )2 >ds 2e2CQT k k0Z t0< 1;PT s >ds = 2e2CQT k k0< 1; >t<1:122Hence we obtain from (4.59) for all 0 <t<T after taking expectationsE[<ut; t >]  <u0; 0 >;i.e.eCQ(T t)E[<ut;(PT t ) >]  eCQT <u0;(PT ) >:Now choose an increasing sequence of non-negative functions  n 2 C1c suchthat  n " 1 for n ! 1. Using the monotone convergence theorem, we obtainfrom the aboveeCQ(T t)E[<ut;1 >] = limn!1eCQ(T t)E[<ut;(PT t n) >] limn!1eCQT <u0;(PT n) >= eCQT <u0;1 >:Hence by the Fubini-Tonelli theorem,E Z t0<us;1 >ds  <u0;1 >Z t0eCQsds<1for all t 0, which proves the claim.Lemma 4.7.2. If < u0;1 >< 1 the weak [0;1]-valued solution to (4.57) isunique in law. If we letRt  exp Z t0Z Q(vs(x))2p2vs(x)(1  vs(x))dW(x;s) 12Z t0Z (1  vs(x)) (Q(vs(x)))22 vs(x)dxds);then dPudPv    Ft= Rt for all t> 0; (4.60)where Ft is the canonical  ltration 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 takeTn = inf(t 0 :Z t0Z (1  us(x)) (Q(us(x)))22 us(x)dx + 1 ds n):Lemma 4.7.1 shows that under PuZ t0Z (1  us(x)) (Q(us(x)))22 us(x)dxds  (CQ)22Z t0<us;1 >ds<1for all t> 0 Pu a.s. and so Tn " 1 Pu-a.s. As in Theorem IV.1.6(a) of [11] thisgives uniqueness of the law Pu of a solution to (4.57). As in Theorem IV.1.6(b)of [11] the fact that Tn " 1 Pv-a.s. (from Lemma 4.7.1) shows that (4.60)de nes a probability Pu which satis es (4.57).123Bibliography[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 convergeto 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.  Ecole d’ ete de proba-bilit es de Saint Flour, XXI (1991), 1{260, Lecture Notes in Math., 1541,Springer, Berlin, 1993. MR1242575[6] Durrett, R. Ten lectures on particle systems. 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Real Analysis, Third edition. Macmillan Publishing Com-pany, New York, 1988. MR1013117[13] Rogers, L.C.G. and Williams, D. Di usions, Markov Processes, andMartingales, vol. 2, Reprint of the second (1994) edition. Cambridge Math-ematical Univ. Press, Cambridge, 2000. MR1780932124[14] Shiga, T. Two contrasting properties of solutions for one-dimensionalstochastic partial di erential equations. Canad. J. Math. (1994) 46, 415{437. MR1271224[15] Walsh, J.B. An Introduction to Stochastic Partial Di erential Equations. Ecole d’ ete de probabilit es de Saint Flour, XIV (1984), 265{439, LectureNotes in Math., 1180, Springer, Berlin, 1986. MR0876085125Chapter 5ConclusionThe last three Chapters investigate di erent models for interacting multi-typepopulations 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 FuturePerspectives of the Manuscripts5.1.1 Degenerate stochastic di erential equations forcatalytic branching networksChapter 2 establishes weak uniqueness for systems of SDEs, modelling cat-alytic branching networks. These networks can be obtained as limit points ofbranching particle systems. Weak uniqueness of the solutions to the SDE showsuniqueness of the limit points and hence implies convergence of the above ap-proximations. It also makes certain additional tools available for analysis ofthe solution, as can be seen in the application of the results of Chapter 2 inChapter 3. For instance, in the proof of existence of a stationary distributionfor the normalized processes in Subsection A.2, weak uniqueness yields that thegenerator A satis es the positive maximum principle.This paper is an extension of Dawson and Perkins [6] (where the networkswere 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 networksit is more e ective to consider every catalyst i with the set Ri of its reactants.In particular, the restriction from Ri to  Ri, including only reactants whose cata-lysts are all zero, turns out to be crucial. As a result, this paper introduces newideas on how to handle networks where there exist catalytic interlinks betweenvertices.As already mentioned in Subsection 1.1 of the introductory Chapter 1, theextension to networks becomes necessary for example in dimensions d  3 inthe renormalization analysis of hierarchically interacting multi-type branchingmodels treated in Dawson, Greven, den Hollander, Sun and Swart [5]. Theconsideration of successive block averages leads to a renormalization transfor-mation on the di usion functions of a system of SDEs similar to (2.1), (2.2).Unfortunately, [5] only show preservation of the continuity of the coe cients of126the SDE under this transformation but not preservation of H older-continuity.In [2], Bass and Perkins prove similar results to [6], i.e. they restrict themselvesto networks with at most one catalyst per reactant, but drop the requirement ofH older-continuity of the coe cients and replace it by continuity only. A futurechallenge would be to investigate how the ideas of my paper can be applied toextend the results of [2] to arbitrary networks. As a  rst step, [2] views the sys-tem of SDEs as a perturbation of a well-behaved system of SDEs with generatorA0. As part of my paper I found an explicit representation of the semigroupPt corresponding to A0 in the extension to the [6] setup. This representationdirectly carries over to the setting of [2]. The modi cation of the remaining rea-soning in [2] to arbitrary networks remains to be done. Here we note that theextension to general graphs in Chapter 2 led to a number of technical problemsin the approach of [6] even after the structure of the generator was resolved.5.1.2 Long-term behaviour of a cyclic catalyticbranching systemThe results of Chapter 2 are used in Chapter 3 to establish weak uniqueness ofthe 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. Suchquestions 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 inthe limit and other results on the  nite time survival-extinction but they werenot able to determine, if the overall sum dies out in the limit or not. I wasable to show that in my setup the sum of all coordinates converges to zero butdoes not hit zero in  nite time a.s. By changing my focus to the normalizedprocesses I showed in particular that the normalized processes converge weaklyto a unique stationary distribution that does not charge the set where at leastone of the coordinates is zero.The weakness of this manuscript lies in the restriction to constant positivecoe cients  i and qji in the SDEs (3.1). It would be of interest to allow thecoe cients to depend on the state of the system. As long as they are uniformlybounded away from zero and in nity, I conjecture a similar behaviour of thesystem but it would be of particular interest to see how the failure of uniformboundedness impacts questions on survival and coexistence. If one considerscoe cients satisfying Hypothesis 2.1.2, the results of Chapter 2 can be appliedto obtain weak uniqueness of the new system of SDEs as a  rst step.Finally, following the approach in Section III.12.2 of [8], one could considerdilution  ows instead of the normalized processes, that is, one removes the excessconcentration so that the total concentration remains constant.1275.1.3 Convergence of rescaled competing speciesprocesses to a class of SPDEsIn Chapter 4, the tightness results obtained yield the relative compactness ofthe approximating particle systems. Limits along subsequences therefore existfor combinations of long-range kernel and  xed kernel interactions in the per-turbations, where a wide class of admissible perturbations was found, includinganalytic functions as they appear in the spatial versions of Lotka-Volterra modelsof Neuhauser and Pacala [11]. It was particularly interesting to see that adding xed kernel perturbations to the long-range case does not impact tightness.In the long-range case I obtain that all subsequential limits satisfy a certainSPDE. It remains open to  nd the form of the limiting equations in the case oflong-range dispersal in the presence of short-range, i.e.  xed kernel competition.If I additionally assume  nite initial mass in the long-range case, weakuniqueness of the limiting SPDE follows. As a consequence of this, the weakuniqueness of the limits of the approximating particle systems follows. It wouldbe of interest to  nd necessary and su cient conditions for weak uniqueness ofthe limiting SPDE. I conjectureR u0(x)(1 u0(x))dx<1 to be a su cient con-dition. If this condition is preserved by the dynamics it su ces for the GirsanovLemma 4.7.2 to hold. In the special case under investigation in Mueller andSowers [9] (see below), R ut(x)(1  ut(x))dx<1;8t> 0 a.s. follows and so weexpect this condition to de ne 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 (m)i 2 R;i = 0;1;m 2 N and their counterparts in the approximating modelsare@u@t = u6 +(1 u)u( 1Xm=0 (m+1)0 um  1Xm=0 (m+1)1 (1  u)m)+p2u(1 u) _W:As I work at the critical range pN, while Cox and Perkins [3] works withlonger range interactions, I obtain a wide class of non-linear drifts. This opensup the possibility to interpret the limiting SPDEs and their behaviour via theirapproximating long-range particle systems and vice versa. For instance, a futurechallenge would be to use properties of the SPDE to obtain results on theapproximating 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 = u6 + (1  u)uf 0   10 +u( 01 + 10)g +p2u(1  u) _Wwith parameters  0; 10; 01 2 R as the limit of spatial versions of the Lotka-Volterra model with competition and fecundity parameters near one. We canrewrite this SPDE to@u@t = u6 + 0(1  u)u2   1u(1  u)2 +p2u(1 u) _W; (5.1)u(0;x)  u0(x)  0;128where  i 2 R;i = 0;1. For  1 =   0 < 0 one obtains after rescaling theKolmogorov-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  0 big enough.A major question is how the change in the drift, in particular, the possiblyadditional zero at  1 0+ 1 2 (0;1), impacts the set of parameters for survival(i.e. lim supt!1 <ut;1 >> 0 with positive probability), coexistence (i.e. thereexists a stationary distribution giving zero mass to the con gurations 0 and 1)and extinction (i.e. lim supt!1 < ut;1 >= 0 with probability 1) and if thereexist phase transitions. Aronson and Weinberger [1] showed for instance inCorollary 3.1(ii) that for  0 < 0; 1 < 0, the corresponding deterministic PDEconverges to the intermediate zero  1 0+ 1 of the drift term uniformly on boundedsets if u0 6 0;1.The author conjectures that there are parameter regions for (5.1) that yieldsurvival and others that yield extinction. To prove survival the author envisionsto apply methods of Mueller and Tribe [10] to the SPDE (5.1). In [10] and Tribe[12], rescaled versions of the SPDE@u@t = u6 + u u2 + pu _W;  > 0 (5.2)were under investigation. Results on the existence of a phase transition betweenextinction and survival in terms of  are obtained in the former paper and theexistence of travelling wave solutions in the latter. Unfortunately, the proof ofextinction in [10] and the proof of existence of travelling waves in [12] relies onthe additive properties of the  uctuation term in (5.2) (also recall the discussionof additive properties of super-Brownian motion in the beginning of the intro-ductory Chapter 1), which makes the application of their methods to SPDEsof the form (5.1) di cult. On the other hand, [10] shows that for  large, thedrift term of the SPDE (5.2) outcompetes the  uctuation term. The proof forsurvival then uses a comparison of u(t;x) to oriented site percolation to provesurvival for  big. It should be possible to apply their reasoning together withthe 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 otherhand seems much more delicate to prove.129Bibliography[1] Aronson, D.G. and Weinberger, H.F. Multidimensional NonlinearDi usion Arising in Population Genetics. Adv. Math. (1978) 30, 33{76.MR0511740[2] Bass, R.F. and Perkins, E.A. Degenerate stochastic di erential 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 convergeto 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. Poincar e 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 di usions. Illinois J. Math.(2006) 50, 323{383 (electronic). MR2247832[7] Fleischmann, K. and Xiong, J. A cyclically catalytic super-brownianmotion. Ann. Probab. (2001) 29, 820{861. MR1849179[8] Hofbauer, J. and Sigmund, K. The Theory of Evolution and DynamicalSystems. 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 KPPEquation with Noise. J. Funct. Anal. (1995) 128, 439{498. MR1319963[10] Mueller, C. and Tribe, R. A phase transition for a stochastic PDErelated 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 theLotka-Volterra model with interspeci c competition. Ann. Appl. Probab.(1999) 9, 1226{1259. MR1728561[12] Tribe, R. A travelling wave solution to the Kolmogorov equation withnoise. Stochastics Stochastics Rep. (1996) 56, 317{340. MR1396765130Appendix AAppendix for Chapter 3A.1 ~a is non-singularCorollary A.1.1. The matrix ~a is non-singular for all x2 ~S, where~S = [0;1]d 1 \(d 1Xi=1xi  1)!n(x : 9i : xi = 0 ord 1Xi=1xi = 1):Proof. Recall that  2 M(d;d) (the space of d d-matrices) and a =   T 2M(d;d). Let  2 M(d 1;d) be constructed from  by deleting the last lineof the matrix (i.e. by deleting the last equation for Y dt of our system of SDEs).Then ~a =   T 2 M(d 1;d 1). Further let ~ 2 M(d 1;d 1) be the matrixobtained from  by deleting the last column.We claim that if ~ is non-singular, then ~a is non-singular as well. Indeed,let v 2 M(d 1;1) denote the last column of  and suppose ~ is non-singular,thendet(~a) = det(  T ) = det(~ ~ T +vvT ) = det ~ ~ T  1 +vT (~ ~ T ) 1v = det ~ ~ T  1+ k ~  1v k2 = (det(~ ))2 (1+ k ~  1v k2) > 0:Recall that for i;j 2 f1;:::;d 1g we have ~ ii(x) = (1 xi)p2 ixixi+1 and~ ij(x) =  xip2 jxjxj+1 if i 6= j, where we set xd  1  Pd 1i=1 xi. Supposethat xi > 0 for all i 2 f1;:::;d 1g and Pd 1i=1 xi < 1. We shall show that inthis case ~ is non-singular. As a  rst step we divide the ith line of ~ by xi fori = 1;:::;d 1. We obtaindet(~ (x))Qd 1i=1 xi= det0BBBB@d1 a2 a3 ::: ad 1a1 d2 a3 ::: ad 1a1 a2 d3 ::: ad 1::: ::: ::: ::: :::a1 a2 a3 ::: dd 11CCCCA= det0BBBBBB@d1  a1 a2  d2 0 ::: 0 00 d2  a2 a3  d3 ::: 0 00 0 d3  a3 ::: 0 0::: ::: ::: ::: ::: :::0 0 0 ::: dd 2  ad 2 ad 1  dd 1a1 a2 a3 ::: ad 2 dd 11CCCCCCA detA1;131where we used the line operations i!i (i+1) on all but the last line and setdi  ai = (1  xi)r2 ixi+1xi+p2 ixixi+1 =r2 ixi+1xi=  aixi: (A.1)Based on the  rst column we can calculate the determinant of ~ .det(A1)  (d1  a1) det(A2) + ( 1)da1d 1Yi=2(ai  di); (A.2)where we obtain the matrix A2 by crossing out the  rst row and column of A1.We obtain recursively for k = 1;:::;d 3, using (A.1) thatdet(Ak) =  akxkdet(Ak+1) +ak( 1)d k+1d 1Yi=k+1aixi: (A.3)By using (A.3) recursively in (A.2) we getdet(A1) = ( 1)d 3d 3Yi=1aixi det(Ad 2) + ( 1)dd 1Yi=1aid 3Xi=1Yj=1;:::;d 1;j6=i1xj:Finally,det(Ad 2) = det dd 2  ad 2 ad 1  dd 1ad 2 dd 1 (A:1)=  ad 2dd 1xd 2  ad 2ad 1xd 1and thus (recall that xi 6= 0 and thus ai 6= 0 for i = 1;:::;d)det(~ (x)) = 0 () det(A1) = 0()d 2Xi=1Yj=1;:::;d 1;j6=i1xj +dd 1ad 1Yj=1;:::;d 1;j6=d 11xj = 0()d 2Xi=1Yj=1;:::;d 1;j6=i1xj  1  xd 1xd 1Yj=1;:::;d 1;j6=d 11xj = 0()d 2Xi=1xi  (1  xd 1) = 0 ()d 1Xi=1xi = xd = 1;which is a contradiction to x2 ~S. Hence det(~ (x)) 6= 0 for all x2 ~S.132A.2 Proof of Proposition 3.2.18Proof. 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 ofYt 2 [0;1]d by t, with 0 =  y for somearbitrary y 2 S. As the state space [0;1]d is compact, f t :  t  1t Rt0  sdsgt 0forms a tight family of distributions.In this case Theorem III.2.2 in Ethier and Kurtz [2] implies that f tgt 0 isrelatively compact for the Prohorov metric. As [0;1]d is Polish, Theorem III.1.7in [2] gives the existence of a limit. Also note that the convergence of a sequenceof f tgt 0 is equivalent to weak convergence by Theorem III.3.1 of [2].Taking this together we  nd a sequence (tn) tending to in nity such that tn converges weakly to a limiting distribution  . The goal will be now to applyTheorem IV.9.17 of [2], where ^C([0;1]d) = C([0;1]d) by de nition (cf. de nitionbefore Lemma IV.2.1) due to the compactness of our state space.To this purpose note  rst that the generator corresponding to (3.15) is givenbyAf(x) =dXi=1bi(x)@if(x) + 12dXi;j=1aij(x)@ijf(x);where a =   T 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 Rogersand Williams [3] yield that the solution is strong Markov. Hence A is thegenerator of a strong Markov process and thus satisties the positive maximumprinciple by [2], Theorem IV.2.2.Moreover, if  is the limit of  tn then for any f 2 C2([0;1]d) we have Af 2C([0;1]d) andE [(Af)(Y0)] = limn!1E tn [(Af)(Y0)]= limn!1 1tnZ tn0E s[(Af)(Y0)]ds= limn!1 1tnZ tn0E 0[(Af)(Ys)]ds= limn!1 1tnE 0 Z tn0(Af)(Ys)ds = limn!1 1tnE 0[f(Ytn)  f(Y0)]= 0:Here we used  tn )  for Af 2 C([0;1]d) in the  rst equality, the de nitionof  t in the second, Af 2 C([0;1]d) in the fourth equality, that Y solves themartingale problem for A in the  fth equality and f bounded in the last.133Finally 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 theexistence of a stationary solution.Uniqueness. Let Yt be the unique strong Markov solution to (3.15). Recallthat we already showed that every equilibrium distribution for A doesn’t putmass on N = fy : 9i : yi = 0g in Proposition 3.2.17 and that Pi Yit = 1for all t  0 in Proposition 3.2.13. Hence, if Yt has two distinct equilibriumdistributions, concentrated on [0;1]d, or to be more precise, onS   [0;1]dn@[0;1]d \(y :Xiyi = 1);then we can  nd two extremal equilibrium distributions and that are singularwith respect to each other (see for instance Exercise 6.9 in Varadhan [5]). As (B) = P (Yt 2 B) = R pt(x;B)d (x), there have to exist x;y 2 S such thatthe transition kernels pt(x;dz) and pt(y;dz) are mutually singular for all t> 0as well. Also, as  respectively  do not put mass on N the same holds forpt(x;dz) respectively pt(y;dz).In what follows we shall consider the process ~Yt  (Y 1t ;:::;Y d 1t ) 2 [0;1]d 1with transition kernels ~pt instead. The martingale problem for the resulting SDEfor ~Y is consequently well-posed as the corresponding martingale problem forY is well-posed.Let ~pt(~x;d~z) and ~pt(~y;d~z) be the resulting transition kernels correspondingto pt(x;dz) and pt(y;dz).For~x2 ~S   [0;1]d 1 \(d 1Xi=1~xi  1)!n(~x : 9i : ~xi = 0 ord 1Xi=1~xi = 1)= [0;1]d 1n@[0;1]d 1 \(~x : 0 <d 1Xi=1~xi < 1);~aij(~x) 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 di usions") of[1] with ~x;~y 2 D,  D  ~S, where D is an arbitrarily  xed ball. Here observethat ~S is an open subset of [0;1]d 1. The Theorem allows us to transport thedi usions started at ~x respectively ~y to a common small neighbourhood withpositive probability. Subsequently we can apply Corollary B.3 (\Transition den-sity for di usions restricted to bounded domains") of [1] to see that ~pt(~x;d~z)and ~pt(~y;d~z) and hence pt(x;dz) and pt(y;dz) cannot be mutually singular forall t> 0.Convergence. Once more, we shall consider the process ~Y instead of Y andthe state space ~S instead of S. Let ~ be the unique equilibrium distribution of~Y corresponding to the unique equilibrium distribution  of Y.134Firstly, note that by Theorem B.4 of [1] the equilibrium distribution ~ assignspositive measure to every open subset of ~S.Secondly we shall show that L( ~Ytj~Y0 = x) ) ~ for all x 2 ~S in three steps,namely by showing  rst that it holds for almost all x 2 ~S w.r.t. ~ . Then weshall extend this result to Lebesgue almost every x 2 ~S and  nally we shallconclude that this implies that it holds for all x2 ~S.To prove the  rst step, we  rst choose x 2 ~S arbitrary but  xed and let Dbe open such that x 2 D,  D  ~S. Recall that ~a is non-singular on ~S and ~S isan open subset of [0;1]d 1. Let ~Yt; ~Zt be two independent copies of the processon [0;1]d 1. Then the joint process (~Yt; ~Zt) is strong Markov and has a uniqueequilibrium distribution given by the product measure ~  ~ . By Theorem 6.9in [5] and the following Remarks the process ( ~Yt; ~Zt) started in equilibrium, i.e.with L((~Yt; ~Zt)) = ~  ~ is ergodic.As the equilibrium distribution ~ assigns positive measure to every opensubset of ~S, ~  ~ assigns positive measure to B (x )  B (x )  D D for  small enough. Therefore (~Yt; ~Zt) visits the set B (x )  B (x ) after any  nitetime T a.s. by the ergodic theorem. We obtain in particular that for almost all(x;x0) w.r.t. ~  ~ , (~Yt; ~Zt) started at (x;x0) visits B (x )  B (x ) after any nite time T a.s. Fix such a (x;x0).In what follows we shall start two independent processes ~Yt and ~Zt withinitial conditions x respectively x0 as above and denote their laws by Px respec-tively Px0.Let the  rst exit time from D be D(!)  infft 0 : !(t) 62Dg:By Corollary B.3, for each  > 0 and z 2D, the measure D (z; )  Pz(! :  < D(!);!( ) 2  )admits a density pD (z; ) with respect to Lebesgue measure. Moreover, (B.2)of the Corollary yields that for  ; su ciently small we have uniformly fory;y0 2B (x ), ZpD (y;u) ^pD (y0;u)du 12; (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 2 B (x ) (and analogously fory0 2B (x ))Py(! : !( ) 2du)  Py(! :  < D(!);!( ) 2du) (A.5)= pD (y;u)du pD (y;u) ^pD (y0;u)du:For y;y0 2 B (x )  xed let  1(y;y0) be the measure on [0;1]d 1  [0;1]d 1de ned by 1(y;y0)(A B)  ZA B pD (y;u) ^pD (y0;u)  pD (y;v) ^pD (y0;v) dudv135for A;B 2 B([0;1]d 1) and observe that 1(y;y0)(A B) =  1(y;y0)(B A): (A.6)Let 2(y;y0)(A B)  Py(! : !( ) 2A)Py0(! : !( ) 2B)   1(y;y0)(A B);which is non-negative by (A.5) and note that by (A.4), 2(y;y0) [0;1]d 1  [0;1]d 1  34 for all y;y0 2B (x ): (A.7)In what follows we shall give the motivation for the later rigorous de nitionsand calculations.We obtain that whenever ( ~Yt; ~Zt), starting at (x;x0) entersB (x ) B (x ) ata random time, sayT1 and through some random point (y;y0) 2B (x )  B (x )we can decompose the conditional law of ( ~Yt; ~Zt) at time T1 + as follows:Py(! : !( ) 2A)Py0(! : !( ) 2B) =  1(y;y0)(A B) + 2(y;y0)(A B): (A.8)Now we shall successively decompose the law of the process ( ~Yt; ~Zt) at timesTk +  ;k 2 N, where Tk is the  rst time after Tk 1 +  that (~Yt; ~Zt) entersB (x )  B (x ) (see (A.9) below). For instance, if at time T1, (~Yt; ~Zt) entersB (x )  B (x ) through some random point (y;y0) 2 B (x )  B (x ), wedecompose the law of (~Yt; ~Zt) at time T1 + into  1(y;y0); 2(y;y0) as above. Nextwe consider (~YT1+ + ; ~ZT1+ + ), starting in  1(y;y0) resp.  2(y;y0). As  1(y;y0) yieldsa common part, we do not decompose the law of ( ~YT1+ + ; ~ZT1+ + ) startingin  1(y;y0) any further. On the other hand, starting in  2(y;y0), we wait until(~YT1+ + ; ~ZT1+ + ) enters B (x ) B (x ) again, say at time T01 = T2  (T1 + )through the point (u;u0) 2 B (x )  B (x ). T01 is  nite  2(y;y0)-a.s. as  2(y;y0)is absolutely continuous with respect to Py(! : !( ) 2  )  Py0(! : !( ) 2  ) by(A.8).Here recall that (y;y0) was the  rst entrance point of B (x )  B (x ) by(~Yt; ~Zt) under Px Px0 and that (~Yt; ~Zt) started at (x;x0) visits B (x ) B (x )after any  nite time T a.s. Hence ( ~Yt; ~Zt) started at (x;x0) visitsB (x ) B (x )again at some  nite random time T2 = T1 + +T01. Thus, starting in  2(y;y0),we can decompose (~YT1+ +T01+ ; ~ZT1+ +T01+ ) in  1(u;u0) and  2(u;u0). Now iteratethe above.To be more explicit, let U0 = 0 and de ne stopping timesTk = infnt Uk 1 : ~Yt; ~Zt 2B (x )  B (x )oand Uk = Tk + (A.9)for k 2 N and  > 0. Then almost surely Tk <1 for all k 2 N.136By the strong Markov property of the process ( ~Yt; ~Zt) we can condition onFU1 and obtain for n2 N arbitrarily  xed, Px  Px0  ~Yt 2A; ~Zt 2B = Px  Px0  ~Yt 2A; ~Zt 2B;t<Un + Px  Px0 h P~YT1(!)  P ~ZT1(!)  ~Yt T1(!) 2A; ~Zt T1(!) 2B;t U1(!) +Un 1     1(t U1(!))i:Here   denotes the shift-operator   (!( )) = !( +  ).Using (A.8) we can rewrite this as Px  Px0  ~Yt 2A; ~Zt 2B (A.10)= Px  Px0  ~Yt 2A; ~Zt 2B;t<Un + Px  Px0 h 1(~YT1(!); ~ZT1(!))(dw;dw0) Pw  Pw0  ~Yt U1(!) 2A;~Zt U1(!) 2B;t U1(!) +Un 1 1(t U1(!))i+ Px  Px0 h 2(~YT1(!); ~ZT1(!))(dw;dw0) Pw  Pw0  ~Yt U1(!) 2A;~Zt U1(!) 2B;t U1(!) +Un 1 1(t U1(!))i:Using (A.6) and the symmetry of Tk and Uk in (~Y; ~Z) we get in particular that    Px  Px0  ~Yt 2A; ~Zt 2B   Px  Px0  ~Yt 2B; ~Zt 2A     2 Px  Px0 (t<Un)+ Px  Px0 h 2(~YT1(!); ~ZT1(!))(dw;dw0) Pw  Pw0  ~Yt U1(!) 2 [0;1]d 1; ~Zt U1(!) 2 [0;1]d 1 i 2 Px  Px0 (t<Un) + 34;the last by (A.7).Ifn 2 we can further condition the inner probability on FU1 and decompose137the last term in (A.10) into Px  Px0 h 2(~YT1(!); ~ZT1(!))(dw;dw0) Pw  Pw0  ~Yt U1(!) 2A; ~Zt U1(!) 2B;t U1(!) +Un 1 1(t U1(!))i= Px  Px0 h 2(~YT1(!); ~ZT1(!))(dw;dw0) Pw  Pw0 h 1(~YT1(!0); ~ZT1(!0))(dz;dz0) Pz  Pz0  ~Yt (U1(!)+U1(!0)) 2A; ~Zt (U1(!)+U1(!0)) 2B;t U1(!) +U1(!0) +Un 2 1(t U1(!) +U1(!0))i1(t U1(!))i+ Px  Px0 h 2(~YT1(!); ~ZT1(!))(dw;dw0) Pw  Pw0 h 2(~YT1(!0); ~ZT1(!0))(dz;dz0) Pz  Pz0  ~Yt (U1(!)+U1(!0)) 2A; ~Zt (U1(!)+U1(!0)) 2B;t U1(!) +U1(!0) +Un 2 1(t U1(!) +U1(!0))i1(t U1(!))i:Using this in (A.10) we obtain with (A.6)    Px  Px0  ~Yt 2A; ~Zt 2B   Px  Px0  ~Yt 2B; ~Zt 2A     2 Px  Px0 (t<Un)+ Px  Px0 h 2(~YT1(!); ~ZT1(!))(dw;dw0) Pw  Pw0 h 2(~YT1(!0); ~ZT1(!0))(dz;dz0) Pz  Pz0  ~Yt (U1(!)+U1(!0)) 2 [0;1]d 1; ~Zt (U1(!)+U1(!0)) 2 [0;1]d 1 ii 2 Px  Px0 (t<Un) + 34 2;the last by (A.7).By iterating the above decomposition we obtain for n2 N  xed,    Px  Px0  ~Yt 2A; ~Zt 2B   Px  Px0  ~Yt 2B; ~Zt 2A     2 Px  Px0 (t<Un) + 34 n:Recall that we have for almost all (x;x0) w.r.t. ~  ~ that almost surely Tk <1for all k 2 N. Hence, to given  > 0 we can choose n 2 N such that  34 n <  2and then choose T > 0 such that Px  Px0 (t<Un) <  4 for all t  T. Weobtain    Px  Px0  ~Yt 2A; ~Zt 2B   Px  Px0  ~Yt 2B; ~Zt 2A    < 138for all t T and thus thatlimt!1 supA;B2B([0;1]d 1)    Px  Px0  ~Yt 2A; ~Zt 2B   Px  Px0  ~Yt 2B; ~Zt 2A    = 0:Choosing A = [0;1]d 1 yieldslimt!1 supB2B([0;1]d 1)   Px0 ~Zt 2B  Px ~Yt 2B    = 0:A simple tightness-argument completes the proof of our  rst step.Next we shall extend our result L( ~Ytj~Y0 = x) ) ~ as t ! 1 for all x 2 ~S,~ -a.s. to Lebesgue almost every x 2 ~S. The proof goes by contradiction. LetA = fx2 ~S : L(~Ytj~Y0 = x) 6) ~ g.We  rst claim that A is Borel-measurable. Indeed, the martingale problemfor ~Y is well-posed and thus the process ~Y is Feller continuous (see for exampleStroock and Varadhan [4], Corollary 11.1.5). By using that the correspondingsemigroup is a contraction we obtain the claim.Suppose by contradiction that A has positive Lebesgue measure. In thiscase there exists a simply connected bounded open domain D  ~S with smoothboundary such that A\D has positive Lebesgue measure. As ~ (A) = 0 bythe step above, ~ (A\D) = 0 follows. If ~Zt is the stationary solution of theSDE in [0;1]d 1 started with initial law ~ , then EhRT0 1( ~Zt 2A\D)dti=0 for all T > 0. On the other hand, by Theorem B.5 (\Occupation timemeasure for uniformly elliptic di usions") of [1], we have for every x 2 D,EhR D0 1(~Yt 2A\D)dtj ~Y0 = xi> 0, where  D = infft  0 : ~Yt 62 Dg. As ~ assigns positive probability to D, we haveZDE Z  D01(~Yt 2A\D)dtj ~Y0 = x ~ (dx) > 0:By the monotone convergence theorem, we can choose T su ciently large suchthat ZDE"Z  D^T01(~Yt 2A\D)dtj ~Y0 = x#~ (dx) > 0:But the l.h.s. is dominated by EhRT0 1( ~Zt 2A\D)dti= 0, which is a contra-diction. Therefore A has Lebesgue measure zero.It remains to show that L(~Ytj~Y0 = x) ) ~ for all x 2 ~S. Indeed, for x 2 ~S,let  > 0 be such that B (x)  ~S. By Corollary B.3 applied to D = B (x), thetransition kernel  B (x)t (x; ) is absolutely continuous w.r.t. Lebesgue measure.As shown above, for Lebesgue almost every y 2 B (x), L(~Yt+sj~Yt = y) ) ~ ass! 1. By observing that  B (x)t (x;B (x)) " 1 as t ! 0 (see (B.3)), we  nallyget L(~Ytj~Y0 = x) ) ~ for arbitrary x2 ~S, which completes our proof.139Bibliography[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. Poincar e Probab. Statist. (2008) 44, 1038{1077.MR2469334[2] Ethier, S.N. and Kurtz, T.G. Markov Processes: Characterization andConvergence. Wiley and Sons, Inc., Hoboken, New Jersey , 2005. MR0838085[3] Rogers, L.C.G. and Williams, D. Di usions, Markov Processes, andMartingales, 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 Di usion 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, NewYork; Amer. Math. Soc., Providence, Rhode Island, 2001. MR1852999140Appendix BAppendix for Chapter 4The following Lemma and Corollary are necessary to prove Lemma 4.4.1 ofChapter 4.Lemma B.0.1. There exists N0 <1 such that for all N  N0;k  1(a)    k(t)  exp  (1 +o(1))kt26N     C1k exp  (1 +o(1)) kt212N fort (1 +o(1))qN3 ,(b) j (t)j  exp  C t212N for t  6N(1+o(1)) 1=2,(c) There exists  > 0 such that j (t)j  1   for t2  6N(1+o(1)) 1=2; N .Proof. The proof mainly follows along the lines of the proof of Lemma 8 inMueller and Tribe [3]. Some small changes ensued due to the di erent setup.Recall the de nition of  (t) from equation (4.38).For (b), we could not  nd the reference mentioned in [3] but the followingreasoning in [3] based on applying Taylor’s theorem at t = 0 works well withoutit.For (a),  rst observe that  k(t) = E eitSk and use Bhattacharya and Rao[1], (8.11), (8.13) and [1], Theorem 8.5. as suggested in [3]. We used thatE[Y1] = E[Y 31 ] = 0.It remains to prove (c). We have to change the proof of [3], Lemma 8(c)slightly, as we used x6 x. We getj (t)j =      12c(N)N1=2X0<jjj c(N)pNeit jN      =      12c(N)N1=2X0<j c(N)pN2Reheit jNi      =     1c(N)N1=2 Re"eit 1N  eit c(N)pN+1N1  eit 1N#     =     1c(N)N1=2 Re"e it 12N eit 1N  eit c(N)pN+1N 2isin t2N #      1c(N)N1=2     22 sin t2N      :141For 1+ c(N)N1=2  t2N   2 with  > 0  xed we get as an upper bound1c(N)N1=2      1sin 1+ c(N)N1=2        11 + < 1;given N big enough. Finally use that 2 <p6 to obtain the claim.Corollary B.0.2. For N  N0, y 2N 1Z we have    NP(Sk = y)  p (1 +o(1)) k3N;y      C1nN expf kC2g +N1=2k 3=2o;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 tot 2h(1 +o(1))qN3 ; Niif we choose  > 0 small enough. Indeed, usingLemma B.0.1(b) we obtainj (t)j  e C t212N  e C N=312N  (1   )as claimed.142Bibliography[1] Bhattacharya, R.N. and Ranga Rao, R. Normal approximation andasymptotic 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 rangecontact and long range voter processes. Probab. Theory Related Fields (1995)102, 519{545. 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