) = X0( d(i and Mp{E) denotes the set of finite Borel measures on a metric space E and it is endowed with the weak topology. T h e o r e m 1.1 ( S u p e r - B r o w n i a n M o t i o n w i t h B r a n c h i n g R a t e 7). Let SQ denote a unit mass at 0. There is a continuous M f ( I R d ) valued, adapted process Xt defined, on a filtered probability space ( f i ,^" , ( ^ ) , P ) such that j. TP(X0 = 60) = 1 2.1f(t>£ C 6 2 (M d ) , then XM = XoW + f\s^)ds:+Zt{4>), ( M F ) A , 7 11 where Zt(* * G Co°°(]Rm)} U {1} m=l We wi l l work on filtered probability space Q = (Sl,T,\Tt),IP)-The following definition is motivated by Theorem (1.2) and by the preceding discussion. D e f i n i t i o n 1.3 ( H i s t o r i c a l S u p e r - B r o w n i a n M o t i o n w i t h Interact ions) . Let Sd denote the space of symmetric positive definite d x d matrices. Suppose a : [0,oo) x D([0,00),MF(Cd)) x Cd —• Sd, b : [0,oo) x D([0,00),MF(Cd)) x Cd —» R d , 7 : [0,oo) x D{[0,oo),MF(Cd)) x Cd —> (0,oo). 13 A predictable process K G C([0,oo),Mp(C d)) on fl satisfies, (MP)ab-r (with initial condition *o) if V ^ € D o Kt(^) = 8o(^)+Zt(^) + j\j[(V^(s,y),b(s,K,y)) ' j d d i=l J=l . where Zt(i>) is a continuous square integrable martingale with square function • (Z$))t=; j* j\(s,K,y)i>(y)2Ks(dy)ds Vi > 0 a.s. Perkins (1995) has shown that (under suitable hypothesis) the above martingale problem is well posed. We expect the solution to be the limit point of a sequence of renormalized interacting B G W branching Brownian motions, just as in the non-interacting case. In this Chapter we introduce such a sequence of systems of particles. We also show that they are tight and that their limit points satisfy the martingale problem ( M P ) 0 J J , i 7 . Our results are a non-trivial extension of Meleard and Roelly (1990). The main difference is that by working in the historical setting the state space for the particle motions becomes Cd. Therefore the convergence of an approximating sequence KN is no longer completely determined by the convergence of the projections / >-> Ifdhf. 1.2 Interactive Branching Particle Systems 1.2.1 The Particle Picture In this Subsection we define a sequence of processes KN which converges weakly to a solution •of the martingale problem ( M P ) 0 j & ) 7 mentioned in the introduction (see Definition 1.3). We begin by introducing a set of labels I. Let / := U ^ o ^ x {!) 2}1, where by convention { 1 , 2 } ° = 0. For any a = (ao, • • • ,ctk) G J we write |a| = fc and a\i = (ao,.... ,OJJ) for i < fc. If a = (ao,... , ctj) G / we denote ai = (ao,... , aj,i), i = 1,2, and ir(a) - (ao,... , cij-i), j > 1.. Let {£a : a G /} be i . i .d . random variables with P [ £ Q = 0] = P [ ^ a = 2] = 1/2. We say that a subset A oi I is a Bienayme-Galton-Watson tree (or B G W tree) with Af roots iff , (i) {a G / : |a| = 0, a|0 = i for some i < N} C A and if a|0 > N then a ^ A. (ii) for any a £ A such that |a| > 1, fTJ^o"1 a^|t •> 0-The family { £ Q : a G /} induces a unique probability distribution I I o n the set of all trees with N roots.. R e m a r k 1.4. The random variable card(A) is the total number of individuals that ever lived in a critical Bienayme-Galton-Watson process. Therefore card(.4) < oo a.s. • The next ingredient we need is a collection {Ba : a.G /} of independent d-dimensional Brownian motions. Define also a family {ea : a G /} of i . i .d . exponential(l) random variables. We 14 assume that the three collections of Bernoullis, Brownian motions and exponentials are mutually independent. We also assume that they are carried by the same,probability space IP). We shall also need drift, diffusion and branching rate coefficients. For the sake of simplicity, throughout the chapter we shall assume that at time t = 0 all the particles are located at the origin. We could have started distributing the initial particles according to any probability distribution, but we want to keep the notation as simple as possible. A l l the proofs can be easily modified to cover such general initial condition. N o t a t i o n 1.5. If E is a topological space and x G c7([0, oo), E), xl denotes the path x stopped at t: xl = x(tA-). If {X(t, u) : t > 0} is a process taking values in a normed linear space (L, || ||) then XI = sup{||Xs|| : 0 < s < t}. If (M,-M) is a measurable space, bM denotes the space of bounded real-valued Af-measurable functions, and M* denotes the universal completion of A l . Cd = C([0, oo),lR d) is endowed with the sup metric p, (Cd) is its canonical filtration, MF(Cd) is the space of finite Borel measures on Cd with the weak topology. It will also be convenient to append an isolated point d to IRd. • Let. L i p l C ^ - ^ i C ^ R r - M o o . ^ l . l ^ - ^ l ^ ^ y ) V x , y G C d } . The Vasershtein metric d = dp on Mp(Cd) is given by d(u, a') = sup{>(4>) - u\cj>)\ : € Lip(Cd)}. , This metric induces the weak topology on Mp(Cd) (Ethier and Kurtz 1986, p. 150, Ex . 2). Suppose that q > 0. Let o :[0,oo) x MF{Cd) x Cd —¥ B d x d , b :[0, oo) x MF(Cd) x Cd —»• R d , : 7::[6, oo) x D{[0, oo),MF(Cd)jx Cd —> [c, oo). Suppose that v, F : [0, oo) [1, oo) are non-decreasing functions, p is an arbitrary but otherwise fixed positive integer. Y : N x [0, oo) x IR -> ]R is defined by (p, t, x) H-> v(t) xp. We will assume that the maps o~, b, 7 have the following properties: B o u n d e d n e s s by the t o t a l mass sup \\b(t,Kt,y)\\ + sup \\o(t,Kt,y)\\ < T(p,t ,X t *( l )> Viv € D([0,00),MF(Cd)), (1.2) yeC yecd ' sup j(t,K,y) < F(t)(l +fK;(l)ds) VK*

? - (ZN(4>))t) : N € N}, ' {/«,... , O ( Z f ( 0 ) 2 - (ZN(

)2 — (Z(4>))t)t>o is a ^"^-martingale. This concludes the proof of the Proposition. • 37 P r o p o s i t i o n 1.29. •{Z())t= fMs,K,y)4>{y)2Ks{dy)ds; (1.38) Jo St() = J j A{K){$)(s,yJKs{dy).ds. V* > 0 a.s. (1.39) (SN, S were defined ai the beginning of the section.) P r o o f Let • Tj:=:mi{s>0:supK?*(l)A [S(l + K»*(l))du>j}. N . Jo Note that by Lemma 1.15 l i m ^ o o 7) = oo a.s. Moreover 7(-, •,•)!(• < Tj) is bounded. There-fore, for any m,m',n,n' € N rtATj Jo K?(~f(s,Kn, •)0(-)2) - - K f h(s\kn\ •)(• f) tATj ds< / k S M ( ( 7 ^ ^ N , - ) - 7 ( 5 , ^ , - ) ) 0 ( - ) 2 ) Jo 1 , + / K?Ms\ Kn\ .)0(-)2). - K?(7(s, Kn', -)0(-)2) Jo ds We estimate rtATj / . K™((y(s,K»,.)-7(s,Kn',.)) 0 a.s. (1:40) Jo . . n-»oo For every integer t there is a j = j(t) such that t A Tj — t. Moreover, the integral in.(1.40) is increasing as a function of t. Therefore \K?fr{s, Kn, •)(-)%- Ks(7(s,Kr)~~ 0). and (QH,nt) = (nHxcd,ntxcd). T h e o r e m 1.31. Any limit point K of the sequence (KN) satisfies the martingale problem: \/4>eD0 Zt() = Kt(cj>) - (0) - f [ A(K)(4>)(S,y)Ks(dy)ds, t > 0, Jo Jc is a continuous square summable (Tf) martingale such that , (Z())t= f [ j(s,K,y)(y)2ds Vt > 0 a.s. Jo Jcd If in addition we assume that the branching rate 7 satisfies the following condition: • There are h : [0,oo) x ilfj —> IR^ and f : [0, 00) x f2# —> IR which are (Tit)-predictable and satisfy the same Lipschitz condition (1-4) as b, o and \\h(t,K,y)\\ + \f(t,K,y)\ < Y(p,Kf(l)). Moreover, if Q! = (ft! ,T' ,(T[),Q) is a filtered probability space carrying on 39 (Tl)-predictable processes (Kt : t > 0) with sample paths in a.s. and an JR.d-valued (ft)-predictable process (Yt:t> 0) such that fort > 0 Yt = Y0 + Mt + ^Ms^K^ds where M is a continuous local martingale such that (Mx,M^)t = JQt(oa*)(s,K,Y)ijds • a.s., then ^(t,K,Y)=-y(0,K,Y) + f h(s, K,Y)dY(s) + f f(s,K,Y)ds Vt > 0 Q - a.s: Jo Jo then the martingale problem is well posed. . P r o o f . The first part of the theorem follows from the fact that Z?{4>) = '(4>) - (f)(0)-'((f)) together with Propositions 1.28, 1.29 and equation (1.36). The uniqueness part is a consequence of Theorem 5.6 (Perkins 1995.) • R e m a r k 1.32. (a) The additional technical condition needed in Theorem 1.31 to ensure that the martingale problem is well posed is restrictive. However it is satisfied by a large number of interesting examples (Perkins 1995, p. 50). In particular it holds for the branching rate given in Example (a) (page 16). (b) In Theorem 1.31 we don't need the full strength of condition (Lip,) to prove that the limit points of the sequence (KN) satisfy the martingale problem. Continuity (as opposed to Lipschitz continuity) and boundedness by some power p > 1 of the total mass suffice. • 40 Chapter 2 Path Properties of a One-Dimensional Superdiffusion with Interactions 2.1 Introduction and Statement of Results In this Chapter we study some path properties of the solution of the historical stochastic equation (Perkins 1995, p. 47) in dimension d = 1. H is a one-dimensional historical Brownian motion and X is a superprocess with interactions. (2.1) shows that X and H have the same family structure. However, the path y (a Brownian path) is replaced by a path Y(y) which is a Brownian motion with a drift depending on X. The term 7 is a mass factor, but under suitable hypothesis (see condition (Ay) in page 44) it can be interpreted as a branching rate (Remark 5.2 of Perkins 1995). A n intuitive explanation of (2.1) is given in Chapter 0. The present chapter is organized as follows. In this Section we review some basic results concerning historical stochastic analysis (Perkins 1993, 1995). We define what we mean by a solution of (2.1) and state Theorem 2.9, the main result of the chapter. The reader is referred to Chapter 0 for additional comments and motivation. In Section 2.2 we introduce and examine some auxiliary processes. These results, as well as their proofs, wi l l be needed in the proof of the main theorem. In Section 2.3 we find a generalized Green's function representation of a localized version of the process X analog to (0.4). W i t h this formula we shall be able to estimate the moments of X. In Section 2.4 we prove Theorem 2.9 using the results of the previous sections together with Kolmogorov's criterion. Finally* in Section 2.5 we give a non-trivial example in which the techniques developed in this Chapter can be successfully applied to discover some path properties of a one-dimensional superdiffusion in a random environment. 2.1.1 Main Result We begin by introducing some basic notation and recalling the definition of historical Brownian motion. (2.1) 41 If \i is a measure, /j,((j>) = f dfi. Throughout this chapter C = C1 = C([0, oo),lR) and Ct = C\ = a(ys : s < t), the canonical a-field of C. If y G C we write y* for the path y stopped at t, i.e. yl = y(t A •). Let MAC)1 = {ixeMF{C):y = yt / i - a.a. y}, t>0. . , fiH = { ^ e C ( [ 0 , o o ) , M F ( C ) ) : i i : t G M F ( C , ) ' V*>0}, ift(u;) = u>(t) for w G fi//, •Ht = o(Hs:0~~~~0). fi = ( f i , .F , (Tt),TP) wil l denote a filtered probability space satisfying the usual hypotheses. PifFt) denotes the cr-field of (.^-predictable sets in [0, co) x fi and (fi , :F, (Tt)) = (fi x C,T x C, (Tt x F ix 0 < h < t2 < ... < tk and # € ^ ( M * ) . If V e C, let * ( y ) = * - ( * i , * 2 , ' . . . , * f c ) ( y ) = . * ( y ( t i ) ; . . . , y ( * * ) ) -and denote the first and second order partials of fc-i V * ( t j y ) = E l ( i < « t + i ) * . i + r ( y ( t A « i ) , . . . , y ( t A t f c ) ) ; . . i=0 ^ fc - l fc-i . - # ( t , y ) = E 53l(«< W i A ' t j + i ) * m + u + i ( y ( t A t i ) , ; . . , y ( < A « f c ) ) . m=0 i=0 Let oo L>0 = ( J { * ( t l , * 2 , - , - * m ) : 0 < t i < t2 < - < . < m > * G C 0°°(IRm)} U {1}. ro=l D e f i n i t i o n 2.1 (One-dimens ional H i s t o r i c a l B r o w n i a n M o t i o n ) . Let m be a finite Borel measure on JR. A predictable process (Ht : t > 0) on fi with sample paths on fi# is a one-dimensional historical Brownian motion starting at m iff (Ht) satisfies the following martingale problem: (MP)TAAo is a continuous square integrable ^i-martingale such that (Z?$))t = fi j' $(y)2Hs(dy)ds Vi > 0 a.s. There exists a process H satisfying the conditions of Definition 2.1 and it is unique in law. (Dawson & Perkins 1991). We picture H as an infinitesimal tree of branching one-dimensional Brownian motions. 42 D e f i n i t i o n 2 .2 . Let (Kt : t > 0) be a process o n f i with paths on fi#. A set A C [0, oo) x fi is (K,JP) evanescent (or K evanescent) iff A C A i where A i is (JFt* )-predictable and sup IAJ {U,UI, y) = 0 Kt — a.e. y Vt > 0 P — a.s. 0~~__ 0) bea process on fi with paths on fi//. A map b : [0,oo)xfi —»• JR is if-integrable (respectively K-locally integrable) iff it is (Pf )-predictable and / 0 ' Ks(\b(s)\)ds < oo (respectively, f£ \b(s)\ds < oo Kt — a.a. y) Vt > O P — a.s. a For technical reasons that wil l become apparent later we shall not look directly at (2.1) but rather at a historical version of it. We shall call such version (HSE)b^ and it is defined below. D e f i n i t i o n 2 .4 . Suppose that H is a one-dimensional historical Brownian motion starting at m € MF(JR). Let b : [0,oo) x MF(JR) x IR—> JR, 7 : [0,oo) x MF(SR) x C—> (0, oc). ' Define the projection Ut : fi/f -> MF{Wi) by Ui(K)(A) = Kt({y : y(t) € A}). We say that the pair (K, Y) solves the historical stochastic equation (a) Yt = yt+ f b{s,fls(K),Ys)ds, t > 0, ;° (HSE)bn . ( b ) . ' KM) = j 0(y*)7(t,n.(A'),r)frt(dj/)i ebc. iff • y : [0, co) x fi -¥ JR is (^ t*)-predictable and K : [0, oo) x fi ->• M F ( R ) is (Ji)-predictable. • The map |6(s,u;,j/)| = \b(s,U.s(K)(u),Ys(!y{a) holds up to an i7-evanescent set in [0, oo) x fi, (HSE)bty{b) holds for all cf> G bC Vt > 0 P - a.s. , Pathwise uniqueness holds in (HSE)bty if whenever (K, Y) and [X,Y) are solutions of (HSE) then y* = Yl except on an if-evanescent set in [0, oo) x fi and K = K except on a P-evanescent set in [0, oo) x fi. To each K solution of (HSE)},^ we can associate its one dimensional projection X defined by X. = f l . (A') . ' o R e m a r k 2 .5 . It is easy to modify the above definition in order to give a rigorous description of (2.1). It is then straightforward to verify that if K solves (HSE)bn then its projection X. = fl.(K) solves (2.1). - " The existence and uniqueness of solutions to (HSE)ytb requires some Lipschitz conditions on the coefficients b, 7. Since their arguments include measures we need to introduce an appropriate metric. 43 D e f i n i t i o n 2.6 (Vasershtein M e t r i c ) . If (S, 5) is a metric spacelet - Lip(S) = {(p : S -> JR : < 1, - 4>{z)\ < 6(x,z) V x , z € 5}. The Vasershtein metric is the metric oh M F ( 5 ) defined by ds(u,y) = snp{\u(cp)-i>{(f))\: cpeLip(S)}. This metric induces the weak topology on Mp(S) (e.g. Ethier and Kurtz 1986, p. 150 Exercise , 2). - . - . . • • , ,. " • " . The following conditions on the coefficients appearing in (HSE)bn wil l be frequently required. (IC) m(dx) = p(x)dx, where p£Cj<(JR) is H61der-d for any a < ^ , / R p(x)o?x = 1. There,is a nondecreasing function Y : :[0, oo) . [1, oo) such that (Lip) For any u£ M F ( B ) |&(t,/z,x)| < T(t V/x(l)) V x € R . -"" (2.2) * Let d e denote the Vasershtein metric on M F (1R) . / \b(L/z, x) - b(t, v., 2 )| < Y(t V p(l) V i/(l))(de(/i,u).+ \x - z\). (By) For any X € C([0, oc), Mp(JR.)) and any Vy € C • 7 ( t , X y ) - 1 < T ( t V A 7 ( l ) ) . Moreover, there is a nondecreasing functionT : [0, oo)• —> [1, oo) such that • l(t,X,y) < T{t) (l + pX;(\)ds^ , VA' € C([0,oc),MF(JR)) Vy € C . . • The following Lipschitz condition on 7 is more restrictive than (Lip). (Ry) Suppose that there are two functions h, f : [0,oo) x M F ( M ) x C —¥ JR such that / !/t(<,//,y)| + | / ( t , / / , y ) i < T ( t V / i ( l ) ) V y G C , : . ' '(2.3)' |/ ((/,/z,y) - />(t.,/,y')| < T(t V p( l ) V p'(l))(d e(/i,/i ') + sup |y(s) - y'(a)|). (2.4) ' |/(t,j/,y) - / ( t , / / , y ' ) | < T(t V p( l ) V M ' ( l ) ) (^( / i , / i ' ) + sup |y(.s) - y'(.s)|). ' (2.5) and with the following property. Recall that fit : —> M F (1R) be given by JJt(K)(A)=Kt({y:y(t)eA}). . If a s p a c e d ' — ( f i^ .F ' , (.?>), P ' ) carries an (^-predictable process (Kt) with sample paths in f2# a.s. and an JR-valued (^"^-predictable process (Yt) such that M(t) = Y(t)-y(o)- / 6(s,n,(^),ys)dS, t>o tion then . , 7(t,fl(A'),Y) = 1+ f h(s,U(K),Y)dY(s) + [* f(sM(K),Y)ds V i > 0 P ' - a . s Jo Jo is a Brownian mo 44 R e m a r k 2.7. Condition (R^) seems strange at a first glance. It ensures that 7 can be inter-preted as a branching rate (Remark 5.2 of Perkins 1995). • The next Proposition is fundamental. P r o p o s i t i o n 2.8. Assume b satisfies (Lip) and 7 satisfies (B,), (R-f). Then (HSE^^) has a pathwise unique solution. P r o o f . See Theorem 4.10 of Perkins (1995). • Before stating the main result of this Chapter we give some examples of coefficients 6 and 7 satisfying the basic hypothesis. E x a m p l e , (a) 7 = 1 satisfies (fly) and (JB7). (b) Assume 7 € C^'2([0,00) is a non-negative function such that f^(s, •) and §^(s, '-) are Lipschitz continuous with a uniform Lipschitz constant for s in compacts. Then j(t, X,y) = j(t,y(t)) satisfies (R,) and (f? 7). This is proved in Example 4.4 Perkins (1995). In this case the mass of each particle depends only on time and the position of the particle, and not on the rest of the population. (c) Let ps(x) be the one dimensional heat kernel. Fix e,a > 0 and set 7(t, X, y) = exp ( - j ' Jpe(yt - x)Xs(dx)e-a^ds). It is not obvious that 7 satisfies (R-,)- This is proved in p. 51 of Perkins(1995). (d) Let b : IR —> IR be bounded Lipschitz continuous. Then b(p,x) — Jb(x — z)fj,(dz) satisfies (Lip). See Example 4.2 Perkins (1995) for a proof. In this model a particle at z in a population p, exerts a drift b(x — z)fi(dz) on a particle at x. (e) More examples can be found in Perkins (1995, p.49). •'" ' • The following is the main result of this Chapter. T h e o r e m 2.9. Let H be a one dimensional historical Brownian motion.starting dt m. Assume m satisfies (IC), b satisfies (2.2) and7 satisfies (B7) and (Ay). Let K be a solution of (HSE)^^ and suppose that X is its one-dimensional projection, i.e. X. = fl.(K). (See Definition 2.4-) Then Xt(dx) = u(t,x)dx for all t > 0 IP - a.s. (2.6) where u(-,-) is an a.s. jointly continuous (adapted) function. In addition, u is Holder continuous in t with any exponent a < 1/4 and in x with any exponent a < 1/2 a.s. R e m a r k 2.10. (a) The case 6 = 0 and 7 = 1, i.e. when X is one-dimensional super Brownian motion, of Theorem 2.9 was proved independently by Reimers (1989) and Konno and Shiga (1988). (b) Note that b is not assumed to satisfy (Lip) (we only assume the boundedness condition (2.2)) so pathwise uniqueness may not hold. • 45 2.1.2. Historical Stochastic Calculus The results in this Subsection will be used repeatedly. The basic reference is Chapter 2 of Perkins (1995). D e f i n i t i o n 2.11. Let (E, \\ \\) be a normed linear space. Let / : [0,oo) x fi —> E. A bounded (Jt)-stopping time T is a reducing time for / iff l(t < T)\\f(t, u>,y)\\ is uniformly bounded. The sequence {Tn} reduces / iff each Tn reduces / and Tn | oo P — a.s. If such a sequence exists, we say that / is locally bounded. - • D e f i n i t i o n 2.12. Let m be a finite Borel measure on TR and let 7 : [0, oo) x fi ->• (0, oo), b : [0, oo) x fi -¥ P , g : [0,• oo) x f l TR, be (jFf*)-predictable. Assume that A = (7,6,g7 _ 1 l(g 7^ 0)). is locally bounded. A predictable process (Kt:t> 0) satisfies (MP)™ j . pn if and only if Kt .<= MF(C)1 for all t > 0 a.s. and V ^ G J D Q Z?$) = Kt$)-m($) is a continuous square integrable ^-martingale such that (Z«$))t = fi Jy(s,u>,y)i>(y)2Ks(dy)ds Vt > 0 a.s. a Assume that K satisfies (MP)™ ~ . Let (Tk) be a reducing sequence for A. If n £ Do, 4>n (f) (bounded pointwise convergence) then dominated convergence and local boundedness of 7 imply __

e C%(1R.) jt{4>)' = f Pt(x)p(x)dx + j j Pt.s(j>(ys)£s(l,u,y)ZR(ds,dy) (2.12) for all t e [0,1] P - a.s. P r o o f . F ix t < 1 .and set ip(s,x) = Pt-S (t,yt) = ip(P,Y0) + J i-£(s,ys) + -^(s,ya)j d s + J .-^(s,ys)dy(s) = rb(0,Y0) + fi ?jt(s,ys)dy(s) for -Kj-a.e. y Vt G [0,1] a.s., (since —(s,x) = --^(s,x)). Moreover, by the same lemma £t(l,u,y) = 1 - £s(l,u),y)b(s,uj,y)dy(s) + b(s,uj,y)2£s(l,u,y)ds 54 for Kt-a..e.-y Vt € [0,1] a.s. Integration by parts, justified once more by Ito's lemma 2.17 yields £T{\,LO,y)^{t,yt) = ^(0,y 0 ) + J (£s(l)^(s,ys) - ^{s,ys)£s(\)b(s))dy(s) + ( 6 ( 5 )2 ^ ( 1 )^ (5 ,^ ) - 6(5)^(1)1^(5,^))^ for Kt-&.e. y Vt € [0,1] a.s. Now apply historical Ito's lemma 2.18 to obtain J £t(l)y>(t,yt)Kt(dy) = j 1>(0,x)p(x)dx + J j5,(1,L0,y)^{s,ys)ZK{ds,dy) Vte'[0,1] a.s. Recalling the definition of ip and jn, we see that this last equation is exactly (2.12). • R e m a r k 2.33. If b = 0 and 7 = 1 then Proposition 2.32 provides a very simple proof for the usual Green's function representation of super Brownian motion (compare with that of Konno and Shiga 1988, p 212). P r o p o s i t i o n 2.34. (a) For each 0 > 0 let £t(6,LO,y) = exp (-6IR(b,t,u,y) + (o - y ) j f 0(5,LO,y) 2 ds) . Then for any 6 > 0, £sA.(6) is a P s -martingale starting at 1. (b) There is a function K : [l,oo) x [0,00) —> R.+ , non-decreasing in each variable such that Ss(l)p <*(p,s)£s(p) • (c) The second term on the r.h.s. of (2.12) is a square integrable martingale null at zero. P r o o f , (a) Recall that by Proposition 2.14 nt = yt — yQ — flb(s,LO,y) is a P^ -Brownian motion. Since b is bounded, £sA.{0) is an exponential martingale. (b) We estimate £s(l)p = £s{p)e{^-p)I°ku)2du <£s{p)es^-p)cBu = K(p,s)£s(p). (c) Note that HPt-s^lloo < 00. Moreover P p Ks(£s(l)2)ds^ < pP*[£s(2M2,s)}ds -0 P - a.s, (2.25) Using (2.24), (2.25) and Ito's lemma.for historical integrals 2.18 we obtain (2.23) •. 2.4 Proof of the Main Result In this section we put together the results from Sections 2.2, and 2.3 to prove Theorem 2.9. We begin with some technical lemmas. -L e m m a 2.43 ( K o l m o g o r o v ) . (a) Let [B(x) : x G P d ) be a family of random variables in-dexed by x € P d . Suppose thai there exists a real p > 0 and two constants Co, $ > 0 swc/i that •-'•r . V x , x € P d , E[jB(x) - £(x)|p] < C0|jx - x\\d+3.. ' Then the process (B(x) : .x € P d ) has,a continuous version which is globally Holder with exponent a, for any a < (3/p. • (b) Let I C P . 3 be the product of 3 intervals (either closed, open or semi-open). Let [B(x) : x £ I) be a 3-dimensional random field. Suppose that forany k > 13 there is a, constant Co > 0 . such that " ' ' E[|B(xi ,x 2 ,x 3 ) - B(xi,x3,xi)\k] < Co(|x i - x i | | ^ V | x 2 - x 2 | ^ + |x3 - x 3 | V ) . for any. ( x i , x 2 , x 3 ) , (x i ,X2 , x 3 ) G I such that 0 < |xi — xi|,|x2 —x2|,|x3 — x3| < 1. Then the process (B(x) : x € I) has a continuous version, which is Holder, with exponent a, for any a < 1/4. Moreover, for any X2,x3 fixed, the map x\ i-4 i? (x i ,x 2 , x 3 ) is Holder-a for any a< 1/4 and the map x 2 >-» l ? (x i ,x 2 ,x 3 ) is Holder-a for any a < 1/2. If we know that the process (-B(x) • x € P d ) is continuous to begin with then there is no need to take a version. 63 P r o o f . Although Kolmogorov's theorem is not generally stated in this form, the standard proof (Revuz and Yor 1991) works equally well. • L e m m a 2.44. (a) Denote p'(t,x) = Dxp(t,x). For any 0 < e < 1, t € [0,1] rt roo I \p'(t — s,x + e) — p'(t — s,x)\dxds < C2AA.iy/s. JO J-oo '(b) Let 0 < s < t < 1. Then rs roo I \p'(t~ r i x ) —p'(s ~ r,x)\dxdr < c2.44.2Vt — s. Jo J-oo (c) Let T{n) := mf{t > 0 : Ht(l) > n} A 1. There is a function 0 : N x [0,oo) —• 1R+, non-decreasing in each variable such that X;(l) < e{n,s) on {T(n)>s}. Notice also that K* (1) =X*(1). P r o o f , (a) Estimates (a) and (b) should be well known. We prove them since we don't know a reference. We need the following elementary estimate (e.g. Ladyzenskaja 1968 p. 274) ' \D?D?p{x,t)\.^Cn^-^-^exp^-Cn^y) . (2.26) Let 8 > 0, 8 < t. We estimate rt roo ft-S roo / / \p'(t - s,x + e) - p'{t - s,x)\dxds = / / \p'(t-s,x + e)-p'{t-s,x)\dxds Jo J-00 ' Jo J-00 rt roo + \p'[t - s,x + e) — p'(t - s,x)\dxds • Jt-S J-00 • ' rt—S roo < / \p'(t - s,x +.e) -p'(t - s,x)\dxds -. Jo J-00 rt roo + 2 / \p'(t- s,x)\dxds Jt-S J-00 = Il+h. Use estimate (2.26) to check that rt. ^ -«$ y/t - S < Ci\T8. J 2 < C ft ~T^=ds 64 To estimate I\ we use the fundamental theorem of calculus followed by a linear change of variables, rt—6 roo . rl h = / / £ / D2p(t - s; x + ze)dz JO J-oo' Jo dxds ft—6 rl roo . <£ / / \D2p(t-s,x + JO JO J-oo ze)\dxdzds (by Fubini) ft-8 rl < e I / dzds Jo Jo t-s (using estimate (2.26)) ^ e l o g Q ) . __ 0. Let CT : M F ( P ) x P — > [e,oc), • b : A f F ( P ) x P — > P , 7 : [0,oo) x C ( [ 0 , o o ) , M F ( P ) ) x C - 4 (0,oo). The hypothesis on these coefficients are the following. Here d = d|.| is the Vasershtein metric on A//.-(P.) defined by : -d(p,u) = sup{|M(/) - K / ) ! : : / : P ^ P , H/l loo < 1, 1 / ( 4 - / ( y ) l < > - ^1 Vx ,y G P } . . ,. 81 B o u n d e d n e s s b y t h e t o t a l m a s s . There are non-decreasing functions T, T : [0, oo) -> [l,oo) such that S U P \b(Pix)\ + W(p,x)\ < T(/x(l)). Vp € M f ( l R ) , (3.3) s u p 7 ( f , X , y ) < T ( t ) ( l + PxiMds) V *eC ( [ 0 , o o ) , M F ( P ) ) . (3.4) S/6C1 JO . L i p s c h i t z c o n d i t i o n . |CT . ( /* ,S)-<7(I/,Z)|-H&(^ (3.5) The reader wi l l find many examples of coefficients b, a, 7 satisfying the conditions above in Chapter 1 and in Chapter 4 of (Perkins 1995). We say that (X, Y) is a solution of (HSE)a^y Yt = Y0+ f a(Xs,Ys)dy(s) + ['b(Xs,Ys)ds, (i) Jo ' Jo . • • ' '. Xt(cP) = j ^X^iYtWdy). (ii) iff • Y : [0, 00) x Q, ->• ]R is ( .^-predictable, X : [0, 00) x ft -> M F ( P ) is (.^-predictable, X € C([0,oo),M/.(IR)). • (HSE)ffti,n(ii) holds for all : TR —> TR bounded measurable for all t > 0 P — a.s. and (HSE)a,b,-y{i) holds H - a.e. The stochastic integral in the r.h.s. of (HSE)Ctt,y(i) was defined in Proposition 2.15. Here is the main result of this chapter: T h e o r e m 3.3. Let X be a solution of (HSE)a^a. Then X has a local time Lf(X). The basic tool needed in the proof is a Tanaka-like formula of Perkins (1995). We shall not need its more general form. We will employ the following version. T h e o r e m 3.4. Assume L : [0,oo)xf2 —> [0,oo) is (P?)-predictable, L(-,u,y) is non-decreasing, left continuous for Ht-a.e. y for all t > 0 P — a.s. Also assume f HS{L2S) ds < 00 V O 0 P - a.s. Jo Then there is an a.s. non-decreasing, left continuous [0, oo)-valued (Tt)-predictable process At(L) = J0tHs(dLs) such that Ao(L)=0 and Ht{Lt) = HoiLo) + f fL(s,u,y)ZH(ds,dy) + f Hs{dLs) Vt > 0 a.s. Jo J Jo Moreover if L is continuous H-a.e. then At(L) is a.s. continuous. 82 Proo f . This is a special case of Theorem 2.24 (Perkins 1995). • We illustrate the idea of the proof of Theorem 3.3 with the particular example of super Brownian motion. Recall that if T is a bounded (Tt) stopping time, Bt{w,y) = yt — Vo is a Brownian motion stopped at T on (fi x C,T x C,TPT) (see Proposition 2.14). Let Cf(cv,y) be its local time. C is normalized so that /0* cj){Bs)ds = / R d ( a ) £ " d a . Applying Theorem 3.4 with Ls = we get Ht{Cat)= f I' £asZH(ds,dy) + f Hs(d£as). (3.6) Jo J Jo The second term on the right hand side of (3.6) is precisely the local time of super Brownian motion. This is intuitively clear from the particle picture. A straightforward computation shows that Lf(W).= J0tHs(dCg) satisfies (3.2). Using the representation of the local time furnished by (3.6) and Koimpgorov's criterion we are able to prove that the local time is indeed jointly (Holder) continuous. R e m a r k 3.5. (a) Another way to verify that L\(W) as defined above is in fact the density of occupation is the following. For simplicity we consider only the local time at 0. Let be the A-Green's function • Rx(x) = / e~xt^= dt. Jo By Ito's lemma, the local time of Brownian motion satisfies the Tanaka formula (see e.g. p.14 Adler 1992): C°S = Rx(Bo)-Rx(Bt)+ [\Rx)'(Bs)dBs + X f Rx(Bs)ds.. (3.7) Jo Jo Now, Ht{£t) m a y be computed using the representation (3.7) for C° together with Ito's lemma for historical integrals (Proposition 2.18.) It yields (formally) HT(C°T) = -HT(Rx(yt) - Rx(y0))+ f f [*(Rx)'(yr)dyrZH(ds,dy) Jo J Jo + X f [ I*Rx(yr)drZH(ds,dy) + \ [ f Rx(ys)Hs(dy)ds Jo J Jo Jo J = ~ fo !{RX{Vs) - RX^ZH^ ^ - \ [ j{RX)"(ys)Hs{dy)ds + f f fS(Rxy(yr)dyrZH(ds,dy) + X f f [' Rx(yr)drZH(ds,dy) Jo J Jo Jo J Jo + A^Y Rx(ys)Hs(dy)ds . = J* j (Rx\yo) -Rx{ys) + J\Rx)'(yr)dyr + \ J° Rx(yr)d^J ZH(ds,dy) + j* J (\Rx(yS)-L-(Rx)"(ys^Hs(dy)ds 83 = fi jc°sZH(ds,dy) + fi J (\Rx(ys) - ^(Rx)"(ys)SJHs(dy)ds Hence Ht(C°t)= f [C°sZH(ds,dy) + f Hs(S0)ds (3.8) Jo J Jo (since Rx - ^ ( i ? A ) " + XRX = 60). Comparing (3.6) and (3.8) we see that /„* Hs{dC°a) = fQtHs(S)ds. b) A l l the arguments in this Subsection concerning super Brownian motion are easily made rigorous. However, all these results are elementary and their proofs may be found elsewhere. c) Local time for super Brownian motion is known to exist in dimensions d < 3. The analogous result for interacting superprocesses is currently under investigation. The cases d = 2,3 are harder than the case d = 1, since super Brownian motion does not have, a density in dimensions greater than one. o This chapter is organized as follows. Section 3.2 is devoted to the proof of Theorem (3.3). We define a family of random variables {L^(X)\a € P c , t € P + } and show that it satisfies the conditions of Definition 3.1. 3.2 Existence and Regularity of Local Times Recall that for any bounded (Jr()-stopping time T the process Y defined by (HSE)c^yl is JJ a semimartingale on the space Q = ( f i x C, (Tt x C ^ P ^ ) . We wish to define an (Tt x Ct)-predictable process £"(w,y) which agrees with the semimartingale local time of Y in fi P T —a.s. for all bounded ( .^-stopping times T. We use Tanaka's formula for semimartingales. Definition 3.6. For any a € JR and t > 0 let Cas = \Yt-a\- \Y0 - a\ - f sgn(y s - a)a(Xs, Ys)dy(s) Jo -f sgn(Ys-a)b(Xs,Ys)ds. (3.9) Jo * . ' • .•; Note that all the terms in the left hand side of (3.9) are defined up to (H, P)-evanescent sets in [0, co) x fi. Also, / 0 ' (p(Ys)d(Y)s = /(/>(a)£"da H - a.e. in [0, oo) x fi (see Section 1 of Chapter V I of Revuz-Yor (1991)). ' • We shall need the following technical results. Lemma 3.7. Let T(n)=mi{s>Q:Hs{l)>n}. (3.10) Then there is a function 0 : N x [0, oo) —¥ P + , non-decreasing in each variable such that X*s(l) < 0(n,s) on {T(n)>s}. (3.11) 84 P r o o f . This statement is proved in p. 61 of (Perkins 1995). In the notation therein 0 ( n , s) = Ti{n)eTl^s. The hypothesis (3.4): on 7 is needed here. • R e m a r k 3.8. In our present setting it is not necessarily true that for p € N , E[AT*(1)P] < oo. o L e m m a 3 .9 . Assume T is a bounded (Ft)stopping time and 'ip € b^TAs- Then /•(AT r HTMW) = HTASW) + hp ZH(dr,dy) Vt>s a.s. JsAT J P r o o f . This is a particular case of Proposition 2.7 of Perkins (1995). • L e m m a 3 .10 . Let L : [0, oo) x —• [0, oo) be {Tt)-optional, and let S be an (Tt)-stopping time. Then P[L*(S) >e] = sup-F[L(T) >e] Ve > 0, T o-an + 3 c 2 P rsAT(n) / 0-(Xr,Yr)'' Jo dr + 3 P H sAT(n) sAT(n) b(Xr,Yr)dr (by Burkholder) < 3 p f A T ( n ) [ | y 0 - a | 2 ] + c 2 P rsAT(n) • / T ( X r ( l ) ) : Jo dr 85 - H I rsAT(n) 1 \ + P S A T ( n ) Jo ?(Xr(l))dr2 j (by (3.3)) .< 3 hsxnn)[\Yo - oi 2 ]+c 2p;;r („) ^ r „ . | 2 l ~ H rsAT{n) / T ( G ( n , r ) ) 2 d r Jo + P H sAT{n) rsAT(n) / T(6(n,r ) )dr JO (by Lemma 3.7) . < 3 (VsAT(n)[\Yo -a| 2 ] + c 2 S T ( 0 ( n , S ) ) 2 + s 2 T ( Q ( n , s)) 2 ) C3.12.1 (*,")• Similarly, P if, sAT(n) r-sAT(n) / s g n ( y r - a ) a ( X r , y r ) d y ( r ) Jo < P / rsAT(n) ; HSAT(n) l y a(X r ,y r ) dr = C3.i 2 . 2(5,n). The same reasoning also shows that ftAT(n) P i7 sAT(n) rtAl(n) / s g n ( y s - a ) 6 ( X „ y s ) d 6 Jo < c 3 . i 2 . 3 (s ,n) . As a consequence of (3.9), (3.13), (3.14) and (3.15) we obtain sup P 0 0 and restrict ourselves to the time interval [0,N]. As usual, some localization procedure is needed. We shall stop the processes H, X, Y and C?. at T(n). Note that £ r ^ A . 1S ^ o c a ^ t * m e °* Y?(n) (in Campbell space.) Also, Z^T^ is the orthogonal martingale measure associated with HT(n\ We shall need the following estimates. < L e m m a 3 . 1 2 . For any p > 1 there are constants C3. i2 . i (n ,N,p ) , 0 3 . 1 2 . 2 (n, TV, p), 03.12.3(^1,TV,p) , such that for any (Ft)-stopping time T < T{n) A TV and any x,z € IR, s, t 6 [0, TV] and sup P T [ ( £ £ ) * ] __

{a) j CatAT[n)HtAT[n){dy)da = J J cp(a)C1AT{n)daHtAT{n)(dy) (by Fubini) = J-J 4>{Ys)o2{Xs,Ys)dsHtAT{n){dy) (by the density of occupation formula). We can apply Ito's lemma for historical integrals to this last H-integral to obtain: ftAT(n) n ptAT(n) f ps (3.42) r rt l( ) ptAT(n) f JJ

(Yr)o2(Xr,Yr)drZH(dsdy) ptAT(n) r + Jo J **{a)da. We need to show that P r rtAT(n) r J Jo J (Ca^T(n))2H°mn)(dy)dsti(a < ob. This is easy to check: P j Jo yr(^ A T ( n ) ) 2 /7 , A T ( n ) (dy)d S /i(da) < J J Q VSAT(n)[(£aSAT(n))2}ds»(da) < A ^ ( P ) c 3 . 1 2 . i ( n , / V , 2 ) < oc (by Lemma 3.12). Thus (3.39) holds. . P r o p o s i t i o n 3.19. Lf(X) satisfies the following density of occupation formula. For any non-negative Borel function (a)Lat{X)da a.i P r o o f . By considering functions of the form 4>(a)T^s^R(uj) it follows easily from a monotone class argument and (3.39) that Ji){s,a,uj)Lads{X)da = j j' o-2{Xs,a)^{s,a,u)Xs{dx)ds (3:45) for any measurable ib.: R + x R x Q —> P + . Substituting ip(s,a,oj) = o~~2(Xs,a) (a)Xs(da)ds = f 4>{a) [ o--2{Xs,a)Lads{X)da Jo J Jo = J cp(a)Lat(X)da. (by (3.18)) -P r o o f of T h e o r e m 3.4. Propositions 3.16 and 3.19 show that the family of random variables (Lf(X)) satisfies the conditions (i) and (ii) of Definition 3.1. • 99 Bibliography [1] Adler, R . J . (1992). Superprocess local and intersection local times and their corresponding particle pictures, in Seminar on Stochastic Processes 1992, Birkhauser, Boston. [2] Barlow, M . T . , Yor, M . (1981). Semimartingale inequalities and local times, Z. Wahrschein-lichkeitstheorie verw. Gebiete 55, 237-254. [3] Dawson, D . A . (1993). Measure-valued Markov processes, Ecole d'ete de probabilites de Saint Flour, 1991, Lect. Notes in Math. 1541, Springer, Berlin. [4] Dawson, D . A . , Gartner, J . (1987). Large deviations from the McKean-Vlasov limit for weakly interacting diffusions, Stochastics #0,247-308. [5] Dawson, D.A\, Perkins, E . A . (1991). Historical processes. Mem. Amer. Math. Soc. 454. [6] Dawson, D . A . , Perkins, E . A . (1996). Measure-valued processes and stochastic partial dif-ferential equations. Preprint. [7] Dellacherie, C . , Meyer, P .A . (1982). Probabilities and Potential B, North Holland Mathe-matical Studies No. 72, North Holland, Amsterdam. [8] Durret, R. (1985) Particle systems, random media, large deviations, Contemporary Maths. 41, Amer. Math. Soc, Providence, R. I. [9] Ethier, S .N. , Kurtz , T . G . (1986). Markov processes: characterization and convergence, Wiley, New York. [10] Hart l , D . L . , Clark, A . G . (1989). Principles of population genetics, second edition, Sinauer Associates, Inc., Sunderland, Masachussetts. [11] Jacod, J . , Shiryaev, A . N . (1987). Limit theorems for stochastic processes, Springer-Verlag, New York. [12] Konno, N . and Shiga, T . (1988). Stochastic differential equations for some measure valued diffusions, Probab. T h . Rel. Fields 79, 201-225. . [13] Ladyzenskaja, O . A . , Solonnikov, V . A . and Ural'ceva, N . N . (1968). Linear and quasilinear parabolic equations of parabolic type, Transl. Math. Monographs Vol 23, Amer. Math. Soc. [14] Le Gal l , J . F . Perkins, E . A . , Taylor, S. J . (1995) The packing measure of the support of super-Brownian motion, to appear in Stoch. Process. Applications. [15] Meleard, S. and Roelly, S. (1990). Interacting measure branching processes and the associ-ated partial differential equations., Stochastics and Stochastic Reports. [16] Perkins, E . A . (1988) A space-time property of a class of measure-valued branching diffu-sions, Trans. Amer. Math. Soc. 305, 743-795. [17] Perkins, E . A . (1993) Measure-valued branching diffusions with spatial interactions, Probab. T h . Rel. Fields 94, 189-245. 100 [18] Perkins, E . A . (1995). On the martingale problem for interactive Measure-Valued Branching Diffusions, Memoirs of the Amer. Math. Soc. No. 549, 1-89. [19] Reimers, M . (1989). One-dimensional stochastic partial differential equations and the branching measure diffusion, Probab. T h . Rel. Fields 81, 319-340. [20] Revuz, R . J . and Yor, M . (1991). Continuous martingales and Brownian motion, Springer-Verlag, New York. [21] Rogers, L . C . G . and Williams, D . (1986). Diffusions, Markov processes and martingales, Vol 2., Wiley, New York, [22] Shiga, T . (1994). Two contrasting properties of solutions for one dimensional stochastic partial differential equations, Canadian Journal of Math. Vol. 46 No. 2, 415-437. [23] Sugitani, S. (1988) Some properties for the measure-valued branching diffusion process, J . Math. Soc. Japan 41. 437-462. [24] Sznitman, A-S . (1991). Topics in the Propagation of Chaos, Ecole d'ete de Probabilites de Saint Flour, L . N . M . 1464. [25] Walsh, J .B . (1986). An Introduction to Stochastic Partial Differential Equations, Ecole d'ete de Probabilites de Saint Flour, L . N . M . 1180. [26] Yor, M . (1978). Sur la continuity des temps locaux associes a certaines semi-martingales, in Temps Locaux, Asterisque 52-53. [27] Zvonkin, A . K (1974). A transformation of the phase space of a diffusion process that removes the drift, Math. USSR Sbornik, Vol. 22 No. 1, 129-149. 101 **