Sup erpro cesses and McKean-Vlasov equations with creation of mass L. Overb eck Department of Statistics, University of California, Berkeley, 367, Evans Hall Berkeley, CA 94720, y U.S.A. Abstract Weak solutions of McKean-Vlasov equations with creation of mass are given in terms of sup erpro cesses. The solutions can b e approxi- mated by a sequence of non-interacting sup erpro cesses or by the mean- eld of multityp e sup erpro cesses with mean- eld interaction. The lat- ter approximation is asso ciated with a propagation of chaos statement for weakly interacting multityp e sup erpro cesses. Running title: Sup erpro cesses and McKean-Vlasov equations . 1 Intro duction Sup erpro cesses are useful in solving nonlinear partial di erential equation of 1+ the typ e f = f , 2 0; 1], cf. [Dy]. Wenowchange the p oint of view and showhowtheyprovide sto chastic solutions of nonlinear partial di erential Supp orted byanFellowship of the Deutsche Forschungsgemeinschaft. y On leave from the Universitat Bonn, Institut fur Angewandte Mathematik, Wegelerstr. 6, 53115 Bonn, Germany. 1 equation of McKean-Vlasovtyp e, i.e. wewant to nd weak solutions of d d 2 X X @ @ @ + d x; + bx; : 1.1 = a x; t i t t t t t ij t @t @x @x @x i j i i=1 i;j =1 d Aweak solution = 2 C [0;T];MIR satis es s Z 2 t X X @ @ a f = f + f + d f + b f ds: s ij s t 0 i s s @x @x @x 0 i j i Equation 1.1 generalizes McKean-Vlasov equations of twodi erenttyp es. Firstly,ifb = 0 it is the classical McKean-Vlasov equations asso ciated with d nonlinear IR -valued di usions and studied in many pap ers, e.g. [L,Oel,S1,S2]. If only b dep ends on then this equation is studied in [CR], [COR]. It is used in applications for some biological problems, like conduction of nerve impulse, cf. [GS,SP] and the references therein. Equations like 1.1 with non-linear drift in the di usion and nonlinear reaction term are sometimes called reaction-convection-di usion equation [Lo, p. 287] and can b e found in many biological applications, cf. [M, Section 11.4]. Our main result concerning equation 1.1 is Theorem 3.1 in Section 3 and claims that under Lipschitz conditions on the co ecients of 1.1 a unique solution of 1.1 can b e found as the limit of the interacting one-typ e su- P 1 N i;N 1;N N;N p erpro cess X as N !1where X ;:::;X is a N-typ e i=1 N sup erpro cess with mean- eld interaction. Obviously, this result should b e emb edded in a Propagation of Chaos statementforweakly interacting N- P 1 typ e sup erpro cesses including in particular tightness of i;N . This is X N the topic of Section 2. However, this result do es not necessarily implies that the limit pro cess gives a solution of 1.1, b ecause the application I from d d M M IR to M IR de ned by 1 Z I mf := f md 1.2 fails to b e continuous. Finally, in Prop osition 3.3 we construct a unique solution of 1.1 by a Picard-Lindelof approximation under stronger condition as in Theorem 3.1. This approximation gives a solution of 1.1 as a limit of intensity measures of sup erpro cesses with non-interactive immigration. 2 2 Weakly interacting and non-linear sup erpro cesses In this section we construct a non-linear sup erpro cess as an accumulation p oint of a sequence of weakly interacting multityp e sup erpro cesses. 2.1 Weakly interacting superprocesses First wewant to describ e an N-typ e sup erpro cess with an interaction de- p ending only on the empirical pro cess. We start with real-valued resp. p ositive functions a ;d ; 1 i; j; k d; and b resp. c de ned on ij k d d [0; 1 IR M M IR , where M E denotes the space of probability 1 1 measures on the Polish space E equipp ed with the top ology of weak conver- d gence. De ne for every s 0, m 2 M M IR the time-inhomogeneous 1 2 d op erator Ls; mon C IR by 0 d d 2 X X @ f @f Ls; mf x:= a s; x; m x+ d s; x; m x: 2.1 ij i @x @x @x i j i i;j =1 i=1 The op erator serves as a description of the one-particle motion. Let us de- note the family Ls; m d by L.For the weakly interacting s2[0;1;m2M M IR 1 sup erpro cess the function bs; x; m describ es the mean branching rate and cs; x; m the variance in the branching rate while the empirical distribution of the pro cess equals m. In order to state now the basic de nition we need some notation. Let N 2 IN + d N d N ~ b e xed. If ~ = ;:::; 2 M IR ; f =f ;:::;f 2 B IR ; 1 N 1 N b + where B are b ounded measurable non-negative functions, then b N X 1 d 2 M M IR R~ := 1 i N i=1 R P N ~ fd: The exp onential function e f ; where f = and ~ f := ~ i i i=1 f ~ is de ned by e ~=exp~ f : Finally,we denote for a Polish space E ~ f the space of E -valued continuous paths by C . E N De nition 2.1 We cal l a measure P 2 M C d N an N-typeweakly 1 M IR 3 interacting measure-valued di usion starting at ~ if 0 Z N t X j ~ ~ ~ ~ + X e e X Ls; RX f + 2.2 X e 0 t ~ ~ s j s ~ s f f f 0 j =1 2 ~ ~ bs; RX f cs; RX f ds s j s j N 2 ~ ~ is a P martingale for al l f =f ;:::;f with non-negative f 2 C . X 1 N i 0 denotes the coordinate process on C d N . M IR 2.2 Propagation of Chaos One main issue in this section is the question under which conditions the N 1 1 sequence fP g is P -chaotic with a measure P 2 M C d . N 2IN 1 M IR 1 N De nition 2.2 Let P 2 M C d . We say that the sequence fP g 1 N 2IN M IR 1 is P -chaotic if for every k 2 IN N 1 k 1 P X ;:::;X = P as N k tends to in nity ; 2.3 1 k i=1 cf. [S1,S2]. Intuitively, this means that the interaction disapp ears if the number of typ es 1 N tends to in nity.Typically the measure P is a solution of a non-linear martingale problem. Roughly sp eaking, b ecause the law of large numb ers N 1 1 1 ~ implies RX ! P X , the measure P solves the martingale s s problem characterized by the martingale prop erty of the pro cess Z t 1 1 1 1 f e X e X + f + bs; P X e X X Ls; P X f t f 0 f s s s s 0 1 1 2 cs; P X f ds 2.4 s 2 d for every f 2 C IR . Such a measure is called non-linear sup erpro cess with 0 parameter L;b;c. N 1 According to [S2] an exchangeable sequence fP g is P -chaotic i the N 2IN P 1 N N N ~ j;N converges = distribution 2 M M C d of RX 1 1 X M IR j =1 : N N ~ 1 weakly to . Here fX g is a sequence of random variables on a P N 2IN N 1 N ~ probability space ; F ;P suchthatP X = P . 4 The next theorem gives general conditions under which there is an exchange- N N able solution P to 2.2 and the sequence f g is tight. Additionally N 2IN N we show that all limits p oints of f g are supp orted by the set of so- N 2IN lutions to the martingale problem formulated in 2.4. If E; disaPolish space with metric d we consider on M E theWasserstein metric 1 d ; := supfjf f j; jjf jj 1g 2.5 BL E;d where jjf jj = jjf jj ^ inf fK ; jf x f y jKdx; y 8x; y 2 E g: Set BL 1 = d d and = d d . 1 2 M IR ;d IR ;j:j d IR ;j:j Theorem 2.3 1. Let the functions a ;d ; 1 i; j; k d; c and b satisfy ij k d d the fol lowing assumptions for functions r on [0; 1 IR M M IR 1 jr s; x ;m r s; x ;m jK m ;m +jx x j2.6 1 1 2 2 r 1 1 2 1 2 d r s; x; m < 1 for each m 2 M M IR : 2.7 sup 1 d x2IR Additional ly we assume that one of the fol lowing growth conditions is satis ed: ZZ sup sup jr s; x; mjdxmd K < 1 2.8 0 d s2[0;1 m2M M IR 1 for al l functions b; c; a ;d ; 1 i; j; k d or ij k Z sup jr s; x; mjdx K 1 + K 2.9 0 1 d d IR m2M M IR 1 for al l functions b; c; a ;d ; 1 i; j; k d: ij k Then there exists an exchangeable solution of 2.2.