Superprocesses and Mckean-Vlasov Equations with Creation of Mass

Home , Empirical process, Jump process, Local martingale, Random walk, Semimartingale, Stopping time

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,

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

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.

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

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

Aweak solution = 2 C [0;T];MIR satis es

X X

@ @

a f = f + f + d f + b f ds:

s ij s t 0 i s s

@x @x @x

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

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-

1 N

i;N 1;N N;N

p erpro cess X as N !1where X ;:::;X is a N-typ e

i=1

sup erpro cess with mean- eld interaction. Obviously, this result should b e

emb edded in a Propagation of Chaos statementforweakly interacting N-

typ e sup erpro cesses including in particular tightness of i;N . This is

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

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

measures on the Polish space E equipp ed with the top ology of weak conver-

gence. De ne for every s 0, m 2 M M IR the time-inhomogeneous

2 d

op erator Ls; mon C IR by

d d

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

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

where B are b ounded measurable non-negative functions, then

2 M M IR R~ :=

i=1

fd: The exp onential function e f ; where f = and ~ f :=

~ i i

i=1

is de ned by e ~=exp~ f : Finally,we denote for a Polish space E

the space of E -valued continuous paths by C .

De nition 2.1 We cal l a measure P 2 M C d N an N-typeweakly

M IR 3

interacting measure-valued di usion starting at ~ if

~ ~ ~

~ + X e e X Ls; RX f + 2.2 X e

0 t ~ ~ s j s ~

f f f

j =1

~ ~

bs; RX f cs; RX f ds

s j s

N 2

~ ~

is a P martingale for al l f =f ;:::;f with non-negative f 2 C . X

1 N i

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

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

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

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

1 1 2

cs; P X f ds 2.4

2 d

for every f 2 C IR . Such a measure is called non-linear sup erpro cess with

parameter L;b;c.

N 1

According to [S2] an exchangeable sequence fP g is P -chaotic i the

N 2IN

1 N

N N

j;N converges = distribution 2 M M C d of RX

1 1

M IR

j =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

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

d ; := supfjf f j; jjf jj 1g 2.5

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

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

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

r s; x; m < 1 for each m 2 M M IR : 2.7 sup

x2IR

Additional ly we assume that one of the fol lowing growth conditions is

satis ed:

sup sup jr s; x; mjdxmd K < 1 2.8

s2[0;1

m2M M IR

for al l functions b; c; a ;d ; 1 i; j; k d or

ij k

sup jr s; x; mjdx K 1 + K 2.9

0 1

m2M M IR

for al l functions b; c; a ;d ; 1 i; j; k d:

ij k

Then there exists an exchangeable solution of 2.2.

d 2

IR 2. Assume additional ly that for each f 2 C

N 1 2

sup P [X f ] < 1: 2.10

Then the sequence f g is tight and every accumulation point is

N 2IN

supported by the set of solutions Q 2 M C d of the martingale

M IR

problem 2.4 of a non-linear superprocess. 5

3. If we assume additional ly that the sequence of initial distributions fm g

N d N 1

with m 2 M M IR is m -chaotic and if there exists only one

0 0

solution P to the martingale problem 2.4 with initial distribution

1 N 1

m then fP g is P -chaotic.

N 2IN

Pro of. 1. An exchangeable solution of 2.2 can b e constructed byweak

approximation with interacting multityp e branching di usions, cf. [MR, GL,

O1]. The condition 2.8 resp. 2.9 ensures the tightness of the interact-

ing branching di usions as well as its existence as an accumulation p ointof

a sequence of branching random walks. The latter can b e constructed as

marked p oint pro cesses , cf. [O1], or as in [RR], which are exchangeable

by construction. This construction is sketched in the app endix. The neces-

sary tightness under condition 2.8 resp. 2.9 follows as in the pro of of

tightness of f g , see part 2 of the present pro of.

2. It is well known that the sequence f g is tight if sequence of the inten-

sity measures fI g is tightin M C d , cf. [S1,S2]. The sequence of

N 1

M IR

measures fI g is tight if their one-dimensional pro jections are tight,

N 2IN

1;N

i.e. if, byexchangeability, the laws of X f build a tight sequence of mea-

2 d

sures on C for every f 2 C IR [f1g, cf. [D, Sect. 3.6]. The tightness

1;N

of fX f g may b e deduced by the Aldous-Reb olledo criterion. The

N 2IN

rst condition of that criterion follows by the uniform L -b oundedness of the

initial distribution and 2.9. The second part follows if we can b ound

S +

@ f

1;n N

X a s; RX E 2.11

i;l

s s

@x @x

i l

i;l=1

N N

~ ~

+ d s; RX + bs; RX f ds

s s

i=1

and

S +

1;N N 2

E [j X cs; RX f dsj] 2.12

s s

for a b ounded stopping time S and 0 < <1. Under 2.9 the term 2.11

is b ounded by

S +

0 1;N

K E [ K X 1 + K ds] K E [sup X 1] + 1:

f 0 s 1

f s

sL+1 6

1;N

Applying now the semimartingale decomp osition of X 1 and once again

assumption 2.9 we obtain

1;N

0 1;N 0 1;N

1] + K E [sup X 1] + K dr E [sup X 1] E [X

0 s 1 s

sr st

which yields by Gronwall's lemma and the uniform L -b oundedness of the

initial distribution an upp er b ound for 2.11, which is indep endentof N .

The same pro cedure applies to 2.12. Under assumption 2.8 we rewrite

the term 2.11 by exchangeabilityas

d N

S +

X X

1 @ f

N j;n

a s; RX X E

i;l

s s

N @x @x

i l

j =1

i;l

N N

~ ~

f ds : + bs; RX + d s; x; RX

s s

i=1

It is immediate that the term 2.11 is then b ounded by K K : Together

f 0

with the same calculations for the quadratic variation, this yields that also

in this case the second part of the Aldous-Reb olledo criterion is satis ed and

the rst part can b e proved analogously as under 2.9.

3. Identi cation of the sets of accumulations points of f g .

N 2IN

1 N

Let b e an accumulation p ointoff g .For 0 r <

N 2IN 1 k

d k 2

function F on M C by

M IR

Z Z

F Q = ! s Ls; Q f + bs; Q f e ! t e ! r +

s s f f

C r

M IR

cs; Q f e ! s dsg ! r ;:::;!r Qd! ;

s f 1 k

where Q denotes the distribution of ! s under Q.We will showthat

2 1

F Q dQ=0: 2.13

M C

M IR

The martingale problem 2.2 implies a formula for the quadratic varia-

i;N

tion of the exp onential martingales M [e ] de ned as in 2.2 with f =

f 7

f ;:::;f ;f = f . This formula yields

1 N j ij

i;N j;N 2 N

E [ hM [e ];M [e ]i ] F Q dQ

f f t

i;j =1

j;N N 2 j;N

E [ X cs; R X f e X ds]

s 2f

j =1

000

t r K ! 0asN !1:

The last inequality follows as in part 2 of the pro of by assumption 2.8

resp. 2.9 and by the submartingal prop erty of the total mass pro cess

1;N 2 N

X 1. The assertion follows now, if wehave that F Q dQcon-

2 1

verges as N !1to F Q dQ. Because F is under the conditions

2.8 resp. 2.9 b ounded by K resp. the uniformly integrable random

i;N

variables K K X 1 + K , this can b e achieved, if F is con-

0 1

i=1

tinuous on M C . To see the continuity of let us consider the func-

M IR

R R

tion F Q:= ! dxbs; Q ;xf xe ! : If Q ! Q in Qd!

b s s f s n

C IR

M IR

1 d 1

M C d then Q ! s 2 M M IR converges to Q ! s .Hence

1 n 1

M IR

wehave to show that m ! m in M M IR implies

n 1

Z Z

dxbs; m ;xf xe 2.14 m d j

n f n

d d

IR M IR

Z Z

md dxbs; m; xf xe j!0:

d d

M IR IR

The left hand side of 2.14 is b ounded by

Z Z

m d dxjbs; m ;x bs; m; xjjf xje 2.15

n n f

Z Z

+ j m d md dxbs; m; xf xe j: 2.16

n f

Using the Lipschitz prop ertywe can b ound 2.15 by

Z Z

dxf xe : K m ;m md

f b 1 n

d d

M IR IR

If we consider for the moment the one-p oint compacti cation IR of IR the

integrand is a b ounded continuous function on M IR iff > 0. Hence 8

the second factor is b ounded. The rst factor converges to 0. The term

2.16 converges b ecause dxbs; m; xf xe isalsoaboundedcontin-

uous function if f > 0. The continuityof F follows in the same way.

Hence every limit p oint is supp orted by the set of solutions of the non-

linear martingale problem 2.4, so far viewed as probability measures on

C [0; 1;MIR . Because every such a probability measure is the distribu-

tion of a regular sup erpro cess it do es not charge the compacti cation p oint

and therefore it is indeed a measure on C [0; 1;MIR .

4. The last assertion of the theorem follows nowby standard arguments, cf.

[S1,S2].

Remarks. 1. The assumption 2.9 is the usual b oundedness condition

which is supp osed if one considers weak convergence of measure-valued pro-

cesses whereas condition 2.8 is formulated just for the case of weakly inter-

acting measure-valued pro cesses.

2. A martingale generator A; D A for a non-linear sup erpro cess P is

given by D A =\ nitely based functions" and

AmF = Lmr F +bmr F +cmr F 2.17

: :

F + F

r F := lim ; is the G^ateaux derivative in direction ,the

x 0 x

Dirac measure on x 2 E . Hence, the ow of the one-dimensional marginal

distributions u = P X of a non-linear sup erpro cess solves the non-linear

`partial di erential equation'

u_ = Au u : 2.18

s s s

In the sp ecial case of mean- eld interaction considered in the following section

the intensity measure of the non-linear sup erpro cess provides a solution to

1.1.

3. The question of uniqueness of the martingale problem describ ed in 2.4 is

discussed in detail in [O2] by means of the Sto chastic Calculus on historical

trees, cf. [P]. Wenowgive a simple example, where we can show uniqueness

directly from the martingale problem 2.4.

Example. Let us consider the sequence of \multityp e" sup erpro cesses con-

ditioned on non-extinction, i.e., a ;d ;c do not dep end on m; c =1and

ij i 9

d N

bs; x; m= . Let ~ 2 M IR satisfy 1 = N . Then

0 0 i

i=1

I m1

there exists only one measure such that

2 j;N

~ ~ ~

X ds f f e X + X Lf + X e e

s j ~ P 0 j t ~ ~

j s

f i;N f f

X 1

j =1

i=1

2 d N

is a martingale for all f 2 C IR , namely the additive H-transform of N-

indep endent sup erpro cesses with the space-time harmonic function H s; ~=

1; cf. [O1]. Let the initial condition b e -chaotic with 1 = N

j 0

j =1 0

1, e.g. ~ = ;:::; . The function b satis es assumption 2.9 and

0 0

therefore, by Theorem 2.3, wehavethatevery limit p ointoff g is

N 2IN

supp orted by measures P suchthat

e X e + X Lf + f f e X ds:

f t f 0 s f s

I P X 1

j =1

Such a measure P gives rise to the linear martingale X f f X Lf +

t 0 s

f ds: If we apply this with f =1weobtainE [X 1] = E [X 1]+

P t P 0

I P X 1

t =1+t for every solution P . Therefore P is unique. It coincides with

the sup ercritical time-inhomogeneous sup erpro cess with branching mean

. It is interesting to notice that P is also an H-transform of the 1+

1+s

sup erpro cess, namely with the multiplicative space-time harmonic function

1+s

H s; =e .From the biological p oint of view one can interpret this re-

sult in the following way. If one conditions a particle p opulation on survival,

which exhibits a critical branching b ehaviour and consists of N subp opu-

lations of the same kind, then for large N every subp opulation executes a

. slightly sup ercritical branching with time-decreasing rate 1 +

1+s

3 Mean- eld interaction

Nowwe attack the problem to nd a solution of 1.1. Wehavetointro duce

a sp ecial kind of weakly interaction, namely the mean- eld interaction. We

mo dify the de nition in 2.2 in the following way.We replace the functions

a; d; b and c by functions dep ending on the mean- eld of the empirical distri-

P P

1 N N 1

i;N

bution, e.g. a s; x; i;N in 2.2 is replaced by a x; X .

ij ij

i=1 i=1

N N

A solution of 2.2 is then called a N-typ e sup erpro cess with mean- eld in-

teraction. 10

3.1 Existence

The next theorem gives existence of a sequence of multityp e sup erpro cesses

1 N

1;N N;N

with mean- eld interaction fX :::;X g ,tightness of f i;N g

N 2IN N

i=1

i;N

and a law of large numb ers for the sequence f X g . The pro of is

i=1

sketched b ecause the techniques are the same as in the pro of of Theorem 2.3.

Theorem 3.1 Let the functions a ;d ; 1 i; j; k d; b and c satisfy the

ij k

d d

fol lowing assumptions for functions r on IR M IR

jr x ; r x ; j K ; +jx x j 3.1

1 1 2 2 r 2 1 2 1 2

r x; < 1 for each 2 M IR ; 3.2 sup

x2IR

where is de ned in 2.5. Additional ly we assume that the fol lowing growth

condition is satis ed for al l functions a ;d ; 1 i; j; k d; b and c:

ij k

jr x; jdx K 1 + K : 3.3

0 1

i;N

Suppose also that X 1 is uniformly in L . Then there exists

i=0

1;N N;N

an exchangeable solution X ;:::;X of 2.2 and the distributions of

1 N

i;N

f g build a tight sequenceinM M C d .

N 1 1

X M IR

i=1

i;N

1 N

If additional ly the distribution of X converges to , then every

i=0 0

accumulation point of

i;N 1

P X 3.4

i=0

in M C d isaDirac distribution on a deterministic solution of

M IR

f = f + L f ds: 3.5

t 0 s s

Henceaweak solution of 1.1 is constructed. Final ly, if 1.1 has a unique

solution 2 M [0; 1;MIR , then

i;N 1

X : 3.6 P

i=0 11

Pro of. The construction of an exchangeable solution of 2.2 is the same

as in Theorem 2.3, b ecause the mapping ~ ! is continuous. The

i=0

i;N

tightness of sequence of the distributions of X follows as in Theo-

i=0

rem 2.3 from the growth conditions 3.3. Toprovethatevery limit p ointis

supp orted by solutions to 3.5 wefollow the same approach as in the pro of

of Theorem 2.3 with the function

F = f f + L f dsg ;:::; ;

t r s s r r

2 C d .From the continuity and the uniform integrabilityof F we obtain

M IR

that X f X f + X LX f ds is a martingale under an accumulation

t 0 s s

1 i;N

p oint P 2 M C d of the distributions of X . The quadratic

M IR

i=0

N 1

i;N 2

X dP ! 0. variation of this martingale is zero, b ecause F

i=0

Hence P = , where solves 3.5.

1 N

i;N

Remark. By the martingale prop erty 2.2 the pro cess X is a

i=0

solution of a martingale problem asso ciated with a one-typ e sup erpro cess

with interaction in the sense of [MR,P]. Therefore Theorem 3.1 says that

a solution of 1.1 can b e found byweak approximation with interactive

i;N N

X whose variances vanish as N !1. sup erpro cesses Y =

i=0

3.2 Uniqueness

In this subsection we consider uniqueness of 2.4 with co ecients only de-

p ending on = I m. We rst consider the case where the branching is

critical, i.e. b =0.

Prop osition 3.2 Let As; d be a family of linear operators

s0;2M IR

under such that for every 2 M IR there is a unique P 2 M C

which for al l f 2 C the process N f de nedby

N f =f f As; P f ds 3.7

t t 0 s

0 s

is a martingale and P = , where is the coordinate process C d .

Let m 2 M M IR .

1 12

Then there is a unique probability measure P on C d such that for al l

M IR

2 d

f 2 C IR

2 m m

f e X ds 3.8 f cs; I e X e X + X As; I

f s f t f 0 s

s s

m 1

is a martingale under P , where I = I P X =E [X ].

m m m s

s s

1 2

Pro of. Let P and P be two solutions of 3.8. It is clear that if the

1 2

ow of the corresp onding intensity measures I and I coincide

s0 s0

s s

1 2

then P = P = \sup erpro cess with time-inhomegenous one-particle motion

1 1

generated byAs; I and branching variance cs; I ". For i =1; 2, the

s s

i i 1 i

intensity measure I equals mdP ,whereP 2 M C d isthe

s s

one-particle motion of P with initial distribution which is also the pro cess

i i

asso ciated with As; I . Hence for i =1; 2, the measures P solve3.7

1 2

which implies I = I = distribution of under the unique solution of 3.7.

s s

This proves uniqueness. For existence we see that the sup erpro cess started

at m and with one-particle motion generated by the non-linear op erator

As; P with initial condition dx= mddxsolves 3.8.

The next prop osition generalizes the uniqueness result in [CR] and shows

existence.

Prop osition 3.3 Let us suppose that al l functions r=a ;d ; 1 i; j; k d;

ij k

and b arebounded and satisfy

sup jr ;x r ;xj jj jj 3.9

1 2 1 2

where jj jj denotes the norm of total variation. Then there is a unique solution

F F

s; x= ;x;d s; x=a of 1.1 and the superprocess with coecients a

k s

F F

F F F

d ;x; 1 i; j; k d; b s; x= b ;x and c s; x= c ;x is the

s s s

unique solution of 2.4.

Pro of. The second assertion follows from the rst assertion as in Prop o-

sition 3.2. Concerning the rst assertion we obtain a solution of 1.1 by

n 1

a Picard-Lindelof iteration. := P X where P is the sup erpro-

s s

cess with co ecients a s; x= a ;x;d s; x=d ;x; 1 i; j; k

ij s k s

d; b s; x=b ;xand c s; x= c ;x. The initial p oint of the itera-

s s

tion is the ow of a sup erpro cess with co ecient de ned by a constant ow 13

= 8s with some xed 2 M IR . Wehave

s 0 0

n2 n1 n1 n 0 n1 n

L f ds f L f =K

s s s t t

which yields by 3.9 that

n n1 00 n n1 n2

jj jj K 1jj jjds: 3.10

t t s s s

Because b is b ounded sup 1 is b ounded. From inequality 3.10 we

obtain a solution of 1.1 as n tends to in nity. Uniqueness follows byGrown-

wall`s lemma from the same inequality.

Remark. Despite the fact that we constructed already a solution to 2.4

in two di erentways wewould liketogive a construction as a limit of a

sequence of multityp e sup erpro cesses with mean- eld interaction as in Theo-

rem 2.3 in order to give another simulation pro cedure for the solution of 1.1.

The diculties stem from the fact that the mapping I from M M IR to

M IR fails to b e continuous, b ecause the function ! f iscontinuous

but not b ounded. Instead of that I can only construct for every K 2 IN a

solution P of the stopp ed martingale problem:

t^T

K 1

e X e X + e X X Ls; I P X f + 3.11

f t f 0 f s s

K 1 K 1 2

bs; I P X f cs; I P X f ds

s s

K 2 d K

is a martingale under P for every f 2 C IR and X = X P

s s^T

a.s. with T ! :=infft 0jX ! 1 K g bythechaos-technique as in

K t

Theorem 2.3. Because I cannot prove that the stopp ed martingale problem

3.11 has a unique solution, the techniques in [EK, Theorem 6.3] are not

applicable in order to prove existence of a global solution to 2.4.

A Construction of an exchangeable weakly interact-

ing sup erpro cess

The starting p oint is a construction of an exchangeable weakly interacting

N-typ e branching random walk. 14

d d N d

Let be a kernel from [0; 1 IR M IR to IR ,let >0 and let for

d N

every ~ 2 M IR

A~f x := f y f x s; x; ~; dy A.1

b e the generator of a jump pro cess in IR . The intensity for a particle at

x 2 IR and time s to die without children resp. with twochildren is

describ ed by p s; x; m resp. p s; x; m.

0 2

d N

Let =f!~ = ft ;~ g with 0 t

m m m2IN 1 2 m

We de ne for every! ~ 2 a predictable random measure v 2 M [0; 1

d N

M IR by

i i j p

X d d v ft ;~ g ; ds; d~ :=

j 6=i O m m m2IN

y :

i=1

i i

~ ~ ~

s; X ; dy + p s; X d + p s; X d ds;

s 0 s 2 s +

i i

where X ~! :=X ft ;~ g := ~ , O denotes the zero-

m m m2IN n i

t s

s s

N 1

. Let v ; = ; b e the canonical ;:::;X measure and X =X

t ;~ s

n n

s s

~ ~ ~

random measure on . There exists a probability measure P on suchthat

the stopp ed pro cesses W v v is a uniformly integrable martingale

for every n and every predictable W ~!; s; -\" indicates integration -.

~ ~

For a nitely based function F = ~ f ;:::;~ f , 2 C and ~

1 k

f = f ;:::; f we consider the predictable function W ~!; s; =

1 1 l l

IBRW

~ ~

F X + ~ F X . Then the pro cess M [F ] de ned by

s s

IBRW

~ ~

M [F ] := F X F X

t t 0

Z Z

~ ~ ~

F X + e F X s; X ; dy + ds X

s i y : s s

IR 0

i=1

~ ~ ~ ~ ~

p s; X F X e + p s; X F X + e F X

0 s s i : 2 s s i : s

is a lo cal martingale until T := sup t .Here~ + e means that weadd

n i

in the i-th comp onentofthevector ~ the measure . 15

Because of our assumptions 2.8 or 2.9 wehave P -almost surely that T =

1,we get an interacting branching random walk, if we de ne P as the

~ ~

distribution of X under P on D d N .

M IR

Starting from a sequence of branching random walks fX g , whose ran-

n2IN

dom walks ~ approximates the generator A~we can de ne an N typ e

interacting branching di usion as an accumulation p oint of the sequence

fX g as in [RR,O1]. Then we can rescale the branching b ehaviour of

n2IN

the N typ e interacting branching di usion in order to get an N typ e in-

teracting sup erpro cess as in [MR,O1]. For a general approachtomultityp e

non-interacting sup erpro cesses see [GL].

Acknow ledgement : I wish to thank Sylvie M el eard for helpful discussions on

the chaos-technique.

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