Non-Linear Superprocesses

Home , Hunt process, Local martingale, Semimartingale

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,

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

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.

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

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

f with 2 M E means fd.

De nition 1.1 Let L =Lm; D be a family of linear op-

m2M M E

erators with common domain DC E , b; c measurable functions

on M M E E with c 0. The function b is cal led immigration

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-

M E

tion F and -algebra F generated by the coordinate process X is cal led

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

1 1

X LP X f + bP X f ds

s s s

is a local martingale with increasing process

Z Z

2 1

;xX dxds; 1.2 f xcP X

0 E

where P X 2 M M E denotes the distribution of X under P .

1 s

In terms of partial di erential equation the ow of the one-dimensional

marginals u := P X of a solution of the non-linear martingale problem

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

F 1.4 AmF = Lmr F +bmr F + cmr

: :

F + F

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

N N

X X

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

i=1 j =1

N N

X X

1 1

b i;N f c i;N f ds ;

X X

s s

N N

i=1 i=1

N N N

where e f for ~ = ;:::; 2 M E and ~ := exp

~ i i 1 N

i=1

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

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

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

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

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-

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

pasting on the given path a new path of length t s,which is distributed

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

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

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. Denote by IP the distribution of the

sup erpro cess G over the one-particle motion Y; N starting from G , i.e., the

historical pro cess over the Huntpro cess and an indep endentPoisson pro cess.

IP is a measure on := C [0; 1;MD equipp ed with the canonical

E [0;1]

ltration F and canonical -algebra. The pro cess G is now the canonical

pro cess on . Let us denote by x =y; n a generic elementin D .Let

E [0;1]

on [0; 1 [0; 1]. n also denote the p oint measure

s;n n

st;n 6=n s s

s s

Let b b e a predictable function from [0; 1 E to[0; 1], the candidate

for the emigration term. Because in Prop osition 2.2 and nally in Theorem

2.4 we consider the martingale problem 1.1,1.2 with b instead of b,we

view b now as an emigration rather then an immigration function.

In order to meet the formulation of [EP] we de ne the [0; 1]-valued function

on [0; 1 D by

s; y ; ! = bs; y s;!: 2.1

Further we de ne the following functions

At; x; ! = nfs; z 2]0;t[[0; 1]j s; y ; ! >zg 2.2

B t; x; ! = 1 t; x; ! : 2.3

fA=0g

Let K b e the martingale measure of the historical pro cess G for the de nition

of martingale measures for measure-valued pro cesses cf. [D, Sect. 7] and for 5

historical pro cesses a de nition can b e found in [P1]. Then we can de ne a

dIP

j = new measure on having the lo cal densities

dIP

Z Z

R = expf s; y K ds; dx 2.4

0 D E [0;1]

Z Z

s; y G dxdsg;

0 D E [0;1]

cf. [D, Sect.7],[EP]. Finally let us de ne the measure-valued pro cesses

H = 1 y B t; xG dx 2.5

D E [0;1]

H = 1 y G dx: 2.6

t t

D E [0;1]

From the de nition of IP it is obvious that H is the historical pro cess over

under IP . The following prop osition is basic for us:

Prop osition 2.1 [EP, Theorem 5.1] Under IP the process H is the his-

torical process over .

We need a slightly di erentversion of this result which will b e obtained bya

h h

Girsanov argument. Let L ;DL b e the martingale op erator of the path

pro cess of .

Prop osition 2.2 For every 2 D L the process H is under IP a

semimartingale with increasing process V H sds, where V =

0 s

H L ds is the increasing process of H under IP . The quadratic

variation of the martingale part equals H ds. Hence under IP the

0 s

process H is a historical process with negative immigration , or in other

words with an emigration function .

Pro of. Applying the Girsanov transformation for martingales we can calcu-

late the semimartingale decomp osition of H under IP from the semimartin-

gale decomp osition of H under IP . In order to do that wehave to consider

dIP

the martingale Z of the densities Z = j .LetM denote the martingale

t F

dIP

measure asso ciated with the historical Brownian motion H under IP . Then

wehave

Z Z

Z = expf s; y M ds; dy + 2.7

D E 0 6

Z Z

s; y H dy dsg

0 D E

Z Z Z Z

t t

= expf[ s; y M ds; dy s; y H dy ds]

0 D E 0 D E

Z Z

s; y H dy dsg

0 D E

Z Z Z Z

t t

s; y H dy dsg; = expf s; y N ds; dy

0 D E 0 D E

where N is the martingale measure asso ciated with H under IP , i.e. :N :=

R R

s; y N ds; dy is the martingale in the semimartingale decomp o-

0 D E

sition of H t under IP . This yields in particular, that Z solves under

IP the equation

Z = 1 Z d :N : 2.8

t s s

According to Prop osition 2.1 and Girsanov's theorem e.g. [RY, p.303],

H V 2.9

s s

= H V + < :N ;H V >

is a martingale under IP . The bracket in the last line equals, again according

to [RY, p.303],

< martingale in the decomp osition of H under IP ;

martingale in the decomp osition of H under IP >

= < martingale in the decomp osition of H under IP;

martingale in the decomp osition of H under IP>

= < :K;B :K >

= G B ds

= H ds:

Because the quadratic variation of this martingale remains unchanged under

achange of measure the prop osition is proved. 7

2.3 Comparison of two historical processes with di erent non-interactive

emigration

i i

We consider two function b ;i =1; 2, from [0; 1 E to [0; 1] and de ne

i i i

and A by b as in 2.1 ab ove. Then ;i =1; 2; do not dep end on ! .For a

t t t

measure H on fx jx 2 D g , where x s:=xs;s

and a function f 2 C E we de ne H f := f x H dx.

D t

Lemma 2.3 For every T>0 there exists a constant C < 1 such that

1 2

1 2 2

E [jb s; b s; j]ds IE [ sup jH f H f j ] C

s s T

t t

jjf jj 1

for al l t T , where is the Hunt process generatedbyL.

Pro of. Let us write n = .

t ;Z

i i

i=1

2 1

f j ] f H IE [ sup jH

t t

jjf jj 1

1 2

= IE [ sup f x1 1 G dx ]

A t;x=0 A t;x=0

jjf jj 1

1 2

IE [ j1 1 jG dx]IE [G 1] + t

t 0

A t;x=0 A t;x=0

G dx]IE [G 1] + t+ IE [ 1

t 0

1 2

[ fb t ;y t Z ;i=1;:::;N ;b t ;y t >Z g

i i i j j j

j =1

G dx]IE [G 1] + t: IE [ 1

t 0

1 2

[ fb t ;y t Z ;i=1;:::;N ;b t ;y t >Z g

i i i j j j

j =1

The term IE [ 1 G dx] equals

1 2

[ fb t ;y t Z ;i=1;:::;N ;b t ;y t >Z g

i i i j j j

j =1

1 2

P [ [ fb t ; Z ;i =1;:::;N;b t ; >Z g]; 2.10

i t i j t j

i j

j =1

where Z ;t are uniform distributed on [0; 1] [0;t], N has a Poisson distri-

j j

bution and all random variables are indep endent from each other. Because

1 2

P [b t ; Z

j t j j t j t

j j j

2 1

= b t ; b t ; 1

j t j t

b t ; b t ; 

t t j j

j j

j j 8

we obtain by conditioning that the probability 2.10 is b ounded by

2 1

E [N ] E [b t ; b t ; 1 ]ds=t:

j t j t

b t ; b t ; 

j j

t t

j j

2 1

By the same argument for P [b t ; Z

j t j j t j t

j j j

nally prove the assertion. 

2.4 Non-linear martingale problem

We de ne for p 1 an appropriate Wasserstein metric:

 m ;m := inf d ; Qd; d ; 2.11

p 1 2

E;d

M E M E 

where the in mum is taken over all Q 2 M M E M E whose marginal

distributions are m and m and where for a Polish space E with metric d,

1 2

the metric d on M E is de ned as follows.

E;d

d ; := supfjf f j; jjf jj 1g 2.12

E;d

where

jjf jj = jjf jj ^ inf fK ; jf x f y jKdx; y 8x; y 2 E g:

BL 1

Notice that if we replace d by d ^ 2 in the de nition of then 

1 1

E;d E;d

is smaller than the original and equivalent with the Prohorov metric and

also with d . Recall that byHolder's inequality K if q p

M E ;d q p;q p

E;d

with some constant K .

p;q

1 2

Fix R>0. Let P and P be two solutions of the non-linear martingale

problem 1.1,1.2 with 0 bm; x R, Lm=L and with c =1.

Theorem A Let b : M M E E ! [0;R] satisfy

jbm ;x bm ;xj K m ;m 2.13

1 2 b 1 2

with some constant K . Then there is a unique solution to the non-linear

martingale problem 1.1, 1.2. 9

Pro of. Let us de ne the map on C [0;T];M M E by

u 1

 ; 2.14 u := P X

0sT

u u

where P is the sup erpro cess with immigration function b s; x:=bu ;x

on the canonical space C [0;T];ME with co ordinate pro cess X . De ne

now for u 2 C [0;T];M M E ;i =1; 2; the pro cesses H and H as in

1 t

i i

Prop osition 2.2 and Lemma 2.3 with t; y ; ! = bu ;y and over

R R

.Byanobvious scaling prop ertythe the one-particle motion generated by

sup erpro cesses pro jected down form the pro cesses H have distributions

t0

;i =1; 2, satisfy the assumptions of Prop osition ;i =1; 2. Because H P

2.2 and Lemma 2.3 we can conclude that

2 1 2 2

1 2

 u ; u E [ sup jH f H f j ] 2.15

t t

tR tR

jjf jj 1

1 2 2

s s

 K u ;u ds

t;b

R R

0 2 1 2

 K u ;u ds:

T;b 2 s s

Hence

1 2 00 1 2

sup u ; u K sup u ;u ds: 2.16

2 r r 2

r r

r t r s

A Picard-Lindelof approximation yields that there is a solution u of the

x-p oint equation

u = u: 2.17

1 0 1 0

The approximation starts with u := P X where P is the sup er-

0sT

pro cess with b s; x=bm ;xwithsomem 2 M M E and for n 2 IN

0 0 1

n+1 n

we de ne u = u . Applying successively the inequality 2.16 with

n+1 n F n+1

u and u we obtain that there exists u := lim u , whichsolves

n!1

2.17. By the prop erty 2.17 the sup erpro cess P is a solution of the

non-linear martingale problem 1.1,1.2. The measure P is the unique

i i 1

solution b ecause if we denote by u ;i =1; 2; the owP X oftwo

i 1 2

solutions P of the martingale problem 1.1,1.2 then b oth u and u are

x-p oints of the equation 2.17. The prop erties 2.17 and 2.16 implies by

1 2 i u

;i =1; 2. Gronwall's inequality that u = u and therefore P = P 10

3 Non-linear one-particle motion

Nowwe consider the non-linear martingale problem 1.1,1.2 where E = IR

and Lm equals some nice partial di erential op erator Am bm, see

Theorem B, b elow. As in the last section wehave to b e able to couple two

di erent solutions of the non-linear martingale problem in order to prove

uniqueness. We will use the sto chastic calculus \along historical trees" de-

velop ed byPerkins in [P1,P2]. In order to describ e interacting sup erpro cesses

he constructs a unique solution of a strong integral equation, in whichthe

sto chastic integral is a \H-historical integral".

3.1 Stochastic calculus along historical trees

Let me recall some of the results in [P1,P2] sp ecialized to the case of non-

interactive parameters.

We xa T 0.

Let C = C [0;T];IR , and let C b e the canonical ltration on C , =

C [0;T];MC , = C with pro duct -algebra and let the Campb ell-

^ ^

typ e measure IP b e de ned by IP [A B ]:= IP [1 H B ]IP [H 1] ,where

A T T

the co ordinate pro cess H on the ltered probability space ; H; H ;IP is

the historical Brownian motion with branching rate 1 and with starting p oint

H .For the de nition of H we refer again to [P2, p.3]. Let F := H C .

0 t t t

d dd 0 d d

Let the functions :[0;T] IR ! IR ;d :[0;T] IR ! IR ,and

d d

c :[0;T] IR ! 0; 1beboundedandLipschitz continuous in x 2 IR .

@c @ c

We assume that s; and s; exist and are Lipschitz continuous in

@s @x x

i j

x with a Lipschitz constant uniform in s. De ne the functions a := ,

@c 1 @ c

s; x+ s; xa s; x, hs; x:= r cs; xand g s; x:=

ij x

i;j

@s 2 @x x

i j

0 1 0 1

d := d + ah c and b := g + h d c : 3.1

Theorem 3.1 [P2, Theorems 4.10 and 5.1,Example 4.4]

^ ^

a Let K := 1 c0;Y y H dy where Y : ! IR is F -

0 fY y 2g 0 0 0 0

measurable. Then thereisaF -predictable IR -valuedcontinuous pro-

t 11

cess Y and a F -predictable M C -valuedprocess K such that

Z Z

t t

Y y = Y y + s; Y y dy s+ d s; Y y ds 3.2

t 0 s s

0 0

K ! = Y !; y ct; Y !; yH ! dy ; 3.3

t t t

where the rst equation holds a.s. with respect to the rst component of

IP , i.e. w.r.t. Wiener measure with initial distribution IP [H ].The

second equality holds for al l 2 C C and 0 t T , IP -a.s.

-valuedprojection K of the M C -valuedpro- b We de ne the M IR

cess K by

 K f := f y sK dy ;f 2 C IR : 3.4

s s b

2 d

Under IP we have that for every f 2 C IR the process

M f := K f K f K Asf ds 3.5

t t 0 s

is a martingale with quadratic variation

Z Z

cs; xf x K dxds; 3.6

0 IR

P 2

@ f

1 d

where Asf x= f xbs; x+rf xds; x+ x a s; x

i;j =1

2 @x @x

i j

2 d

for f 2 C IR .

Pro of. The Theorem is a sp ecial case of [P2, Theorems 4.10 and 5.1]. In

Example 4.4 in [P2] the case of non-interactive c is considered. The fact that

the expression \H a:s:" used in Theorem 4.10 in [P2] is equivalent with

\a.s. with resp ect to Wiener measure" if all co ecients of the sto chastic

equation 3.2, 3.3 do not dep end on the the pro cess K follows by Remarks

3.3a and 3.13d in [P1].

3.2 Non-linear martingale problems

In the case of a non-linear martingale problem wewant to consider functions

a; b; c; d in 3.5 and 3.6 which dep end on the external force caused by 12

the distribution of K attimes. Hence we consider b ounded functions

0 d d d

a; b; c; d; d on M M IR IR instead of functions on [0;T] IR . We

assume that the functions are Lipschitz continuous with resp ect to b oth

variables where in the rst variable we use the Wasserstein metric on

M M IR cf.2.11, i.e., we assume

jr m ;x r m ;x jK m ;m +jx x j 3.7

1 1 2 2 r 2 1 2 1 2

for r = d ;;c;g and h. Note that 3.7 is a stronger condition than the

condition 2.13 on b. Caused by the di erentiability assumption for the

function c there is an additional condition on c which will b e formulated in

Theorem B, b elow. Fix now the starting p oint 2 M IR . The martingale

problem in question is to nd a probability measure on C [0;T];MIR 

2 d

such that for every f 2 C IR the pro cess

M f := X f f X AP X f ds 3.8

t t s

is a lo cal martingale with quadratic variation

Z Z

2 1

f xcP X X dxds; 3.9

0 IR

2 P

@ f

x. a m; x where Amf x=f xbm; x+ rf x dm; x

i;j =1

2 @x @x

i j

First we prove uniqueness.

Theorem B Let us suppose that there exist bounded and Lipschitz contin-

d d

uous functions c~ ; c~ ; c~ ; 1 i; j; k d,onM M IR IR such that for

0 1 jk 1

the ow u := P X of every solution P of 3.8, 3.9 we have

@ @

cu ;x=c ~ u ;x; cu ;x=c ~u ;x 3.10

s 0 s s i s

@s @x

cu ;x=c ~ u ;x and

s jk s

@x @x

j k

for 1 i; j; k d. Assume 3.7. Then there exists at most one proba-

d 0

bility measure P on C [0;T];MIR which solves 3.8,3.9 with d = d +

1 0 1

ah c ;b =g + hd c , where

hm; x = ~c m; x;:::;c~ m; x and 3.11

1 d

c~ m; xa m; x: 3.12 g m; x = c~ m; x+

jk jk 0

j;k =1 13

Pro of. Let P ;i =1; 2, b e two solutions of 3.8, 3.9. De ne the cor-

i i 1

resp onding ows by u = P X . We can apply Theorem 3.1 with

s s

i i 0;i 0 i i i

the functions s; x= u ;x, d s; x=d u ;x, c s; x=cu ;x

s s s

i i i i

g s; x=g u ;xand h s; x= hu ;x: Then the distributions of the pro-

s s

i i

cesses K ;i =1; 2; as de ned in 3.3, 3.4 with these functions equals P .

By Theorem 3.1b wehavethat

2 1 2 1 2 2

u ;u IP [ sup jK f K f j ]

s s t t

jjf jj 1

1 1 1 2 2 2 2

 IP [ sup jf Y t; y cu ;Y t; y f Y t; y cu ;Y t; y jH dy ]:

t t

jjf jj 1

This can b e b ounded by

1 2

IP [ fjjcjj jY t; y Y t; y j + 3.13

1 1 2 2 2

jcu ;Y t; y cu ;Y t; y jgH dy ]:

t t

i i 0 i i

Because Y t Y 0 d u ;Y s; y ds are continuous martingales for

j j 0 s

1;2

i i

we ;Y s; y ds and b ecause c 2 C a u 1 i d with covariation

b s

haveby the It^o-formula that

Z Z

t t

i i i i i

ct; Y t; y = c0;Y 0;y + hs; Y s; y dY s; y + g s; Y s; y ds:

0 0

Hence 3.13 equals

 

Z Z

1 1 2 2

IP u ;Y s; y u ;Y s; y dy s+

s s

0 1 1 0 2 2

d u ;Y s; y d u ;Y s; y ds jjcjj +

s s

2 1 2 1

;Y s; y dy s+ ;Y s; y h u h u

s s

 

2 1 0 2 0 1

;Y s; y ds H dy : ;Y s; y hd + g u hd + g u

s s

By Cauchy-Schwarz and the formula for the second moment of a sup erpro cess

this is b ounded by

 

Z Z

1 1 2 2

 u ;Y s; y u IP ;Y s; y dy s+

s s

0 14

0 1 1 0 2 2

d u ;Y s; y d u ;Y s; y ds jjcjj +

s s

1 1 2 2

h u ;Y s; y h u ;Y s; y dy s+

s s

 

0 1 1 0 2 2

hd + g u ;Y s; y hd + g u H dy ;Y s; y ds

s s

IP [H 1] + t

 

Z Z

1 1 2 2

= E u ;Y s; W u ;Y s; W dW s+

s s

0 1 1 0 2 2

d u s; W d u s; W ds jjcjj + ;Y ;Y

s s

2 1 2 1

;Y s; W dW s+ ;Y s; W h u h u

s s

 

0 1 1 0 2 2

hd + g u ;Y s; W hd + g u ;Y s; W ds

s s

IP [H 1] + t;

where W is a Brownian motion with initial distribution IP [H ] and Y s; W 

i i

is a solution of 3.2 with W instead of y and s; W = u ;W . Because

s s

hd + g and h also satisfy 3.7 we can b ound the last expression by

2 2 2 2 2

;tK ;K gjjcjj _ 1 4IP [H 1] + tmaxfK ;tK

0 0

hd +g h 1 d

1 2 2 1 2

E [jY s; W Y s; W j ]ds + u ;u ds

s s

1 2 2 1 2 0

E [jY s; W Y s; W j ]ds + u ;u ds: K

s s T

It remains to provethat

2 1 2 2 1 2

ds 3.14 ;u E [sup jY s; W Y s; W j ] K u 

s s 2

st

with a nite constant K .We de ne similarly as in [F],

0 1 1 0 2 2

A := d u ;Y s; W d u ;Y s; W ds

s s

2 1 2 1

;Y s; W dW s: ;Y s; W u u M :=

s s

0 15

By Burkholder-Davis-Gundy's inequalitywe obtain

E [sup jM j ]

st

d d

X X

1 1 2 2 2

 K 2 E [ j u ;Y s; W u ;Y s; W j ds

ij ij

s s

i=1 j =1

2 1 2 2

 K 2K tE [sup jY s; W Y s; W j ]

st

2 2 1 2

+K 2K u ;u ds

 2 s s

with some constant K 2. For A we obtain

 

Z Z

t t

2 2 1 2 1 2 2

E [sup jA j ] K E u ;u ds + jY s; W Y s; W jds

s 2

d s s

0 0

st

2 2 1 2 2 2 1 2 2

 2K t u ;u ds +2K t E [sup jY s; W Y s; W j ]:

0 0

d 2 s s d

st

Therefore

1 2 2 2 2

E [sup jY s; W Y s; W j ] 2E [sup jM j ]+2E [sup jA j ]

s s

st st st

2 2 1 2 2

 2K 2K +4K ttE [sup jY s; W Y s; W j ]

 d

st

2 1 2 2 2

 u ;u ds: t 2K 2K +4K

2 s s d

^ 1wehave 3.14. This implies by the previous Hence for t<

+4K 2K 2K

calculations that

2 1 2 0 2 1 2

 u ;u K u ;u ds

2 t t T 2 s s

1 2

for small t.Gronwall's lemma yields that u = u for small t. Exploring now

t t

the Markov prop ertyofthetwo solutions we obtain uniqueness for all t T ,

cf. [F], and the assertion is proved. 

Generally, existence of a solution to 3.8, 3.9 is proved byapproximation

with weakly interacting N-typ e sup erpro cesses, cf. the app endix. In order

to prove an existence result with the present techniques wehavetobemore

sp eci c ab out the function c, e.g. it suces that c is a nitely based function

with nitely basedbase functions. 16

3 k +d

Corollary 3.2 Let cm; x= mF ;:::;mF ;x with 2 C IR 

1 k

and F = f ;:::;f such that f ;j =1;:::;k ; ;i =1;:::;k.

i i 1i k i ji i i

We keep on assuming 3.7 and the boundedness of the functions a ;d ;d ; 1 

ij k

i; j; k d and b. Then there exists a unique solution of 3.8, 3.9.

Pro of. Uniqueness follows by Theorem B, if we takefor~c appropriate dif-

ferentiations of c.LetP denote the sup erpro cess with parameters dep end-

d 0

ing on the ow u 2 C [0;T]:M M IR , e.g. a s; x= a u ;xand

1 ij s

c s; x= u F ;:::;u F ;x. The starting p oint of the Picard-Lindelof

s 1 s k

0 u

approximation as in Theorem A is now u the ow of the sup erpro cess P

with parameter dep ending on some constant ow u = m for all s. De ne

n+1 n u 1

u = u =P X .Wehaveby the b oundedness assumptions

0sT

that

@c @c

n n n

u u u n1

j c s; x c s; y j sup E [Au F X ]K jx y j

i t 

s s

tT

 sup KE[H 1]jx y j

tT

with a nite constant K = K , where Am is de ned in 1.3.

; ;f ;a ;d ;b;c

i i ij

Hence the Lipschitz condition for in the Remark following Theorem 3.1 is

proved. It is straight forward to see that the other conditions for Theorem 3.1

n n+1

u u

are all satis ed. Hence we can construct P and P as a strong solution

of a sto chastic equation driven by a historical pro cess. Pro ceeding nowasin

n n+1

1 u 2 u

the pro of of Theorem B with P = P and P = P we are led to

2 n+1 n 0 2 n n1

 u ;u K u ;u ds;

2 t t T 2 s s

F F F u

which nally yields a solution u of u =u . The sup erpro cess P

solves 3.8,3.9. 

Of course, all assumptions on c are satis ed for constant c.

Corollary 3.3 Assume c is constant and and d arebounded and Lips-

chitz continuous in m; x with respect to in the rst component. Then

there exists a unique probability measure P on C [0;T];MIR which solves

3.8,3.9 with b =0.

Examples. Assume that c satis es the assumption of Theorem B. Let us

now give examples for whichwe can satisfy condition 3.1 in Theorem 3.1. 17

First of all a necessary condition is that hd + g = hah c + cb with h and g

as in 3.12 and 3.11, i.e.

d d

X X

cm; x c~ m; xa m; x+ c~ m; xd m; x 3.15

jk jk i i

i=1

j;k =1

= c m; xbm; x+ c~ m; xa m; x~c m; x:

j jk k

j;k =1

 Hence if c; a; d are given with c strickly p ositive, a p ossible choice is

d d

X X

bm; x = cm; x c~ m; xa m; x+ c~ m; xd m; x

jk jk i i

i=1

j;k =1

c~ m; xa m; x~c : cm; x

j jk k

j;k =1

It is clear that b is Lipschitz continuous and b ounded ifc; ~ a ;d are as

jk i

0 1

a c~ is also Lipschitz well. Under the same conditions d := d c

ij j

j =1 i

continuous and b ounded. Then the functions a; b; c; d; d satisfy all

assumptions of Theorem B.

 If a and b are given, a p ossible choice for the functions d and d is

d d

X X

d m; x= c~ m; x c~ m; xa m; x~c m; x

j j j jk k

j =1

j;k =1

cm; xbm; x c~ m; xa m; x

jk jk

j;k =1

0 1

and d = d c a c~ for 1 j d.

j jk i

j i=1

 If b =0 and a; d are given then c has to satisfy

d d

X X

c~ m; xd m; x c~ m; xa m; x+ cm; x

i i jk jk

i=1

j;k =1

= c~ m; xa m; x~c m; x; 3.16

j jk k

j;k =1

which seems to b e very restrictive. 18

3 d

 If cm; x= c xwithc 2 C IR then the functions in 3.10

0 0

are computed as follows: c m; x= 0;c m; x= x; c~ m; x=

0 i jk

@ c

x; 1 i; j; k d. But nevertheless the conditions an a; b and

@x @x

d are not much simpli ed. Therefore it seems that only a constant

branching variance c leads to reasonable concrete examples, cf. Corol-

lary 3.3.

A Propagation of chaos for weakly interacting sup er-

pro cesses

P 2 P

@ f @f

Theorem A.1 Let Lmf x:= a m; x x+ d m; x x be

ij i

@x @x @x

i j i

asecond order partial di erential equation operator on IR .

1. Let the functions a ;d ; 1 i; j; k d; c and b satisfy the fol lowing as-

ij k

d d

sumptions for functions r on M M IR IR

jr m ;x r m ;x jK m ;m +jx x j A.1

1 1 2 2 r 1 1 2 1 2

sup r m; x < 1 for each m 2 M M IR ; A.2

x2IR

where the Wasserstein metric = d d is de ned in 2.12, below.

M IR ;d 

IR ;j:j

Additional ly we assume that the the vectors X are exchangeable and that

one of the fol lowing growth conditions is satis ed:

sup jr m; xjdxmd K < 1 A.3

m2M M IR 

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

ij k

0 00

sup jr m; xjdx K 1 + K A.4

r r

m2M M IR 

for al l functions b; c; a ;d ; 1 i; j; k d: Then there exists an exchangeable

ij k

solution of the martingale problem associated with 1.5.

1;N

d 2 2

IR we have sup P [X f ] < 2. Assume additional ly that for each f 2 C

b N

1. Then the sequence f g M M C d of distributions of

N 2IN 1 1

M IR 

N 1

 j;N is tight and every accumulation point is supportedbythe

j =1

set of solutions of the martingale problem 1.1, 1.2. 19

3. If we assume nal ly that there exists only one solution P to the martingale

problem 1.1, 1.2 then the \Propagation of Chaos" holds.

Sketch of pro of. An exchangeable solution of 1.5 can b e constructed by

weak approximation with interacting multityp e branching di usions. The

condition A.3 resp. A.4 ensures the tightness of the interacting branch-

ing di usions as well as its existence as an accumulation p oint of a sequence

of branching random walks. The latter can b e constructed as marked p oint

pro cesses, which are exchangeable by construction. It is wellknown that the

N N

sequence f g is tight if sequence of the intensity measures fI g 

N N 2IN

N i;N 1;N

M C de ned by I F := E [F X ] = E [F X ] are

M IR 

i=1

tight. By wellknown criteria for tightness of measure-valued pro cesses, cf.

1;N

[D], weonlyhavetoshow that the distributions of fX f g are tightfor

d 2

IR . The latter follows bytightness criteria as in [EK], e.g. each f 2 C

the Aldous-Reb olledo criterium from the growth conditions A.3 or A.4.

The identi cation of the limit p oints of f g follows from the fact that

N 2IN

2 1

Q dQ= 0 with

M C 

M IR 

 

Z Z

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

f f s s

r C

M IR 

 

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

s f 1 k

By assumption A.1 is continuous and b ounded by the uniform integrable

i;N

1 N

function K K X 1 + K . The last part of the Theorem follows by

0 1

i=1 T

standard arguments of the \Propagation of Chaos" techniques, cf. [S1,S2].

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