getdoc43c2. 1329KB Jun 04 2011 12:04:19 AM

(1)

E l e c t ro n ic

Jo ur n

a l o

f P

r o b

a b i l i t y

Vol. 3 (1998) Paper no. 5, pages 1–120.

Journal URL: http://www.math.washington.edu/˜ejpecp/

Paper URL: http://www.math.washington.edu/˜ejpecp/EjpVol3/paper5.abs.html

Collision local times, historical stochastic calculus,

and competing superprocesses

Steven N. Evans and Edwin A. Perkins

Department of Statistics #3860, University of California at Berkeley 367 Evans Hall, Berkeley CA 94720-3860, USA

Department of Mathematics, University of British Columbia Vancouver BC V6T 1Z2, CANADA

evans@stat.berkeley.edu perkins@heron.math.ubc.ca

Abstract: _{Branching measure–valued diffusion models are investigated that can be regarded as pairs of historical Brownian}

motions modified by a competitive interaction mechanism under which individuals from each population have their longevity or fertility adversely affected by collisions with individuals from the other population. For 3 or fewer spatial dimensions, such processes are constructed using a new fixed–point technique as the unique solution of a strong equation driven by another pair of more explicitly constructible measure–valued diffusions. This existence and uniqueness is used to establish well–posedness of the related martingale problem and hence the strong Markov property for solutions. Previous work of the authors has shown that in 4 or more dimensions models with the analogous definition do not exist.

The definition of the model and its study require a thorough understanding of random measures called collision local times. These gauge the extent to which two measure–valued processes or anRd_{–valued process and a measure–valued process “collide”}

as time evolves. This study and the substantial amount of related historical stochastic calculus that is developed are germane to several other problems beyond the scope of the paper. Moreover, new results are obtained on the branching particle systems imbedded in historical processes and on the existence and regularity of superprocesses with immigration, and these are also of independent interest.

Keywords: super-process, super-Brownian motion, interaction, local time, historical process, measure-valued Markov branching process, stochastic calculus, martingale measure, random measure AMS subject classification: 60K35, 60J55, 60H99, 60G57

Date of submission: August 26, 1997 Date of publication in EJP: April 8, 1998

(2)

Introduction

1.1

Background

Consider populations of two different species which migrate, reproduce and compete for the same resources. As a result, when one individual gets close to one or more individuals of the other species its life expectancy decreases or its ability to successfully reproduce is diminished. Continuous space point processes have recently been used by mathematical biologists to model such competing species (see [3], [4] and [28]). One goal of this work was to demonstrate the importance of incorporating realistic spatial structure into population biology models (in contrast to the classical approach that seeks to model total population size directly and usually involves implicit “stirring” assumptions).

Although the biologists are dealing with organisms of a given size and finite interaction range, from a mathematical perspective it is natural to consider a scaling limit in which the interaction becomes purely local and the total population becomes large. In the regime of critical or near critical branching the scaling limit without the inter–species competition is a superprocess (super–Brownian motion if the migration is given by a random walk), and so when competition is present one may expect a scaling limit that is a pair of locally interacting super–Brownian motions. The purpose of this paper is to continue the analysis of such models that was begun in [20].

Super–Brownian models in which the mortality or fertility of individuals is subject to local effects are relatively easy to construct and analyse in one dimension. This is for two reasons. Firstly, one– dimensional super–Brownian motion takes values in the set of measures which are absolutely continuous with respect to Lebesgue measure, and so describing the extent to which two populations collide is easy. Secondly, the interacting model solves a martingale problem that looks like the one for a pair of independent super–Brownian motions except for the addition of tame “drift” terms. The law of such a process can therefore be constructed by using Dawson’s Girsanov theorem (see [5]) to produce an absolutely continuous change in the law of a pair of independent super–Brownian motions (see Section 2 of [20]).

Neither of these features is present in higher dimensions: super–Brownian motion takes Lebesgue– singular values and the model we wish to construct has a law that is not absolutely continuous with respect to that of a pair of independent super–Brownian motions (see Theorem 3.11 of [20]). A substantial body of new techniques is therefore needed.

(4)

are studied in [10].

1.2

Historical Brownian motion

We begin with some general notation and the definition of historical Brownian motion. The process of interest to us can be thought of as a pair of “competing” historical Brownian motions.

WriteC for the spaceC(R+,Rd_{) of continuous functions from}_R

+ intoRd. Givenψ∈Cb2(Rnd,R)

(_≡ functions on Rnd with bounded partial derivatives of order 2 or less) and 0 _≤ t1 < . . . tn, let

¯

ψ= ¯ψ(t1, . . . , tn)(·) :C→Rbe given by ¯ψ(y) =ψ(y(t1), . . . , y(tn)). Put ¯y(t) = (y(t∧t1), . . . , y(t∧tn))

and

¯ ∆

2ψ¯(t, y) = 1 2

d

X

i=1

n−1

X

k=0

n−1

X

ℓ=0

1_{t < tk+1∧tℓ+1}ψkd+i,ℓd+i(¯y(t))

(hereψαβ, 1≤α, β≤nd, are the second order partial derivatives ofψ.)

Set

DS ={ψ¯(t1, . . . , tn) : 0≤t1< . . . < tn, n∈N, ψ∈CK∞(Rnd)} ∪ {1},

whereC∞

K(Rnd) is the space ofC∞functions with compact support. Setyt(·) =y(· ∧t) fory∈C, put

D1={φ:R+×C→R : φ(t, y) =φ1(t)φ2(yt), φ1∈C1(R+), φ2∈DS},

and letDST denote the linear span ofD1. Forφ∈D1 withφ(t, y) =φ1(t)φ2(yt) (whereφ1,φ2are as

above) define

Aφ(s, y) =φ′₁(s)φ2(ys) +φ1(s)∆¯

2φ2(s, y), (1.1)

and extendAlinearly toDST. SetSo={(t, y)∈R+×C:yt=y}.We identifyφ∈DST andAφwith

their restrictions toSo_.

Given a measurable space (Σ,_A), letMF(Σ) denote the space of finite measures on Σ. The

nota-tionb_A(respectively,bp_A) denotes the set of bounded (respectively, bounded positive)_A–measurable functions. We will often use the functional notationρ(f) to denoteR fdρforρ_∈MF(Σ). When Σ is

separable metric space and_A is the corresponding Borelσ–field (which we will denote by _B(Σ)), we will equipMF(Σ) with the topology of weak convergence.

PutMF(C)t={µ∈MF(C) :yt=y, µ−a.e.y}fort≥0. Write ΩH[τ,∞[ for the space of

contin-uous,MF(C)–valued paths,h, on [τ,∞[ such thatht∈MF(C)tfor allt≥τ. Let (Ω,F,(Ft)t≥τ,P) be

a filtered probability space with the filtration (_Ft)t≥τ right–continuous and theσ–fieldF universally

(5)

on (Ω,_F,(_Ft)t≥τ,P)started at(τ, µ) if (Ht)t≥τ is a (Ft)t≥τ–predictable processes with sample paths

almost surely in ΩH[τ,∞[ such that for allφ∈DS

Mt(φ) =Ht(φ)−µ(φ)−

Z t

τ

Hs(

¯ ∆

2φs)ds, t≥τ is a continuous (_Ft)t≥τ–martingale for whichMτ(φ) = 0 and

hM(φ)it=

Z t

τ

Z

C

φ(y)2Hs(dy)ds.

By Theorem 1.3 of [31] this martingale problem uniquely determines the law ofH,Qτ,µ, on ΩH[τ,∞[.

A simple application of Itˆo’s lemma shows that forφ_∈DST,

Mt(φ) =Ht(φt)−µ(φτ)−

Z t

τ

Hs(Aφs)ds, t≥τ

is a continuous (_Ft)t≥τ–martingale for whichMτ(φ) = 0,

hM(φ)it=

Z t

τ

Z

C

φ(s, y)2Hs(dy)ds.

(Use the fact thatφ2(y) =φ2(yt₎_{, H}

t-a.a. y, ∀t≥τ, a.s.,∀φ2∈DS.)

Let ΩX[τ,∞[=C([τ,∞[,MF(Rd)) and define Γ : ΩH[τ,∞[→ΩX[τ,∞[ by putting

Γ(h)t(φ) =

Z

φ(yt)ht(dy).

If m(A) = µ(yτ ∈ A) and H is as above then X = Γ(H) is a super–Brownian motion on

(Ω,F,(Ft)t≥τ,P) starting at m at time τ with law on ΩX[τ,∞[ that we will write as Pτ,m. This

follows from the martingale problem characterisation of super–Brownian motion (see Ch. 6 of [5]).

1.3

Collision local times

In order describe how the martingale problem for a pair of independent historical Brownian motions has to be modified in order to reflect decreased longevity/fertility due to local collisions, we need to define an object that measures such collisions. Suppose that (K1

t)t≥τ and (Kt2)t≥τ are two predictable,

MF(C)–valued processes on (Ω,F,(Ft)t≥τ,P) such that almost surely Ki has paths in ΩH[τ,∞[ for

i= 1,2. Forǫ >0 define a continuous,MF(C)–valued process (Lǫt(K1, K2))t≥τ by setting

Lǫ_t(K1, K2)(φ) =

Z t

τ

Z Z

pǫ(y1(s)−y2(s))Ks2(dy2)

φ(y1)Ks1(dy1)ds,

(6)

We say that K1 _and _K2 _{have a} _{field–field collision local time} _or _FFCLT ₍_L

t(K1, K2))t≥τ if

(Lt(K1, K2))t≥τ is an (Ft)t≥τ–predictable,MF(C)–valued process with sample paths almost surely in

ΩH[τ,∞[ such that

lim

ǫ↓0L

ǫ

t(K1, K2)(φ) =Lt(K1, K2)(φ)

in probability for allt_≥τ and all bounded continuous functionsφonC.

If (Lt(K1, K2))t≥τ exists, then it is unique up to evanescent sets. Almost surely for allτ ≤s < t

and allA_{∈ C}we haveLs(K1, K2)(A)≤Lt(K1, K2)(A), and so there is an almost surely unique Borel

random measure on ]τ,∞[×C that we will denote as L(K1, K2) such that L(K1, K2)(]s, t]×A) =

Lt(K1, K2)(A)−Ls(K1, K2)(A). Adjoin an isolated point ∆ to MF(C) to form M∆_F(C) The same

notation as above will be used for M∆_F(C)–valued processes, with ∆ treated as the 0 measure for purposes of integration.

Remark 1.1 Define continuous MF(Rd)–valued processes (Wt1)t≥τ and (Wt2)t≥τ by setting Wi =

Γ(Ki_{). The continuous}_M

F(Rd)–valued process (Lt(W1, W2))t≥τ defined by

Lt(W1, W2)(φ) =

Z t

τ

Z

φ(y(s))L(K1, K2)(ds, dy)

is the collision local time ofW1andW2defined and studied in [2] and [20]. Note thatLt(W1, W2) =

Lt(W2, W1) (see Section 1 of [2]), but it is certainly not the case that Lt(K1, K2) =Lt(K2, K1) in

general. Note also that we perhaps should writeL(K1_{, W}2_{) for}_L₍_K1_{, K}2_{), as this object only depends}

onK2 _through_W2_.

A substantial portion of this paper is devoted to investigating the properties of FFCLT’s (see Section 3). One of the advances over what was accomplished in [2] and [20] (and the key to the rest of the paper) is the demonstration that, loosely speaking, when (K1_{, K}2_{) are, in a suitable sense, “sub–}

populations” of a pair of independent historical Brownian motions with d _≤3, then L(K1_{, W}2₎₍_dt₎

can be written as R ℓ(y, W2₎₍_dt₎_K1

t(dy), where ℓ(y, W2) is a measure called the path-field collision

local time(PFCLT) that lives on the set of timest such thaty(t) is in the support ofW2

t. In essence,

if y is chosen according to Wiener measure, then ℓ(y, W2_{) is the inhomogeneous additive functional}

with Revuz measureW2

t at timet. Our main tools for studying PFCLT’s is a Tanaka–type formula for

ℓ(y, W2_{) in the case when}_y_{is chosen according to Wiener measure and}_W2_{is a suitable sub-population}

(7)

1.4

Smooth measures

The last ingredient we need before describing our interacting historical processes model is a suitable state–space and path–space for the model. As one might imagine, if the initial disposition of the two populations is not sufficiently “dispersed”, then the killing mechanism we wish to introduce may be so intense that the process undergoes some sort of initial catastrophe rather than evolving continuously away from its starting state. On the other hand, we can’t be too restrictive in our class of initial measures, because in order to have a reasonable strong Markov property we need the process to take values in this class at all times.

Let MF S(Rd) ⊂ MF(Rd) be the set of measures, µ, such that R₀1r1−dsupxµ(B(x, r))dr < ∞,

where B(x, r) is the closed ball of radius r centred at x _∈ Rd. Write MF S(C)t for the subset of

MF(C)tconsisting of measuresµwith the property thatµ(y(t)∈ ·)∈MF S(Rd), that is

Z 1

0

r1−d sup

x∈Rd

µ(_{y:_|y(t)₋x_{| ≤}r_})dr <_∞.

Following the pattern of notation used in Appendix A set

S_C′ =_{(t, µ)_∈R₊_×MF(C) :µ∈MF S(C)t},

ˆ

S ={(t, µ1, µ2)∈R₊_×MF(C)×MF(C) :µi∈MF S(C)t, i= 1,2}

and

Ω′

C={ω∈C(R+,M∆F(C)) :αC(ω)<∞, βC(ω) =∞}

where

αC= inf{t:ω(t)6= ∆}andβC= inf{t≥αC: (t, ω(t))∈/SC′ } (inf∅=∞),

and we use the notationC(R+,M∆

F(C)) for the subset of the spaceD(R+,M∆F(C)) of c`adl`agM∆F(C)–

valued functions consisting of functions h such that ∆ _{∈ {}/ h(t₋), h(t)_} implies h(t₋) = h(t) for all

t >0. Let_F′

C be the trace of the universally measurable subsets ofC(R+, MF∆(C)) on Ω′C

1.5

Competing species martingale problem

Here then is a martingale problem for two interacting historical processes for whom inter–species collisions are unfavourable. Let ( ˆΩ,Fˆ,( ˆFt)t≥τ,Pˆ) be a filtered probability space with the filtration

( ˆ_Ft)t≥τ right–continuous and the σ-field ˆF universally complete. Suppose that r1, r2, τ ∈ R+ and µ1_{, µ}2 _∈ _M

F(C)τ. Suppressing dependence on (r1, r2), we say that the pair ( ˆH1,Hˆ2) satisfies the

martingale problemM PdH(τ, µ1, µ2) if:

(i) ( ˆHi

(8)

(ii) ˆHi

t= ∆,∀t < τ a.s.,i= 1,2,

(iii) ( ˆHi

t)t≥τ is ( ˆFt)t≥τ–predictable,i= 1,2,

(iv) L( ˆH1_,_Hˆ2_{) and}_L_{( ˆ}_H2_,_Hˆ1_{) exist,}

(v) for allφ1, φ2∈DS

ˆ

M1

t(φ1) = ˆHt1(φ1)−µ1(φ1)−

Z t

τ

ˆ

H1

s(

¯ ∆

2φ1(s))ds−r1

Z t

τ

Z

φ1(y)L( ˆH1,Hˆ2)(ds, dy), t≥τ,

and ˆ

M_t2(φ2) = ˆH_t2(φ2)−µ2(φ2)−

Z t

τ

ˆ

H_s2(∆¯

2φ2(s))ds−r2

Z t

τ

Z

φ2(y)L( ˆH2,Hˆ1)(ds, dy), t≥τ,

are continuous ( ˆ_Ft)t≥τ–martingales for which ˆMτi(φi) = 0,

hMˆi(φi)it=

Z t

τ

Z

φi(y)2Hˆsi(dy)ds, ∀t≥τ, a.s.

and_hMˆ1₍_φ

1),Mˆ2(φ2)i= 0.

A simple application of Itˆo’s lemma shows that if ( ˆH1_,_Hˆ2_{) satisfies}_{M P}d

H(τ, µ1, µ2) and φ1, φ2∈

DST, then

ˆ

M1

t(φ1) = ˆHt1(φ1(t))−µ1(φ1(τ))−

Z t

τ

ˆ

H1

s(Aφ1(s))ds−r1

Z t

τ

Z

φ1(s, y)L( ˆH1,Hˆ2)(ds, dy), t≥τ,

and ˆ

M_t2(φ2) = ˆH_t2(φ2(t))₋µ2(φ2(τ))₋

Z t

τ

ˆ

H_s2(Aφ2(s))ds₋r2

Z t

τ

Z

φ2(s, y)L( ˆH2,Hˆ1)(ds, dy), t_≥τ,

are continuous ( ˆFt)t≥τ–martingales for which ˆMτi(φi) = 0,

hMˆi(φi)it=

Z t

τ

Z

φi(s, y)2Hˆsi(dy)ds

and_hMˆ1₍_φ

(9)

Remark 1.2 Suppose that the pair ( ˆH1_,_Hˆ2_{) solves the martingale problem}_{M P}_d

H(τ, µ1, µ2) and ˆXi=

Γ( ˆHi_{). Set} _νi _{= ˆ}_Xi

τ, so that νi(φ) =

R

φ(y(τ))µi₍_dy_{). Then for all bounded, continuous functions}

f1, f2 with bounded, continuous first and second order partial derivatives we have ˆ

Z_t1(f1) = ˆX_t1(f1)₋ν1(f1)₋

Z t

τ

ˆ

X_s1(∆

2f1)ds−r1Lt( ˆX

1_,_X_ˆ2₎₍_f1₎_{, t}

≥τ

and

ˆ

Z_t2(f2) = ˆXt2(f2)−ν2(f2)−

Z t

τ

ˆ

X_s2(∆

2f2)ds−r2Lt( ˆX

2_,_X_ˆ1₎₍_f

2), t≥τ

are continuous ( ˆ_Ft)t≥τ–martingales for which ˆZτi(fi) = 0 and

hZˆi(fi),Zˆj(fj)it=δij

Z t

τ

ˆ

X_si((fi)2)ds.

Denote this latter martingale problem as M PdX(τ, ν1, ν2). Thus M PdX(τ, ν1, ν2) is the “two–way

killing” martingale problem discussed in [20]. There it was shown that solutions exist if and only if

d_≤3, and so throughout this work we will always assume

d≤3.

Roughly speaking, the two populations don’t collide enough in higher dimensions to give a non-trivial interaction — even though a non-trivial FFCLT exists for two independent super Brownian motions whend≤5 (see [2]).

Remark 1.3 For d≤3 the description of solutions ( ˆH1_,_Hˆ2_{) to} _{M P}_d

H(τ, µ1, µ2) developed in this

paper can be summarised (very loosely) as follows:

• There is a pair (H1_{, H}2_{) of independent historical Brownian motions such that ˆ}_Hi

t≤Hti for allt.

• If we “choose a pathy_∈C according toHi

t”, then y is a Brownian path stopped att.

• Any such path is either absent from the support of ˆHi

t or present with the same infinitesimal mass

(a little more precisely, the Radon–Nikodym derivative of ˆHi

tagainstHti is{0,1}–valued).

• The pathsy in the support ofHi

t that are also in the support of ˆHti are the ones that have survived

being killed off according to the random measureriℓ(y,Xˆj) (wherej= 3−i).

(10)

be established for models with other local interactions, such as the one obtained by modifying the above intuitive description so that particles are killed as soon as they encounter the other population (that is, by settingri=∞). This will be discussed in a forthcoming paper with Martin Barlow.

Our major result concerning M PdH(τ, µ1, µ2) is the following.

Theorem 1.4 Let r1, r2≥0and (τ, µ1_{, µ}2₎_∈_Sˆ_.

(a) There is a solution ( ˆH1_,_Hˆ2₎_of_{M P}_d

H(τ, µ1, µ2).

(b) There is a lawPˆτ,µ1_,µ2

on(Ω′

C×Ω′C,FC′ × FC′ )which is the law of any solution ofM PdH(τ, µ1, µ2).

Moreover, for any A_{∈ F}′

C× FC′ the map (τ, µ1, µ2)7→Pˆτ,µ

1_,µ2

(A) is Borel measurable fromSˆtoR. (c) Let Hˆ = ( ˆH1_,_H_ˆ2₎_{be a solution of} _{M P}_d

H(τ, µ1, µ2)on ( ˆΩ,Fˆ,( ˆFt)t≥τ,P)ˆ and let T ≥τ be an a.s.

finite ( ˆFt)t≥τ–stopping time. Then

ˆ

Phφ( ˆHT+·)|FˆT

i

(ω) =

Z

φ(ω′(·+T(ω))) ˆPT(ω),HˆT(ω)₍_dω′₎_, _forˆ_P_-a.e. _ω_∈_Ωˆ

for any bounded, measurable functionφonC(R+,MF(C)×MF(C)).

Remark 1.5 If r1 = 0 or r2 = 0, then one may easily adapt the methods of [20] (Section 4) to see thatM PdH(τ, µ1, µ2) is well-posed, and so we will always assume r1, r2>0.

1.6

Approximating systems

Remark 1.3 and the way we make it precise in the course of this paper provides a strong justification for our claim that M PdH(τ, µ1, µ2) and M PdX(τ, ν1, ν2) realistically capture certain features of two

evolving populations subject to purely local inter-species competition. However, additional support for this assertion and a firmer connection with more biologically realistic models would be established if we could show that models satisfyingM PdX(τ, ν1, ν2) arise as high–density limits of discrete branching

particle systems that incorporate mutually disadvantageous encounters between two populations. One such class of models is given by the following family of two–type, long–range contact processes.

Consider two types of particles that occupy sites on the rescaled integer latticeN−1 2M−1

N Zd, where

N, MN∈N. Several particles of one type can occcupy a site, but particles of different types cannot

co-exist at a site. At rateN, a particle of either type at sitex∈N−1 2M−1

N Zdeither dies (with probability

1

2) or (also with probability 12) it selects a site at random from the set of sites

{y_∈N−1 2M−1

N Zd:|yi−xi| ≤

√

6N−1

2, 1≤i≤d}

and produces an offspring there if no particle of the other type is already occupying the site. Define a pair of c`adl`agMF(Rd)–valued processes ( ˆXN,1,XˆN,2) by

ˆ

(11)

Increasing MN lessens the rate at which attempted births fail because of inter–species collisions.

Heuristic calculations show that ifd_≤3,MNis taken as√6r−

1

dN

2

d−

1

2 withr >0 fixed and ˆXN,i 0 →νi

asN → ∞,i= 1,2, then ( ˆXN,1_,_XˆN,2_{) converges in distribution to a solution of}_{M P}_d

X(0, ν1, ν2) with

r1=r2=r. IfMN grows slower than this critical rate, then we expect that the limit exists and solves

a martingale problem in whichL( ˆXi_,_Xˆj₎₍_{dt, dx}_{) is replaced by a more singular measure living on the}

times and places of “collisions”. In the extreme case whenMN = 1 for allN, we expect the limit to be

the model discussed above in which collisions of particles with the other population are immediately fatal. On the other hand, ifMN grows faster than the critical rate, then we expect the interaction to

disappear and the limit to be just a pair of independent super–Brownian motions.

Moreover, just as super–Brownian motion is “universal” in that many superficially different se-quences of branching particle systems converge to it, we expect that a large class of models that incorporate near–critical branching, spatial motion converging to Brownian motion and some form of inter–species competition will converge in the high–density limit to a solution ofM PdX(τ, ν1, ν2). We

leave the investigation of such questions to future work.

There are also other simple, not–so–singular, measure–valued diffusion models that converge to so-lutions ofM PdX(τ, ν1, ν2) andM PdH(τ, µ1, µ2). For example, letM Pd

ǫ

X(τ, ν1, ν2) andM Pd ǫ

H(τ, µ1, µ2)

denote the analogously defined martingale problems with the field–field collision local times L(·,·) replaced by the approximationsLǫ₍_·_,_·_{). Dawson’s Girsanov theorem shows that} _{M P}_dǫ

X(τ, ν1, ν2) is

well–posed for ν1_{, ν}2 _∈ _M

F(Rd) (see Theorem 2.3 (b) in [20]). We now set τ = 0 and identify

MF S(Rd) with MF S(C)0. Let ˆQν

1_,ν2

ǫ denote the law on ΩX ×ΩX (where ΩX ≡ ΩX[0,∞[) of the

solution to M Pdǫ_X(0, ν1_{, ν}2_{). The same Girsanov argument shows that the law, ˆ}_Pν1_,ν2

ǫ on ΩH×ΩH

(where ΩH ≡ΩH[0,∞[), of any solution toM Pd ǫ

H(0, ν1, ν2) is unique. If ν1, ν2 ∈ MF S(Rd), then

{Qˆν_ǫ1,ν2 :ǫ_∈[0,1]_}is tight and any subsequential weak limit satisfiesM PdX(0, ν1, ν2) (see Theorem 3.6

of [20]). Theorem 1.4 allows us to strengthen this conclusion. We let ( ˆH1_,_Hˆ2_{) denote the coordinate}

maps on ΩH×ΩH in the following result, which will be proved at the end of Section 7.

Theorem 1.6 Assume ν1_{, ν}2_∈_M

F S(Rd).

(a)Qˆν1_,ν2

ǫ converges weakly toPˆ0,ν

1_,ν2

((Γ( ˆH1₎_,_{Γ( ˆ}_H2₎₎_{∈ ·}₎_on _Ω

X×ΩX asǫ↓0.

(b)Pˆν_ǫ1,ν2 converges weakly toPˆ0,ν1,ν2 onΩH×ΩH asǫ↓0.

1.7

Overview of the paper

Our overall strategy for proving Theorem 1.4 is to make rigorous sense of the intuition laid out in Remark 1.3.

(12)

the path–field collision local time as an inhomogeneous additive functional of Brownian motion with super-Brownian motion as the Revuz measure.

The most naive idea for making sense of the notion that “a path in the support ofHi

tthat is also in

the support of ˆHi

tis one that has survived being killing according to a multiple of its path–field collision

local time against ˆXj _{= Γ( ˆ}_Hj_{) (}_j _{= 3}₋_i_{)” is to somehow equip each path} _y _{in the support of}_Hi t

with the points of a Poisson process on [τ,_∞[ with intensityriℓ(y,Xˆj) and then to kill off paths that

receive one or more points. Of course, there is something a little circular in this, because it appears we already need to have ( ˆH1_,_Hˆ2_{) in order to define such Poisson processes.}

We proceed a little more obliquely by, loosely speaking, constructing a pair of independent historical Brownian motions (H1_{, H}2_{) in which each path}_y _{in the support of}_Hi

t is equipped with a number of

R₊_×[0,1]–valued marks. Conditional on (H1_{, H}2_{), the marks are laid down according to a Poisson}

process with intensityriℓ(y, Hj)⊗m, wherej= 3−iandmis Lebesgue measure on [0,1]. Moreover,

the marks inherit the branching structure of (H1_{, H}2_{). For example, if}_y

1 in the support ofHti1 and y2 in the support of Hti2 are such that that y

s

1 =ys2 for some s≤ t1∧t2 and the two paths diverge

after time s, then the corresponding marks coincide up to time s but are laid down independently (conditional on (H1_{, H}2_{)) thereafter. We call this pair of historical Brownian motions with added}

marks thedriving process. With the aid of the driving process, we can define ( ˆH1_,_Hˆ2_{) as the pair such}

that if we kill a pathy in the support ofHi

t at the first time thaty has an attached mark (u, z) for

which the Radon–Nikodym derivative (dℓ(y,Xˆj₎_/dℓ₍_{y, X}j₎₎₍_u_{) is greater than} _z_{, then we recover ˆ}_Hi t.

We call this implicit definition of ( ˆH1_,_Hˆ2_{) the} _{strong equation}_{. We establish pathwise existence and}

uniqueness of solutions to the strong equation in Section 5 using a fixed–point argument, and show that the unique solution satisfiesM PdH(τ, µ1, µ2).

The key to proving uniqueness of solutions toM PdH(τ, µ1, µ2) (that is, Theorem 1.4(b)) is then to

show that given any solution such solution ( ˆH1_,_Hˆ2_{) we can, with extra randomness, build a driving}

process such that the ( ˆH1_,_Hˆ2_{) is the solution of the strong equation with respect to the driving process.}

This is carried out in two stages in Sections 6 and 7. Part of the argument depends on working with processes such as (H_ti,ǫ)t≥τ+ǫ, whereHti,ǫ(φ) =

R

φ(yt−ǫ₎_Hi

t(dy). The random measure Hti,ǫis atomic,

the corresponding set–valued process of supports is a branching particle system andHi,ǫ_{converges to}

Hi _as _ǫ _↓_{0. Several results of this sort are in [9], but we need to obtain a number of stronger and}

more general results. The advantage of working with such approximating, embedded particle systems is that attaching the necessary marks to paths in order to reconstruct the driving processes (or, rather, their approximating particle systems) is reduced to a fairly straightforward exercise involving a finite number of branching paths.

In Section 8 we observe that, as expected, this uniqueness translates into the strong Markov property Theorem 1.4(c), and this in turn gives the existence of strong Markov solutions toM PdX(τ, ν1, ν2).

We remark that throughout the paper we are continually dealing with processes of the formGt(φt),

(13)

measures on some space of paths and φtis defined as the result of performing some sort of Lebesgue

or stochastic integration along a path (possibly with a random integrand). We develop a number of new stochastic calculus tools to obtain the the semimartingale decompositions of such processes. This sort ofhistorical stochastic calculushas proven to be useful in a number of other contexts such as superprocesses with spatial interations in [30] and [31], nonlinear superprocesses with McKean–Vlasov– like mean field interactions in [29], and explicit stochastic integral representations of functionals of superprocesses in [21]. Also, in Sections 4 and 6 we require new results on the existence and regularity of superprocesses with immigration for quite general spatial motions. These results are presented in Appendix A. The main novelty lies in establishing Hunt and sample-path continuity properties and in deriving semimartingale decompositions.

Unfortunately, we are unable to show uniqueness of solutions for M PdX(τ, ν1, ν2). This has been

accomplished recently by Leonid Mytnik [27] using a novel and complex duality argument.

The methods developed in this work have proved useful in other settings. For example, they also apply to a related model in which the masses of particles are reduced by inter–species collisions. Rather than state the martingale problem for this model, we go straight to the corresponding strong equation. Here the “driving process” is just a pair (H1_{, H}2_{) of independent historical Brownian motions and the}

strong equation is

ˆ

H_ti(φ) =

Z

φ(y) exp(₋riℓt(y,Hˆj))Hti(dy)

fori= 1,2 andj= 3−i. This model will be investigated in a forthcoming paper with Martin Barlow, where it will be shown that a fixed–point technique similar to the one used here suffices to establish existence and uniqueness of solutions. Because of their utility in other contexts, we emphasise the tools for dealing with strong equations and the historical stochastic calculus associated with collision local times as an important feature of this work.

This is a rather large paper with a (necessarily) extensive amount of notation. To assist the reader, we have collected some of the notation together with where it first appears into a table in Appendix D.

Acknowledgement.It is a great pleasure to thank Martin Barlow for a number of valuable contribu-tions to this work, Loren Pitt for a helpful discussion concerning moment problems, and an anonymous referee whose careful scrutiny added substantially to the readability of the paper.

2

Brownian collision local times

Let (_Mt)t≥τ denote the universal completion of the canonical right–continuous filtration on ΩX[τ,∞[,

where τ ≥ 0 is fixed. Write (Ct)t≥τ for the canonical filtration on C([τ,∞[,Rd). Put ˆΩX[τ,∞[=

(14)

Forǫ >0, define a continuous ( ˆ_Mt)t≥0–predictable process on ˆΩX[τ,∞[ by

ℓǫ_τ,t(y, ν) =

Z t

τ

Z

pǫ(y(s)−x)νs(dx)ds, t≥τ.

Dependence onτ will usually be suppressed. Note that the expression forℓǫ

τ,t(y, ν) still makes sense

whenever ν : [τ,_∞[_→ MF(Rd) is such that s 7→ νs(A) is Borel measurable for all A ∈ B(Rd) and

sup_τ_≤_s_≤_tνs(1)<∞,∀t >0. We will use this generality on occasion in the next section.

Notation 2.1 LetC↑be the set of nondecreasing continuous functions,f(t), onR+such thatf(0) = 0.

Ifg_∈C↑, let

Mg_{F S}(Rd) ={µ∈MF(Rd) :

Z ǫ

0

r1−dsup

x µ(B(x, r))dr≤g(ǫ), ∀ǫ∈[0,1]}.

Recall from Section 1 thatMF S(Rd) is the set of measures,µ, such thatR₀1r1−dsupxµ(B(x, r))dr <∞.

Finally let

¯

g(ν, ǫ) = sup

τ′_≥_τ,x_∈Rd

Z ǫ

0

ps(x−y)ντ′₊_s(dy)ds

and

ΩXS[τ,∞[={ν∈ΩX[τ,∞[: lim

ǫ↓0¯g(ν, ǫ) = 0 andνt= 0 fortsufficiently large}.

LetW = (T, B0) be a space–time Brownian motion, that is, a diffusion with state–spaceR+×Rd and

laws

Qs,x(W _∈A) =Px(_{y: (s+_·, y(_·))_∈A_}),

forAa Borel subset ofC(R+,R+×Rd),s≥0 andx∈Rd, wherePxis Wiener measure starting atx.

LetB(t) =B0(t−T(0)),t≥T(0), be Brownian motion starting atxat times.

The following result is an easy consequence of Theorem 4.1 and Proposition 4.7 of [20] and their proofs. Recall from Subsection 1.2 thatPτ,m_{is the law of super-Brownian motion started at time}_τ _at

the measurem.

Theorem 2.2 (a) For each path ν in ΩXS[τ,∞[, there is a (Ct)_t_≥_τ–predictable map ℓˆt(y, ν) on

[τ,∞[×C([τ,∞[,Rd)such that ℓˆτ+·(y, ν)∈C↑,Qτ,x-a.s., for each xinRd and

lim

ǫ↓0_xsup_∈Rd

Qτ,x

sup

t≥τ

ℓǫ

t(B, ν)−ℓˆt(B, ν)

2

= 0. (2.1)

(b) For each g ∈ C↑ there is Borel subset, ΩgXS[τ,∞[, of ΩXS[τ,∞[, a sequence ǫn ↓ 0, and an

(15)

(i) S_g_∈_C_↑Ωg_XS[τ,_∞[= ΩXS[τ,∞[and Pτ,m(ΩgXS[τ,∞[) = 1,∀m∈M g F S(Rd).

(ii) Ifν ∈Ωg_XS[τ,∞[, thenℓt(B, ν) = ˆℓt(B, ν),∀t≥τ,Qτ,x-a.s.,∀x∈Rd and so (2.1) holds with ℓ

in place ofℓˆ.

(iii) Let (Bt)t≥τ be a Brownian motion on (Ω,F,Ft,P′)starting with law µ at time τ, and let τ ≤

T _{≤ ∞} be an(_Ft)t≥τ–stopping time. Then for allν in ΩgXS[τ,∞[,

lim

ǫ↓0P

′

sup

τ≤t≤T , t<∞

ℓǫ

t(BT, ν)−ℓt(BT, ν)

2

= 0

and

lim

n→∞_τ_≤_t_≤sup_{T , t<}_∞

ℓǫn

t (BT, ν)−ℓt(BT, ν)

= 0 P′-a.s.

(iv) For all(t, y, ν),ℓt(y, ν) =ℓt(yt, νt), whereνst=νsfors≤t andνst= 0 otherwise.

Proof. Part (a) is immediate from the results cited above.

For (b), one easily can use these arguments to define an increasing sequence of Borel subsets

{Ωg,n_XS[τ,_∞[: n _∈ N_}such that Ω_XSg [τ,_∞[= S_nΩg,n_XS[τ,_∞[ satisfies (i), and a decreasing sequence of positive numbers ǫn↓0 such that

sup

Qτ,x

sup

t≥τ

ℓǫ

t(y, ν)−ℓδt(y, ν)

2

: 0< ǫ, δ_≤ǫn, x∈Rd, ν ∈Ωg,nXS[τ,∞[

<2−n_. _(2.2)

(An explicit definition of Ωg,n_XS[τ,∞[ is given below.) Now set

ℓt(y, ν) =

(

limn→∞ℓǫtn(y, ν), if the limit exists,

0, otherwise. (2.3)

Property (ii) is then clear. We need only comment on the extension to stopped Brownian motions in (iii). Note that

ℓǫ

t(BT, ν) =ℓǫt(B, ν) and ℓt(BT, ν) =ℓt(B, ν) for all τ≤t≤T, t <∞ a.s.

Hence the general case follows at once from the T_{≡ ∞}case which is obvious from (2.2).

Remark 2.3 In fact one may take

Ωg_XS[τ,∞[=nν ∈ΩXS[τ,∞[ : ¯g(ν, ǫ)≤N

(16)

for some appropriate universal constantc and

Ωg,n_XS[τ,_∞[=

ν_∈ΩXS[τ,∞[ :

Z ∞

τ

νs(1)ds≤Kn, g¯(ν, ǫ)≤Kn(ǫ1/4+g(cǫ1/4)), ∀ǫ∈[0,1]

for a suitable sequence _{Kn}. Note that ν1 ∈ ΩgXS[τ,∞[, ν2 ∈ ΩX[τ,∞[ and ν2 ≤ ν1 imply ν2 ∈

Ωg_XS[τ,∞[. In practice we will work with a fixedg and hence a fixed version ofℓt(y, ν) given by (2.3).

It will, however, be convenient on occasion to use a subsequence of _{ǫn} in (2.3) to define another

version ofℓt(y, ν).

We now want to extend ℓt(y, ν) to the case whenν =X is random. This extension in non-trivial.

It is not hard to see that ifX = Γ(H),H is a historical Brownian motion andy “is chosen according toHT” for someT ≥τ (see the definition of Campbell measure in the next section), then (yt)t≥τ will

be a Brownian motion stopped at timeT butℓǫ

t(y, X) will not converge in any reasonable manner.

Notation 2.4 Puthα(x) = 1₂R₀∞e−αs/2ps(x)dswhereps(x) = (2πs)−d/2exp{−|x|2/2s}, withα >0

ifd_≤2 andα_≥0 ifd= 3. Set h0(x) = 1 + log+(1/_|x_|) ifd= 2 andh0(x) = 1 ifd= 1. We abuse the notation and writehα(x) =hα(|x|) at times. Then we of course have:

h0(x) =c|x|−1, d= 3

c1(α) 1 + log+(1/|x|)≤hα(x)≤c2(α) 1 + log+(1/|x|, d= 2, hα(x)≤c(α), d= 1.

(2.4)

Definition 2.5 Fix a filtered probability space (Ω,_F,(_Ft)t≥τ,P′). Given an (Ft)t≥τ–stopping time

τ≤T ≤ ∞, anRd_{–valued process (}_B

t)t≥τ = (B1t, . . . , Btd)t≥τ is an (Ft)t≥τ–Brownian motion stopped

atT if (Bi

t−Biτ)t≥τ is a continuous (Ft)t≥τ–martingale for eachi and

hBi_{, B}j_i

t=δij(t∧T −τ) ∀i, j, t≥τ, a.s.

Here, and elsewhere, _hM, N_i_t_{≡ h}M₋Mτ, N−Nτi_t.

The following Tanaka–type representation is the main result of this section.

Theorem 2.6 Let (Xt)t≥τ and (At)t≥τ be right–continuous, (Ft)t≥τ–adapted, MF(Rd)–valued

pro-cesses on (Ω,F,(Ft)t≥τ,P′). Assume A has non-decreasing paths a.s. and Aτ = 0. Let (Bt)t≥τ be a

(d-dimensional)(_Ft)t≥τ–Brownian motion stopped at T. Assume

(i) For all φ ∈ C∞

K(Rd), the process Mt(φ) = Xt(φ)−Xτ(φ)−R_τtXs

∆φ

2

ds−At(φ), t ≥ τ,

is a continuous (Ft)t≥τ–martingale such that hM(φ)it = R_τtXs(φ2)ds and hM(φ), Bjit = 0,

(17)

(ii) Bτ has lawm and Xτ =µa.s., whereµ∈MF(Rd)with R Rh0(x−y)µ(dx)2m(dy)<∞.

Then there is an a.s. continuous, non–decreasing, square–integrable (_Ft)t≥τ–predictable process,

ℓt(B, X), which satisfies

sup

τ≤t≤u∧T|

ℓǫ_t(B, X)₋ℓt(B, X)| L

2

−→0 asǫ_↓0, _∀u > τ, (2.5)

and

Z

hα(x−Bt)Xt(dx) =

Z

hα(x−Bτ)µ(dx)

+α

Z t

τ

Z

hα(x−Bs)Xs(dx)ds−

Z t

τ

Z

hα(x−Bs)A(ds, dx)

+

Z t

τ

Z

hα(x−Bs)M(ds, dx)−

Z t

τ

Z

~

∇hα(x−Bs)Xs(dx)

·dBs

−ℓt(B, X), ∀τ≤t≤T, a.s.

(2.6)

Each of the terms, Tt, in (2.6) satisfiesP′supτ≤t≤u∧T|Tt|2<∞, the fourth and fifth terms on the

right are continuous martingales, all the terms on the right are a.s. continuous except perhaps the third term which will also be a.s. continuous if At(Rd)is.

Remark 2.7 (a) The martingales Mt(φ) in hypothesis (i) extend to a family of continuous local

martingales, Mt(φ) = R_τtR φ(s, ω, x)M(ds, dx), where φ is P(Ft)× B(Rd) measurable (P(Ft) is the

predictableσ-field on [τ,_∞[_×Ω) and_hM(φ)_i_t=R_τtR φ(s, ω, x)2_X

s(dx)ds <∞,∀t > τ, a.s. (see Ch. 2

of [32] and Section 2 of [30]). This extension is used in the Tanaka Formula (2.6).

(b) The construction in Section 5 of [2] will allow us to construct a super-Brownian motion (X0

t)t≥τ

such thatX0

τ =µand Xt0 ≥Xt, ∀t≥τ, a.s. If M0 is the orthogonal martingale measure associated

withX0_{, then this construction also gives}

hM0(φ), Bjit= 0, ∀t≥τ, 1≤j≤d, a.s., ∀φ∈CK∞(Rd). (2.7)

(c) Theorem 2.6 remains valid if the condition on_hM(φ)_iin hypothesis (i) is weakened to_hM(φ)_i_t_≤

γR_τtXs(φ2)ds ∀t ≥τ a.s. and allφ inCK∞(Rd) for someγ >0. We must, however, now assume the

existence of X0 _{as in (b) and satisfying (2.7). The proof is virtually the same. This extension has}

proved useful in the two–type model discussed at the end of Section 1 in which inter–species collisions reduce the masses of the colliding particles.

In this work we will usually assumeµ∈MF S(Rd) and so the following result shows that hypothesis

(18)

Corollary 2.8 If µ _∈ MF S(Rd), then hypothesis (ii) of Theorem 2.6 is valid for any law m. In

particular, the conclusion of Theorem 2.6 will hold whenever hypothesis (i) of that result does.

Proof. Letνy([0, r[) =µ(B(y, r)). Integration by parts shows that ify ∈Rd, then

Z

h0(x−y)dµ(x) =

Z ∞

0

h0(r)νy(dr)

≤

Z 1

0

h0(r)νy(dr) +h0(1)µ(Rd)

≤2h0(1)µ(Rd) +

Z 1

0

cr1−dνy([0, r])dr

=c(µ)<_∞

because of the hypothesis onµ. The result follows.

Proof of Theorem 2.6 By adding an independent Brownian motion to B after time T to obtain a Brownian motion ˜B (not stopped at T!) and setting ℓt(B, X) = ℓt∧T( ˜B, y), one easily derives the

general case from theT _{≡ ∞}case. Hence we may setT _{≡ ∞}in what follows.

Choose g∈C_K∞(Rd) such that1{|x| ≤ 1} ≤g(x)≤1{|x| ≤2}and letgn(x) =g(x/n). Consider

hypothesis (i) withφ=gn and useXt ≤Xt0 (see Remark 2.7(b)) and dominated convergence to see

that each of the terms in (i), except perhaps At(gn) converges inL1 as n→ ∞. We therefore deduce

At(gn)L

1

−→At(1), (i) holds forφ≡1 and

P′[At(1)]<∞, ∀t≥τ. (2.8)

Let φ(x, y) = φ1(x)φ2(y) for φi ∈ CK∞(Rd). Write ˜∆φ for the 2d–dimensional Laplacian and

~

∇2φ∈Rd denotes the partial gradient with respect to the lastdvariables. Then hypothesis (i), Itˆo’s

Lemma and integration by parts give that

Z

φ(x, Bt)Xt(dx) =

Z

φ(x, Bτ)µ(dx) +

Z t

τ

Z _∆_˜

2φ(x, Bs)Xs(dx)ds

−

Z t

τ

Z

φ(x, Bs)A(ds, dr) +

Z t

τ

Z

φ(x, Bs)M(ds, dx)

+

Z t

τ

Z

~

∇2φ(x, Bs)Xs(dx)

·dBs, ∀t≥τ, a.s.

(2.9)

This remains valid for φ in the algebra A of linear combinations of the above form. Now let φ ∈

C2

(19)

finds there are_{φn}inAsuch thatφn→φand ˜∆φn→∆˜φuniformly asn→ ∞. Each of the terms in

(2.9) withφ=φn converges inL1to the corresponding term withφ(useX ≤X0 again, and also (2.8)

to handle the integrals with respect toA) except perhaps the last. To analyze this term first note that the stochastic integral representation property of super-Brownian motion (Theorem 1.1 of [20]) and ordinary Brownian motion, and (2.7) show thatX0_and_B_{are independent. If}_U

K = inf{t:Xt0(1)≥K}

andψn=φ−φn, then forK > µ(1)

P′

"Z t

τ

1_{s_≤UK}

Z ~

∇2ψn(x, Bs)Xs(dx)

2 ds #

≤KP′

Z t

τ

Z

|∇~2ψn(x, Bs)|

2

X_s0(dx)ds

=KP

Z t

τ |

~

∇2ψn( ¯Bs)|

2 ds

,

(2.10)

where ¯B is a 2d–dimensional Brownian motion starting at timeτwith lawµ_⊗munderP (note thatµ

is not, in general, a probability measure and so we are using the term “law” somewhat loosely). Using Itˆo’s Lemma we see that the right side of (2.10) equals

KP



 ψn( ¯Bt)−ψn( ¯Bτ)−

Z t

τ

˜ ∆

2ψn( ¯Bs)ds

!2

_→0 as n→ ∞.

We now may letn_{→ ∞}in (2.9) (withφ=φn) to see that (2.9) holds for allφinCK2(Rd) (the result

is valid for allt_≥τ by right–continuity). By a truncation argument, as in the derivation of (2.8), one readily sees that (2.9) holds forφinC2

b(R2d) (that is, forφbounded with bounded continuous partial

derivatives of order 2 or less). The bound X _≤X0 _{easily shows that each of the stochastic integrals}

areL2 _martingales.

Let ¯Rαbe theα-resolvent for 2d–dimensional Brownian motion, set ¯pǫ(x, y) =pǫ(x−y) and define

¯

hǫ(x, y) = ¯Rαp¯ǫ(x, y) =

Z ∞

0

e−αtp2t+ǫ(x−y)dt=

1 2

Z ∞

ǫ

e−αs/2ps(x−y)ds eαǫ/2. (2.11)

Then ¯hǫ∈Cb2(R2d) satisfies

˜ ∆

2¯hǫ=α¯hǫ−pǫ. Therefore (2.9) gives

Z

¯

hǫ(x, Bt)Xt(dx) =

Z

¯

hǫ(x, Bτ)µ(dx) +α

Z t

τ

Z

¯

hǫ(x, Bs)Xs(dx)ds

−

Z t

τ

Z

¯

hǫ(x, Bs)A(ds, dx) +

Z t

τ

Z

¯

hǫ(x, Bs)M(ds, dx)

+

Z t

τ

Z

~

∇2¯hǫ(x, Bs)Xs(dx)·dBs−ℓǫt(B, X).

(2.12)

We now derive (2.6) and the existence ofℓt(B, X) by establishing the convergence of each of the terms

in (2.12), except perhaps for the last. Equation (2.11) shows that

(20)

Monotone convergence shows that the left side of (2.12) and the first three terms on the right side converge to the corresponding integrals withhα(x−y) in place of ¯hǫ(x, y). The limit of the first term

in the right is even finite by hypothesis (ii). Before dealing with the finiteness of the other terms, consider the two martingale terms in (2.12).

Lemma 2.9 For eacht_≥τ,

lim

ǫ↓0P

′Z t

τ

Z

¯

hǫ(x, Bs)−hα(x−Bs)2Xs(dx)ds

= 0.

Proof. The integrand converges pointwise to zero on _{(s, x) : x ₆= Bs} (by (2.13)). This set is of

full measure with respect to P′_[_X0

s(dx)]ds (recall the independence of X0 and B) and hence also

P′[Xs(dx)]ds. Therefore the result will follow from the bound in (2.13) and dominated convergence

once we show

P′

Z t

τ

Z

hα(x−Bs)2Xs0(dx)ds

<_∞. (2.14)

Note that if

f(u) =

    

u−1/2_, _d_{= 3}_,

1 + log+(1/u), d= 2,

1, d= 1,

then

hα(x)2=

1 2

Z ∞

0

Z ∞

0

1_{s_≤t_}e−α(s+t)/2_p

s(x)pt(x)dtds

≤c

Z ∞

0

Z ∞

0

1_{s_≤t_}e−α(s+t)/2ps(x)t−d/2dtds

=c

Z ∞

0

Z ∞

0

1{u≤v}e−α(u+v)p2u(x)v−d/2dvdu

≤c(α)

Z ∞

0

f(u)e−αup2u(x)du

(where we make the substitutionss = 2uand t = 2v to get the second equality). The expression in (2.14) is therefore bounded by

c(α)

Z t

τ

Z ∞

0

f(u)e−αu

ZZZZ

p2u(x−y)ps−τ(x−x0)ps−τ(y−y0)dxdy µ(dx0)m(dy0)du ds

=c(α)

ZZ Z t−τ

0

Z ∞

0

f(u)e−α(u+w)p2(u+w)(x0−y0)du eαwdw

(21)

A simple change of variables shows the term in square brackets is bounded by

c(α, t)

Z ∞

0

Z v

(v−(t−τ))+

f(u)du e−αv_p

2v(x0−y0)dv

≤c′(α, t)hα(x0−y0).

Now (2.14) follows from the above and hypothesis (ii) of Theorem 2.6 (in fact, a weakerL1_condition

suffices here).

Lemma 2.10 Let (Bt)t≥0 be a d-dimensional Brownian motion starting at xunder Px. For θ >0

there are constantsc1=c1(θ, d)such that for all x, y, y′_∈Rd and t >0the following hold. (a)Px_[_|_B

t−y|−θ]≤c1 |y−x|−θ∧t−θ/2 for0< θ < d.

(b)Pxlog+(1/(Bt−y))≤c1t−θ/2.

(c)Px₍_|_B

t−y|−θ∧t−θ/2)(|Bt−y′|−θ∧t−θ/2)≤c1 |y−x|−θ∧t−θ/2 |y′−x|−θ∧t−θ/2.

Proof. We may assumex= 0 and set P =P0_.

(a) A simple scaling argument reduces the proof to thet= 1 case. Then

P[_|B1−y|−θ] =θ

Z ∞

0

r−1−θ_P_[_|_B

1−y|< r]dr

≤c

Z ∞

0

1

|y|

2 ∨1≤r

r−1−θdr+

Z ∞

0

1

r_≤ |y|

2 ∨1

r−1−θe−|y|2/8rddr

≤ch_|y_|−θ_∧1 +e−|y|2/8(1_{∨ |}y_|)d−θi

≤c _|y_|−θ_∧1.

(22)

(c) Again it suffices to considert= 1. Then

P(_|B1−y|−θ∧1)(|B1−y′|−θ∧1)

=θ2Z ∞ 1

Z ∞

1

r−θ−1_r′−θ−1_P

{|B1−y|< r,|B1−y′|< r′}drdr′

≤c

Z ∞

1

Z ∞

1

r−θ−1r′−θ−1

"

1

|y_|

2 ≤r,

|y′_|

2 ≤r

′

+1

|y|

2 > r,

|y′|

2 ≤r

′

e−|y|2/8rd+1

|y|

2 ≤r,

|y′|

2 > r

′

e−|y′|2/8r′d

+1

|y|

2 > r,

|y′_|

2 > r

′ _e−|y|2_/₈

rd∧e−|y′|2/8r′d

#

drdr′

≤c

(_|y_|−θ_∧1)(_|y′_|−θ_∧1) + (_|y′_|−θ_∧1)_|y_|de−|y|2/8+ (_|y_|−θ_∧1)_|y′_|de−|y′|2/8

+ exp ₋(_|y_|2_{∨ |}y′_|2)/8 Z

∞

1

Z ∞

1

(rd_∨r′d)r−θ−1r′−θ−11

r < |y|

2 , r <

|y′_|

2

drdr′

≤c(|y|−θ_∧₁₎₍_|_y′_|−θ_∧_{1) + exp}_{−₍_|_y_|2_{∨ |}_y′_|2₎_/₈₎_|_y_|d_|_y′_|d

≤c(_|y_|−θ_∧₁₎₍_|_y′_|−θ_∧₁₎_.

Lemma 2.11 For each t_≥τ,

lim

ǫ↓0P

′"Z t

τ

Z

|∇~2¯hǫ(x, Bs) +∇~hα(x−Bs)|Xs(dx)

2 ds

#

= 0.

Proof. Note that

~

∇2¯hǫ(x, y) =−

1 2

Z ∞

ǫ

e−αs/2_∇_~_p

s(x−y)ds eαǫ/2

→ −∇~hα(x−y) as ǫ↓0 if x6=y.

An elementary calculation (use_|_∇~ps(x)|=|x|s−1ps(x)), now shows that

|∇~2¯hǫ(x, y)| ≤c|x−y|1−d ∀x, y∈Rd, ǫ∈]0,1] and somec >0.

Using dominated convergence, as in the proof of Lemma 2.9, we see that it suffices to show that for

d= 2 ord= 3

P′

"Z t

τ

Z

|x−Bs|1−dXs0(dx)

2 ds

#

(23)

If, as above, Px _{is Wiener measure starting at} _x _{at time 0 and} _Pν ₌ R_Px_ν₍_dx_{), then (see, for}

example, (A.15) withL= 0)

P

"Z

|x−y|1−d_X0

s(dx)

2#

=Pµ|Bs−τ−y|1−d

2

+

Z s−τ

0

PµhPBr_|_B

s−τ−r−y|1−d2

i

dr

≤c

"Z

|y−x|1−d_∧₍_s₋_τ₎(1−d)/2_µ₍_dx₎2₊Z s−τ 0

Z

|y−Br|2−2d∧(s−τ−r)1−ddPµdr

#

,

(2.16)

where Lemma 2.10 is used in the last inequality.

Use this to write (2.15) asI+II where (sett0=t−τ and recallX0 andB are independent)

I=

Z t0

0

ZZ

Pmh|Bs−x|1−d∧s(1−d)/2 |Bs−x′|1−d∧s(1−d)/2

i

µ(dx)µ(dx′)ds

and

II=

ZZ Z t0

0

Z s

0

P0_|_B

r+s+y0−x0|2−2d∧(s−r)1−ddr ds

m(dy0)µ(dx0).

If

f(w) =

(

|w_|−2_, _d_{= 3}_,

1 + log+(1/_|w_|), d= 2,

then a change of variables (u=s₋r, v=s+r) shows the integral in square brackets inII equals 1

2

Z

Rd

nZ t0

0

Z v

0

pv(w−(y0−x0)) (|w|2∨u)1−ddu dv

+

Z 2t0

t0

Z 2t0−v

0

pv(w−(y0−x0)) (|w|2∨u)1−ddu dv

o

dw

≤c(t0)

Z

Rd

Z 2t0

0

pv(w−(y0−x0))|w|4−2d+f(w) dv dw.

≤c(t0)

Z 2t0

0

Px0−y0_[_f₍_B

v)]dv.

Ifd= 3, then Lemma 2.10(a) bounds the above by

c(t0)

Z 2t0

0 |

x0−y0|−2∧v−1dv≤c(t0)

1 + log+ 2t0/|x0−y0|2.

(24)

Turning to I, we may use Lemma 2.10(c) to bound it by

2c1

ZZZ Z t0

0 |

y₋x_{| ∨}√s1−d _|y₋x′_{| ∨}√_s1−d_ds

1_{|y₋x_{| ≤ |}y₋x′_|}_m₍_dy₎_µ₍_dx₎_µ₍_dx′₎_.

Consider onlyd= 3 asd= 2 is even easier. In this case the above equals

c(t0)

ZZZ

2|x′−y|−2_{+ 2}_|_x′₋_y_|−2_log+|x′−y|

|x₋y_|

1{|x−y| ≤ |x′−y|}m(dy)µ(dx)µ(dx′)

≤c(t0)

ZZZ

|x′−y|−1_|_x₋_y_|−1

1 + |x−y|

|x′₋_y_|log

+ |x′−y|

|x−y|

1_{|x₋y_{| ≤ |}x′₋y_|}m(dy)µ(dx)µ(dx′)

≤c(t0)

Z Z

h0(|x−y|)µ(dx)

2 m(dy)

which is finite by hypothesis (ii) of Theorem 2.6. This showsI is finite and hence proves (2.15) and completes the proof.

Lemma 2.12 (a)limǫ↓0P′hR_τtR h¯ǫ(x, Bs)−hα(x−Bs)Xs(dx)2 ds

i

= 0,∀t > τ. (b)limǫ↓0P′

hRt τ

R

|¯hǫ(x, Bs)−hα(x−Bs)|A(ds, dx)2

i

= 0,_∀t > τ. (c)limǫ↓0P′

h

sup_τ_≤_t_≤_uR¯hǫ(x, Bt)−hα(x−Bt)Xt(dx)2

i

= 0,_∀u > τ.

Proof. (a) As in the proof of Lemma 2.9 (see (2.14)), it suffices to prove

P′

Z t

τ

Z

hα(x−Bs)Xs0(dx)2ds

<∞.

This is immediate from (2.4) and (2.15) (the cased= 1 being trivial). (b) The decomposition (2.12) shows that

Z t

τ

Z

¯

hǫ(x, Bs)A(ds, dx)≤

Z

¯

Z t

τ

Z

¯

hǫ(x, Bs)Xs(dx)ds

+

Z t

τ

¯

hǫ(x, Bs)M(ds, dx) +

Z t

τ

Z

~

∇2¯hǫ(x, Bs)Xs(dx)·dBs.

Lemmas 2.9, 2.11, part (a) and hypothesis (ii) of Theorem 2.6 (and (2.13)) show that the right side of the above converges inL2 _{to the corresponding expression with}_h

(25)

(2.13) and Fatou’s Lemma to see thatP′hR_τtRhα(x−Bs)A(ds, dx)2

i

<_∞. This shows thatA(ds, dx) does not charge{(s, x) :Bs=x}a.s. and therefore (by (2.13))

lim

ǫ↓0

¯

hǫ(x, Bs)−hα(x−Bs) = 0, A(ds, dx)−a.e., P′−a.s.

(Implicit in the above isd >1, but this last assertion is trivial ifd= 1.) The result now follows from (2.13), the above square–integrability and dominated convergence.

(c) Argue as in the proof of Lemma 5.6(b) of [2] (see the last displayed equation in the proof) to see that foru > τ fixed

lim

δ↓0_τ_≤sup_t_≤_u

Z

hα(x−Bt)1{|x−Bt| ≤δ}Xt0(dx) = 0, a.s. (2.17)

We have sup

τ≤t≤u

Z ¯

hǫ(x, Bt)−hα(x−Bt)Xt(dx)

≤ sup

τ≤t≤u

Xt(1)

sup_{|¯hǫ(x, y)−hα(x−y)|:|x−y|> δ}

+c sup

τ≤t≤u

Z

hα(x−Bt)1(|x−Bt| ≤δ)Xt0(dx) (by (2.13)).

Use (2.17) and the uniform convergence of ¯hǫ(x, y) to hα(x−y) on {(x, y) :|x−y|> δ}as ǫ↓0 (see

Lemma 5.3 of [2]) to see that the left side of the above approaches 0 a.s. asǫ_↓0. If we could show P′

sup

τ≤t≤u

Z

hα(x−Bt)Xt(dx)2

<∞, (2.18)

then the result would follow from the above, (2.13) and dominated convergence. To prove (2.18) use (2.12) to see that

sup

τ≤t≤u

Z

¯

hǫ(x, Bt)Xt(dx)≤

Z

¯

Z t

τ

Z

¯

hǫ(x, Bs)Xs(dx)ds

+ sup

τ≤t≤u

Z t τ Z ¯

hǫ(x, Bs)M(ds dx)

+ Z t τ Z ~

∇2¯hǫ(x, Bs)Xs(dx)·dBs

.

The right side converges inL2 _{to the corresponding expression with}_h

α(x−Bs) in place of ¯hǫ(x, Bs)

by hypothesis (ii) of Theorem 2.6, part (a), Lemmas 2.9 and 2.11, and Doob’s maximalL2 inequality. An elementary argument using Fatou’s Lemma now gives (2.18) and completes the proof.

Proof of Theorem 2.6 (continued)Return now to (2.12). Lemmas 2.9, 2.11, 2.12 and hypothesis (ii) of Theorem 2.6 (with (2.13)) show that if Tǫ

(26)

terms on the right side andTtis the corresponding term withhα(x−y) in place of ¯hǫ(x, y) (the initial

condition is independent oft), then for anyu > τ, sup_τ_≤_t_≤_u|Tǫ t−Tt|L

2

−→0 asǫ↓0. Therefore there is an a.s. continuous, non–decreasing, square–integrable (_Ft)t≥τ–predictable process (ℓt(B, X))t≥τ such

that (2.5) and (2.6) hold withT _{≡ ∞}. IfAt(Rd) is a.s. continuous then so isR_τtRh¯ǫ(x, Bs)A(ds, dx)

and hence the same is true ofR_τtR hα(x−Bs)A(ds, dx) by Lemma 2.12(b). The remaining assertions

in the Theorem are now obvious.

Remark 2.13 There is some possible confusion over the notationℓt(y, ν) in Theorem 2.2(b) and the

notationℓt(B, X)(ω) in Theorem 2.6. We claim that given a process X and a time T as in Theorem

2.6 we may constructℓt(y, ν) as in Theorem 2.2 (and more specifically (2.3)) so that

ℓt(B, X)(ω) =ℓt(y, ν)|(y,ν)=(B·(ω),X·(ω)), ∀τ≤t≤T, P

′_-a.s. _(2.19)

Given_{ǫn}as in (2.3) we may (by (2.5)) choose a subsequence {ǫnk}such that

ℓt(B, X) = lim k→∞ℓ

ǫnk

t (B, X), ∀τ ≤t≤T, P′-a.s.

Now replace_{ǫn}with{ǫnk}in (2.3) to defineℓt(y, ν) so that (2.19) holds. Clearly this argument can accommodate countably many{Xn}in (2.19).

3

Historical collision local times

3.1

Path–field collision local times

Let (Ω,_H,(_Ht)t≥τ,P) be a filtered probability space such that (Ht)t≥τ is right–continuous and the

σ–field H is universally complete. Let (H1_{, H}2_{) be a pair of independent historical Brownian}

mo-tions starting at (τ, µ1_{) and (}_{τ, µ}2_{), respectively, defined on (Ω}_,_H_,₍_H

t)t≥τ,P) and with corresponding

martingale measures (M1_{, M}2_).

Definition 3.1 Write_M(H1_{, H}2_{) for the collection of pairs of predictable,} _M

F(C)–valued processes

(G1_{, G}2_{) with values in Ω}

H[τ,∞[ such that there are nondecreasing, predictable, MF(C)–valued

pro-cesses (A1_{, A}2_{), null at} _τ_{, and with sample paths almost surely in Ω}

H[τ,∞[ such that: Git ≤ Hti,

i= 1,2, for allt_≥τ, and for allφ1_{, φ}2_∈_D

ST we have that

N_t1(φ1) =G1_t(φ1_t)₋µ1(φ1_τ)₋

Z t

τ

Gs(Aφ1s)ds−

Z t

τ

Z

φ1(s, y)A1(ds, dy) and

N_t2(φ2) =G2_t(φ2_t)−µ2(φ2_τ)−

Z t

τ

Gs(Aφ2s)ds−

Z t

τ

Z

(27)

are continuous (_Ht)t≥τ–martingales null atτ for which

hNi(φi)it=

Z t

τ

Z

φi_s(y)2Gi_s(dy)ds

and_hNi₍_φi₎_{, N}j₍_φj₎_i₌_h_Mi₍_φi₎_{, N}j₍_φj₎_i_{= 0 for}_i₆₌_j_.

Let _C denote the Borel σ-field on C and, as in Section 2, write (_Ct)t≥0 for the canonical

filtra-tion. Using the extension procedure in Section 2 of [32], we can construct orthogonal martingale measures Ni _{(defined on appropriate (}_C

t× Ht)t≥τ–predictable integrands) that have the following

properties. If γi _{is (}_C

t× Ht)t≥τ–predictable and R_τtR γi(s, y)2Gis(dy)ds < ∞ for all t ≥ τ, P-a.s.

(respectively, P[R_τtRγi₍_{s, y}₎2_Gi

s(dy)ds] < ∞ for all t ≥ τ), then

Rt τ

R

γi₍_{s, y}₎_dNi₍_{s, y}_{) is a}

con-tinuous local martingale (respectively, a concon-tinuous square - integrable martingale). We have that

Ni t(φi) =

Rt

τφi(s, y)dNi(ds, dy). Moreover, h

R·

τ

R

γi₍_{s, y}₎_dNi₍_{s, y}₎_i t =

Rt τ

R

γi₍_{s, y}₎2_Gi

s(dy)ds and

hRτ·

R

γ1₍_{s, y}₎_dN1₍_{s, y}₎_,R·

τ

R

γ2₍_{s, y}₎_dN2₍_{s, y}₎_i _{= 0. The analogous extensions of course also hold for} Mi _and_hR·

τ

R

γi₍_{s, y}₎_dNi₍_{s, y}₎_,R· τ

R

γj₍_{s, y}₎_dMj₍_{s, y}₎_i_{= 0 for}_i₆₌_j_.

For the rest of this section we will assume that (H1_{, H}2_{) is a pair of independent historical}

Brownian motions starting at (τ, µ1_{) and (}_{τ, µ}2_{), respectively, with} _µi _∈ _M

F S(C)τ, i = 1,2, and

(G1_{, G}2₎_{∈ M}₍_H1_{, H}2_).

We first extend the results in Section 2 concerning ℓt(y, ν) andℓt(y, X) to the setting where y is

chosen according toHi _{instead of being a fixed Brownian path.}

Definition 3.2 Given a bounded (_Ht)t≥τ–stopping time T ≥ τ, the normalised Campbell measure

associated withHi _{is the probability measure ¯}_PHi

T on on (C×Ω,C × H)≡( ˆΩ,Hˆ) given by

Z

γ(y, ω) ¯PH_Ti(dy, dω) =

Z Z

γ(y, ω)H_Ti(dy)

P(dω)/µi(1).

Let ˆHt=Ct× Ht and ˆH∗t denotes its universal completion.

Definition 3.3 Say that Λ_⊂[τ,_∞[_×Ω isˆ Hi_{–evanescent if Λ} _⊂_Λ

1 where Λ1 is ( ˆH∗t)t≥τ–predictable

and

sup

τ≤u≤t

1Λ1(u, y, ω) = 0, Hti-a.a.y, ∀t≥τ, P′−a.s.

Say that a property holdsHi_{-a.e. iff it holds off an}_Hi_{-evanescent set.}

Definition 3.4 Let (Xt)t≥τ be an (Ht)t≥τ–optional,MF(Rd)–valued process on ˆΩ such that

sup

τ≤s≤t

(28)

Apath–field collision local time(or PFCLT) forHi_{with respect to}_X_{is an ( ˆ}_H

t)t≥τ–predictable process

(t, y, ω)_→ℓt(y, X)(ω) such that for any bounded (Ht)t≥τ–stopping timeT ≥τ,

sup

τ≤t≤T|

ℓǫ_t(y, X(ω))₋ℓt(y, X)(ω)| →0 in ¯PH

i

T -probability asǫ↓0.

If (Gt)t≥τ is anMF(C)–valued process andX= Γ(G) is as above, we writeℓt(y, G) (andℓǫt(y, G)) for

ℓt(y, X) (andℓǫt(y, X)) and call the former the PFCLT forHi with respect toG(if it exists).

Remark 3.5 A simple application of the section theorem shows that the PFCLT forHi _{with respect}

toX is unique up toHi-evanescent sets. To see this note that ifℓ′t is another PFCLT then

{(t, ω) :t_≥τ, H_ti( sup

τ≤s≤t|

ℓs(y, X)−ℓ′s(y, X)|>0)>0}

is (_Ht)t≥τ–predictable by Corollary 3.6 of [30] (use the fact that if f is bounded and ( ˆHt∗)t≥τ–

predictable, then so is f∗_{, where} _f∗

t = supτ≤s≤tfs = supτ≤s<tfs∨ft) and therefore is evanescent

(in the usual sense) by the above definition and the section theorem. A slightly more involved applica-tion of the secapplica-tion theorem shows thatℓ·(y, X)|[τ,t] is non-decreasing and continuousHti-a.a. y,∀t≥τ,

a.s. (that is,Hi_-a.e.).

Lemma 3.6 Let T _≥ τ be a bounded (_Ht)t≥τ–stopping time and set X = Γ(G2). Under P¯H

1

T ,

Bt(y, ω) = yt is a ( ˆHt)t≥τ–Brownian motion stopped at T and (B, X) satisfies the hypotheses of

Theorem 2.6 on ( ˆΩ,_Hˆ,( ˆ_Ht)t≥τ,P¯H

1

T ).

Proof. Theorem 2.6 of [30] gives the first assertion about B. To prove thatX satisfies hypothesis (i), fixt0 large andφ∈CK∞(Rd), and set ¯φ(y) =φ(y(t0)). Ifν(A) =µ2({y :yτ ∈A}), then the definition

ofM(H1_{, H}2_{) shows that for}_τ _≤_t_≤_t0_, Xt(φ) =ν(φ) +

Z t

τ

Xs(

∆φ

2 )ds+N

2

t( ¯φ)−At(φ),

whereAt(φ) =R_τtR φ(y(s))A2(ds, dy) andNt2( ¯φ)≡Mt(φ) is an (Ht)t≥τ–martingale underPsatisfying

hM(φ)_it=

Z t

τ

Z

φ(y(t0))2G2_s(dy)ds=

Z t

τ

Xs(φ2)ds, τ ≤t≤t0.

Ifτ≤s < t≤t0, A∈ Hsandf is bounded andCs-measurable, then

µ1(1)¯PH_T1[(Mt(φ)−Ms(φ)) 1Af(y)] =PHT1(f)(Mt(φ)−Ms(φ))1A

=P

"

(H_s1_∧_T(f) +

Z T

s∧T

Z

f(y)M1(ds, dy))(N_t2( ¯φ)−N_s2( ¯φ))1A

#

(29)

the last becauseN2 _and_M1 _{are orthogonal. This shows}_M

t(φ) is an ( ˆHt)t≥τ–martingale under ¯PH

1

T .

If ˆBt=Bt−Bτ and As=A∩ {T > s}, then

getdoc43c2. 1329KB Jun 04 2011 12:04:19 AM

Collision local times, historical stochastic calculus,

Contents

Introduction

Historical Brownian motion

Collision local times

Smooth measures

Competing species martingale problem

Approximating systems

Overview of the paper

Brownian collision local times

Historical collision local times