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Non-Linear Superprocesses Non-linear sup erpro cesses L. Overb eck Lab oratoire de Probabilit es, Universit eParis VI, eme 4, Place Jussieu, Tour 56, 3 Etage, 75252 Paris Cedex 05, y France January 27, 1995 Abstract Non-linear martingale problems in the McKean-Vlasov sense for sup erpro cesses are studied. The sto chastic calculus on historical trees is used in order to show that there is a unique solution of the non-linear martingale problems under Lipschitz conditions on the co ecients. Mathematics Subject Classi cation 1991: 60G57, 60K35, 60J80. 1 Intro duction Non-linear di usions, also called McKean-Vlasov pro cesses, are di usion pro- cesses which are asso ciated with non-linear second order partial di erential d equation. IR -valued McKean-Vlasov di usions are studied in detail in many pap ers, e.g. [F,Oel,S1,S2]. The main issues are approximation by a sequence of weakly interacting di usions, asso ciated large deviations and uctuations Supp orted byanEC-Fellowship under Contract No. ERBCHBICT930682 and par- tially by the Sonderforschungsb ereich 256. y On leave from the Universitat Bonn, Institut fur Angewandte Mathematik, Wegelerstr. 6, 53115 Bonn, Germany. 1 and nally uniqueness and existence of the non-linear martingale problem asso ciated with McKean-Vlasov pro cess. In this pap er we fo cus on the latter question in the set-up of branching measure-valued di usions pro cesses, also called sup erpro cesses. For an ex- cellentintro duction to the theory of sup erpro cesses we refer to [D]. In order to formulate the basic de nition we need to intro duce some notation. The space of nite resp. probability measures over a Polish space E is denoted by M E resp. M E and is equipp ed with the weak top ology. The space 1 of continuous resp. c adl ag E valued paths is denoted by C resp. D E E and C E is the set of b ounded continuous functions on E . The expression b R f with 2 M E means fd. 1 De nition 1.1 Let L =Lm; D be a family of linear op- m2M M E 1 erators with common domain DC E , b; c measurable functions b on M M E E with c 0. The function b is cal led immigration 1 function and the function c measures the variance in the branching be- havior. Fix 2 M E .Ameasure P on C ; F ; F with canonical ltra- t M E tion F and -algebra F generated by the coordinate process X is cal led t a non-linear superprocess with parameter L;b;c startedfrom , if for each f 2D the process M f de nedby M f := X f f 1.1 t t Z t 1 1 X LP X f + bP X f ds s s s 0 is a local martingale with increasing process Z Z t 2 1 ;xX dxds; 1.2 f xcP X s s 0 E 1 where P X 2 M M E denotes the distribution of X under P . 1 s s In terms of partial di erential equation the ow of the one-dimensional 1 marginals u := P X of a solution of the non-linear martingale problem s s 1.1,1.2 solves the weak non-linear equation u_ = A u u ; 1.3 s s s 2 where for nice functions F on M E 2 F 1.4 AmF = Lmr F +bmr F + cmr : : : F + F x with r F := lim . This is one motivation for the study of x 0 non-linear sup erpro cesses from the p oint of view of partial di erential equa- tions. Another motivation is a kind of Law of Large Numb ers for weakly interacting N-typ e sup erpro cesses, whichprovides also an pro of of the exis- tence of a non-linear sup erpro cess. Aweakly interacting N-typ e sup erpro- N 1 N ~ =X ;:::;X 2 C N is characterized by the martingale cesses X M E prop erty of the pro cesses Z N N t X X 1 N N N j ~ ~ ~ e X e X + e i;N f + 1.5 X X L ~ ~ ~ j t 0 s s f f X f s N 0 i=1 j =1 N N X X 1 1 2 b i;N f c i;N f ds ; j j X X s s N N t0 i=1 i=1 P N N N N where e f for ~ = ;:::; 2 M E and ~ := exp ~ i i 1 N i=1 f N ~ f =f ;:::;f 2 C E : The actual pro of the approximation result is 1 N b based on the Propagation of Chaos techniques, cf. [S1,S2]. It needs some machinery on tightness of measure-valued pro cesses. I state the result and an outline of the pro of in the app endix. For details I refer to [O1]. In the ac- companying pap ers [O1,O2] I study the large deviations and the uctuations asso ciated with the approximationiftheweakly interacting sup erpro cesses are sup erpro cesses with mean- eld interaction. The main result of the present pap er is the pro of that there is a unique solu- tion to 1.1,1.2 under Lipschitz conditions on the parameter L;b;c. The i pro of relies on the fact that for two sup erpro cesses P ;i =1; 2; with di erent ~ ~ ~ parameters there exits a ltered probability space ; F ; F ;IP onwhichwe t i i can de ne pro cesses X with distribution P ;i =1; 2. This follows from the sto chastic calculus along historical trees, recently develop ed bySteven N. Evans and Ed A. Perkins [P1,P2,EP]. Once this is established the pro of of existence and uniqueness is carried out by a Picard-Lindelof approximation. Basically, there are two di erentcases. First, if only b dep ends on m 2 ~ ~ ~ M M E then ; F ; F ;IP is the canonical space of a marked historical 1 t 3 pro cess as in [EP], cf. Theorem A in Section 2. If all parameters dep end on m 2 M M E , we assume that Lm is a nice di erential op erator on 1 d IR and that the co ecients of L, b and c are strongly related, cf. Theorem ~ ~ ~ 3.1 and Theorem B in Section 3. Then wecancho ose ; F ; F ;IP asthe t canonical space of the historical Brownian motion. In b oth cases it turns out that the historical pro cess plays the same role for non-linear sup erpro cesses as the Brownian motion plays for non-linear di u- d sions on IR , namely as a driving term for strong stochastic equations.The fundamental role of the historical pro cess also b ecomes apparentinseveral other pap ers, e.g. in [P1,P2], where interacting measure-valued pro cesses are considered, in [EP], where a Clark-typ e formula for measure-valued pro cesses is proved, in [LG], where the connections to Brownian excursions are inves- tigated, and in [Dy2] , where the relations to quasi-linear partial di erential equation are explored. 2 Non-linearity in the immigration function In this section we consider the case in which Lm=L is a generator of a time-homogeneous Hunt pro cess indep endentofm and c = 1. Hence the non-linearity app ears only in the immigration function b. Because weneed the historical pro cess from now on I will shortly describ e it. 2.1 Historical process The historical pro cess over a one-particle motion , e.g. over a Hunt pro cess with state space E , can b e seen as the sup erpro cess constructed over the path-process of the one-particle motion. A path pro cess is a path-valued pro cess and evolves from a path of length s to a path of length t> sby s pasting on the given path a new path of length t s,which is distributed s as the underlying one-particle motion started from s. By construction this is a time-inhomogeneous Markov pro cess with state-space D and it has E h h a generator L ;DL in the sense of martingale problems, cf. e.g.[P1,P2]. If we sup erp ose a critical branching mechanism to this path-pro cess and take the usual \sup erpro cess limit" we arrive at the historical pro cess, which can then b e viewed as the solution of the martingale problem describ ed in h 1.1,1.2 with cm= 1;bm=0;Lm=L . It is called \historical" 4 b ecause every particle carries all the information ab out the places it and its anchestor visited. Additionally one can reconstruct from this information the genealogy of a present particle byinvestigation of the overlap of the paths of two di erent particles. Because we only use the historical pro cess as a to ol and we will not prove theorems ab out it we will omit an exact de nition and refer to [D, Sect. 12] or [DP,P1,P2,Dy1]. 2.2 Superprocesses with emigration as functionals of the marked historical process Let X =Y; N 2 D D E [0; 1] denote the path pro cess of the Hunt pro cess generated by L and an indep endentPoisson pro cess with uniform jumps on [0; 1], i.e., N is the path pro cess of a Poisson p oint measure on [0; 1 [0; 1] with intensity ds dx.
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