Non-linear Neumann's condition for the heat equation : a probabilistic representation using catalytic super-brownian motion

www.elsevier.com/locate/anihpb

Non-linear Neumann’s condition for the heat equation:

a probabilistic representation using catalytic super-Brownian motion

Jean-François Delmas

, Pascal Vogt

aENPC-CERMICS, 6-8 av. Blaise Pascal, Champs-sur-Marne, 77455 Marne La Vallée, France bUniversity of Bath, Department of Mathematical Sciences, Bath BA2 7AY, UK Received 17 October 2003; received in revised form 31 March 2004; accepted 2 May 2004

Available online 23 June 2005

Abstract

LetDbe a bounded domain inR^d with smooth boundary∂D. We give a probabilistic representation formula for the non- negative solution of the mixed Dirichlet non-linear Neumann boundary value problem (DNP)

u=0 inD,

u=ϕ onF₂,

∂nu+2u²=0 onF₁,

where(F₁, F₂)is a non-trivial partition of∂D,ϕis a non-negative, bounded and continuous function defined onF₂, and∂_n denotes the outward normal derivative on the boundary ofD.

To solve the DNP, we consider a catalytic super-Brownian motion with underlying motion a Brownian motion reflected on∂D, killed when it reachesF₂and catalysed by the setF₁, i.e. the branching rate is given by the local time of the paths onF₁. Then we prove that the log-Laplace transform ofϕintegrated with respect to the exit measure of the catalytic process onF₂, is a non-negative weak solution of the DNP.

In a second part we show that we still have a probabilistic representation formula if the Dirichlet condition onF₂is replaced by a Neumann condition.

2005 Elsevier SAS. All rights reserved.

Résumé

SoitDun domaine borné deR^dde frontière,∂D, régulière. Nous présentons une formule de représentation probabiliste des solutions positives du problème non linéaire mixte Dirichlet–Neumann (DNP)

✩ The research was partially supported by the National German Merit Foundation and the EPSRC.

* Corresponding author.

E-mail addresses: [email protected] (J.-F. Delmas), [email protected] (P. Vogt).

URLs: http://cermics.enpc.fr/ delmas/home.html (J.-F. Delmas), http://www.bath.ac.uk/ maspv (P. Vogt).

0246-0203/$ – see front matter 2005 Elsevier SAS. All rights reserved.

doi:10.1016/j.anihpb.2004.05.007

u=0 inD, u=ϕ onF₂,

∂nu+2u²=0 onF₁,

où(F₁, F₂)est une partition non triviale de∂D,ϕune fonction positive bornée et continue définie surF₂, et où∂_ndésigne la dérivée normale extérieur sur∂D.

Pour résoudre DNP, nous considérons un superprocessus avec catalyse surF₁, où le processus sous-jacent est le mouvement brownien dansD, réfléchi sur∂D, et tué quand il atteintF₂. Le mécanisme de branchement est donné par le temps local du mouvement brownien surF₁. Nous montrons que la transformée de log-Laplace de la fonctionϕintégrée contre la mesure de sortie du superprocessus surF₂, est une solution de DNP en un sens faible.

Dans une deuxième partie, nous donnons également une formule de représentation quand la condition de Dirichlet est rem- placée par une condition de Neumann.

2005 Elsevier SAS. All rights reserved.

MSC: 60J55; 60J80; 60H30; 60G57; 35C15; 35J65

Keywords: Non-linear boundary value problem; Collision local time; Catalytic super-Brownian motion; Exit-measure

1. Introduction

Super-Brownian motions are measure valued stochastic processes. Since the works of Dynkin, Kuznetsov and Le Gall (see for example the monograph [8], and the references therein), the log-Laplace transform of the super- Brownian motion appears to be a powerful tool to study the non-linear PDEu=u²in a domainD. In particular, using a probabilistic representation formula, it is possible to describe all the non-negative solutions of this non linear PDE.

Super-Brownian motion represents a cloud of infinitesimal particles which evolve according to independent Brownian motions and are subject to a critical branching mechanism. Roughly speaking the spatial motion appears in the PDE through its infinitesimal generator, which in our case is the Laplacianu. The branching mechanism is responsible for the non-linear term,u²in our case. Since the early nineties, models appeared where the branching occurs only in a subset of the space called the catalytic set. Such models are called catalytic super-Brownian motion (see for example the survey [10]). Outside the catalytic set, the catalytic super-Brownian motion has a density w.r.t.

the Lebesgue measure and this density solves the heat equation (with random boundary condition on the catalytic set). In particular, the non-linear phenomenon is located on the catalytic set.

In March 1999, during the Seminar on Stochastic Processes in Toronto, Dynkin asked if one could use a cat- alytic super-Brownian motion to give a probabilistic representation for solutions of the mixed Dirichlet non-linear Neumann boundary value problem (DNP)

u=0 inD, u=ϕ onF₂,

∂_nu+2u²=0 onF₁,

(1) whereDis a smooth domain,(F₁, F₂)is a non-trivial partition of∂D, and∂_ndenotes the outward normal derivative on the boundary ofD. In this paper, we give such a representation formula. Instead of building the catalytic super- Brownian motion as a limit of branching particle systems, we use the construction introduced in [13] based on collision local time. From this construction, we derive a representation formula for non-negative solutions of (1) with Dirichlet or Neumann condition onF₂.

Let us describe more precisely the content of the paper. We consider a reflected Brownian motion inD,B= (Bt, t0). (This process can be used to give probabilistic representation formula of the heat equation inDwith linear Neumann boundary conditions, see [16].) In Section 2, we recall some facts on excursion theory from [12], introducing the family ofσ-finite measures (H^x, x∈F₁)which describe the “law” of the excursion of B inD

started fromx∈F₁. IfLdenotes the associated capacitary local time onF₁(see Section 2.1 for a precise definition), we prove thatLhas a density, sayρ, with respect to the local time ofBonF₁.

In Section 3, we consider underP^Xν,(Xt, t0)a superprocess started at the initial measureν, with quadratic branching mechanism and underlying motion a processξ =(ξt, t 0). The processξ is, up to a random time change, the trace onF1of B before it hitsF2. More precisely, letl^∗=(l_t^∗, t0)be the local time onF1 ofB before it hitsF₂,l^∗,−1 its right-continuous inverse, and setξ_t =(l_t^∗,−1, B

l_t^∗,−1). In particular,X_t takes values in Mf(R₊×F₁), the set of finite measures onR₊×F₁. Then we consider the total occupation measureΓ (dr,dx)= _∞

0 ds X_s(dr,dx). From this, we introduce in Section 4.1 the random measure,Z^Dir, onF₂defined for any non- negative functionϕonF₂by

Z^Dir, ϕ

=

Γ (dr,dx), ρ(x)H^x

ϕ(e_τ2)1_{τ2<∞}

,

whereτ2is the hitting time ofF2for the excursioneunderH^x. Intuitively, the measureZ^Dir describes the death positions of infinitesimal particles released from the catalyst at time drand position dxaccording to the random measureρ(x)Γ (dr,dx), performing independent Brownian excursions outsideF₁killed when they first reachF₂. Let us assume the measureν∈Mf(R+×F₁)is of the formδ₀⊗η, whereδ₀is the Dirac mass at 0 andηa finite measure onF₁. Then the random measureZ^Dir corresponds to the so-called exit measure ofDof the catalytic superprocess with catalytic setF₁, quadratic branching mechanism and initial measureη. If the initial measure is not supported byF₁, then one has to make some slight modification to get the exit measure (see Definition 4.1).

LetP^Z_δx denote the law of the exit measure,Z^Dir, whenη=δ_x, the Dirac mass atx∈D.

In Sections 4.2 and 4.3, we study the properties of the log-Laplace transform,w, of the measureZ^Dir, defined by

w(x)= −logE^Z_δx exp−

Z^Dir, ϕ .

In particular, we prove thatwis a solution of the DNP in a weak sense, see Definition 4.10 and Theorem 4.13.

In Section 5, using techniques developed in [2], we replace the Dirichlet condition onF₂by a Neumann con- dition. In particular, we are able to give in Theorem 5.18 a similar representation formula for solutions to the

PDE u=0 inD,

∂_nu−2ϕ=0 onF₂,

∂_nu+2u²=0 onF₁.

Those two representation formulas for Dirichlet or Neumann condition onF₂could be presented in an unified way, but at a cost of more complex notations. Therefore, we choose to keep the notations as simple as possible, and treat the two conditions in apparently different ways.

Eventually, we collect in the appendix some results on reflected Brownian motion inD.

2. Notations

IfEis a polish space, letB(E)denote its Borelσ-field as well as the set of real measurable functions defined onE. LetB₊(E)(resp.C(E)) be the subset ofB(E)of non-negative (resp. continuous) functions. Forϕ∈B(E) bounded, we write ϕ_∞=sup_x∈E|ϕ(x)|. Let Mf(E)be the set of finite measures on E, endowed with the topology of weak convergence. For ν∈Mf(E) andϕ ∈B(E)bounded or non-negative, we write ν, ϕ for

Eν(dx) ϕ(x). IfAis a Borel subset ofR^d, letA¯denote its closure.

LetDbe a bounded domain, i.e. a connected open subset ofR^d,d2, withC³-boundary∂D. LetC^p(D)(resp.

C^p(D)) be the set of continuous functions defined onD(resp.D) of class C^p. Let(nx, x∈∂D)be the outward unit normal vector field and∂nf (x):= ∇f, nx denote the outward unit normal derivative on∂Datx of a function

f ∈C¹(D). LetF₁andF₂two relatively open subsets of∂D. We assume thatF₁andF₂are non-empty, disjoint and thatF₁∪ F₂=∂D. We also assume that the relative boundary ofF₁is equal to the relative boundary ofF₂, and that it is either empty or aC²-manifold of codimension 2. We shall denote it by∂F. For example, the condition

∂F= ∅can be achieved ifDis a region between two concentric sphere, withF₁being one sphere andF₂the other one.

LetB=(B_t, t0)be a reflecting Brownian motion inD, with normal reflection, started atx∈ DunderPx. Let(Ft, t0)be the filtration generated byBcompleted the usual way. See Section 6.1 in the appendix for some properties ofB. We say a property holds a.s. if it holdsPx-a.s. for allx∈ D. Fort >0, letp_t(x, y)denote the transition density ofB. There exists a unique continuous additive functional=(_t, t0)ofB called the local time on∂D, such that for everyϕ∈B₊(R₊× D)andx∈ D,

Ex

∞ 0

d_sϕ(s, B_s) = ∞

0

ds

∂D

σ (dy) ϕ(s, y)p_s(x, y), (2)

whereσ is the surface measure on∂D. In other words,σ is the Revuz-measure of the continuous additive func- tional . Denote by| · | the Euclidean norm inR^d, and forx ∈ D, let d(x, ∂D)=inf{|x−y|: y ∈∂D}. The continuous additive functionalcan be constructed explicitly as

t= lim

n→∞

1 ε_n

t

0

ds1_{{d(Bs,∂D)εn}}, (3)

where the limit exists for allt0,Px-a.s., for some positive sequence(ε_n, n1)decreasing to zero which does not depend onx∈ D(see Theorem 7.2 in [15]).

2.1. Local times onF₁

A key-rôle is played by the exit systems, introduced by Maisonneuve in [12]. In particular, we shall need the last exit decomposition ofBout ofF₁.

Fori=1,2, letτ_i=inf{t >0: B_t∈F_i}be the first hitting time ofF_i, with the convention that inf∅ = +∞. Notice the stopping timesτ_i are finite a.s. (see Lemma 6.3). LetF₁^r be the set of regular points ofF₁, i.e.F₁^r:=

{x∈ D:Px(τ₁=0)=1}. Since∂Dand∂F are smooth, we haveF₁^r= F₁. We set M:= {t >0, B_t∈ F₁}.

So,M is almost surely a closed subset of (0,∞). Furthermore the set M is optional and time homogeneous.

Following [12], we set R:=inf{s >0: s∈M}, R_t:=inf{s >0:s+t∈M}, G:= {t >0:R_t−=0, R_t>0}.

Notice thatR=τ1a.s. The setG, is the set of left endpoints in(0,∞)of the intervals contiguous toM. NoticeG is countable andG⊂Ma.s. SinceF₁^r is regular for itself, we deduce thatG= {t∈G,PB_t(R=0)=1}. Following [12], there exists a continuous additive functionalL=(L_t, t0)ofB, such that for allx∈ D,

Ex

_∞

0

e^−t dLt =Ex

e^−τ1 .

The Revuz measure,µ, ofLis such that for any functionϕ∈B₊(R₊× D), Ex

∞ 0

ϕ(s, B_s)dL_s = ∞

0

ds

µ(dz)p_s(x, z)ϕ(s, z).

Notice the measure µ is the 1-capacitary measure of the set F₁. Hence, we call the additive functional L the

“capacitary local time” onF₁.

The capacitary local timeLis called in [12] the local time onF1. However, the so called local time onF1,¹_s, is defined by d¹_s=1F₁(B_s)d_s (this correspond to∂Dreplaced byF₁in (3)). In factLand¹do not coincide in general. However, in our setting, the next lemma implies thatLis absolutely continuous with respect to¹. Recall thatσ, the Revuz measure of, is also the surface measure on∂D.

Lemma 2.1. There existsρ∈B₊(R^d), such that µ(dx)=ρ(x)1F₁(x)σ (dx).

The proof of this lemma is postponed to Section 6.4 of the appendix.

In the particular case, where D⊆R^d, d2, is an open ball of radius r, and F₁=∂D, we deduce from Proposition 1.9 in [14] that

µ(dy)= 2π^d/2r^d−2 (d/2−1)σ (dy),

where denotes the Gamma-function. Notice that the density ofµ with respect toσ depends on the curvature of∂D.

2.2. Exit formula out ofF1and applications

Letδbe a cemetery point added toR^d, letD=D(R+,R^d∪ {δ})be the set of càdlàg functions defined onR+, and let[δ]be the constant functiont→δ. Fors >0, leti_s:D→Dbe the family of translation operators defined by,

i_s(e)(t )=e(t+s) for 0t < R_s, i_s(e)(t )=δ fortR_s.

Moreover, let(Q¹_t, t0)be the transition kernels of the reflected Brownian motion killed onF₁. We recall the exit formula (see Proposition 9.2 in [12]).

Theorem 2.2 (Maisonneuve). There exists a family of universally measurableσ-finite measures(H^x, x∈F1), on (D,B(D)), such that for any non-negative predictable processZ=(Z_s, s0), w.r.t. the filtration generated byB, and for any functionf ∈B₊(D), such thatf ([δ])=0, we have the exit formula:

Ex

s∈G

Z_sf ◦i_s(B)

=Ex

∞ 0

Z_sH^Bs[f]dL_s .

Fori=1,2 ande∈D, letτ_i(e)be the first hitting time ofF_i: τ_i(e)=inf

s >0: e(s)∈F_i .

We use the convention that e_+∞ =δ and we always write τ_i for τ_i(e) as well as e_s for e(s), when there is no ambiguity. We now give particular applications, we shall use later. Let ϕ ∈ B₊(R^d). For θ 0, set f (e)=e^{−θ τ2}ϕ(eτ₂)1_{τ₂<∞}andZs(e)=e^{−θ s}1_{τ₂>s}. From Theorem 2.2, we have

Ex

τ₂ 0

e^{−θ s}H^Bs

e^{−θ τ2}ϕ(eτ₂)1_{τ₂<∞}

dLs =Ex

s∈G

e^{−θ s}1_{τ₂>s}f ◦is(B)

=Ex

1_{τ2>τ1}e^{−θ τ2}ϕ(B_τ2)

, (4)

sinceτ₂◦i_s+s=τ₂on{τ₂> s}. Withθ=0, we get, asτ₂<∞a.s., Ex

τ2

0

H^Bs

ϕ(e_τ2)1_{τ2<∞}

dL_s =Ex

1_{τ2>τ1}ϕ(B_τ2)

. (5)

LetZs=e^{−θ s}andf be defined byf (e)=_∞

0 dt e^{−θ t}ϕ(et), with=(e)given by (3) whereBis replaced bye. We obtain

Ex

∞ 0

dL_s e^{−θ s}H^Bs ∞

0

d_t e^{−θ t}ϕ(e_t)

=Ex

s∈G

e^{−θ s}

Rs

0

d_t◦i_s(B)e^{−θ t}ϕ

i_s(B)(t )

=Ex

s∈G

s+τ₁◦i_s(B)

s

d_t e^{−θ t}ϕ(B_t)1_F2(B_t)

=Ex

∞ τ₁

d_t e^{−θ t}ϕ(B_t)1_F2(B_t) , (6) where we used for the second equality that d_t=d_t1_F2(B_t)fort /∈M, andτ₁=inf{s;s∈G}a.s. for the third.

Using a monotone class argument, Theorem 2.2 implies that for all predictable processesZ=(Z_s, s0)and for any functionf∈B₊(R₊×D)such thatf (·,[δ])=0,

Ex s∈G

Zsf (s,·)◦is(B)

=Ex

_∞

0

ZsH^Bs f (s,·)

dLs .

SettingZ_s =1_{τ2>s} and for fixedt >0, f (s, e)=1_{{0<t−s<τ2}}φ◦i_t−s(e), whereφ(e):=1_{τ2<+∞}ϕ(e₀), we deduce that

Ex

τ2

0

1_{s<t}H^Bs

1_{{t−s<τ2<+∞}}ϕ(e_t−s)

dL_s =Ex

1_{τ1<t <τ₂}ϕ(B_t)

. (7)

3. F1-catalytic super-Brownian motion

In this section we construct a catalytic super-Brownian motion inDwith catalytic setF₁and underlying motion a reflected Brownian motionB, killed when it first hitsF₂. Even if the construction of this catalytic superprocess is not explicitly needed to solve the boundary value problem, it gives insights in the underlying ideas. Our construction is motivated by the methods developed in [13].

Recall thatτ₂denotes the first hitting time ofF₂by the reflected Brownian motionB. Consider the local time ^∗=(^∗_t, t0)onF₁ofBkilled onF₂. It is defined by

d^∗_t =1_{t <τ₂}dt.

Let^∗,−1denote the right continuous inverse of the continuous additive functional^∗, i.e.

^∗_t^,−1:=inf

s0:^∗_s > t , with the convention that inf∅ = +∞.

LetE=(R+×F1)∪ {δ}, where δ is a cemetery point. We define the E-valued time-homogeneous Markov processξ =(ξt, t0)by

ξt:=

(^∗_t^,−1, B◦^∗_t^,−1) if^∗_t^,−1<∞,

δ otherwise,

and denote byP^ξ_t,xˆ its law started atxˆ∈Eat timet0. We also writeP^ξ_xˆforP^ξ_0,xˆ. Forν∈Mf(E)andt0, let P^Xt,ν denote the law of the quadratic (non-catalytic) superprocessX=(X_s, st )with spatial motionξ, starting atνat timet. We shall writeP^X_ν forP^X_0,ν. Recall thatXis anMf(E)-valued Markov process. Its total occupation measureΓ, defined underP^X_t,ν, by

Γ (dr,dx):=

∞

t

dsX_s(dr,dx),

plays the key-rôle in the construction of theF₁-catalytic super-Brownian motion.

Lemma 3.1. Letφ∈B₊(E). The functionvdefined onEby E^Xν

exp−Γ, φ

=exp−ν, v, (8)

is a non-negative solution of the integral equation

v(s, x)+Ex

τ₂ 0

drv²(r+s, Br) =Ex

τ₂ 0

drφ(r+s, Br) , (9)

wheres0 andx∈F₁. Ifφ(·, x)= ˜φ(x)does not depend on time, we get that fors0,v(s, x)= ˜v(x), wherev˜ is a non-negative solution onF₁of

˜

v(x)+Ex

τ₂ 0

d_rv˜²(B_r) =Ex

τ₂ 0

d_rφ(B˜ _r) . (10)

Remark 3.2. It is not clear if the integral equations (9) or (10) have a unique solution. From the previous lemma, we can compute the first moment ofΓ:

E^Xν

Γ, φ

=

ν(ds,dx)Ex

^τ2

0

d_rφ(r+s, B_r) . (11)

Proof of Lemma 3.1. As a special case of the weighted occupation time formula (see e.g. [11], II. 3) we have for all non-negative, bounded and measurable functionsφ andh on(R₊×F₁)∪ {δ}andR₊respectively, with φ(δ)=0 and such thathhas compact support,

E^Xt,ν

exp−

∞ t

dsh(s)X_s, φ =exp−ν, v_t,

wherevis the unique, non-negative solution of the integral equation fort0 andxˆ∈E, v_t(x)ˆ +E^ξ_t,xˆ

∞ t

dsv_s²(ξ_s) =E^ξ_t,xˆ ∞

t

dsh(s)φ(ξ_s) .

By substitution (^∗_r=s), we have withxˆ=(s, x)∈E, and therefore^∗_t^,−1=s, that v_t(s, x)+E^ξ_t,(s,x)

∞ s

d^∗_rv²_∗

r(r, B_r) =E^ξ_t,(s,x) ∞

s

d^∗_rh(^∗_r)φ(r, B_r) . Using the time homogeneity ofξandB, this last equation can be written as

v_t(s, x)+Ex

∞ 0

d^∗_rv_l²_∗

r+t(r+s, B_r) =Ex

∞ 0

d^∗_rh(^∗_r+t)φ(r+s, B_r) . (12) Using the time homogeneity of the processX, we also get that

E^Xt,ν

exp−

∞

t

dsh(s)X_s, φ =E^Xν

exp−

∞

0

dsh(s+t )X_s, φ .

In particular, the functionv^T defined fort∈ [0, T]by the equation, E^X_ν

exp−

T−t

0

dsX_s, φ =exp− ν, v^T_t

,

is the only non-negative solution of (12), withh(t )=1_[0,T](t ). By monotone convergence, lettingT tend to+∞, we get thatv^T_t increases point-wise to a functionv, independent oft, defined by (8), and v is a non-negative solution of

v(s, x)+Ex

∞ 0

d^∗_rv²(r+s, B_r) =Ex

∞ 0

d^∗_rφ(r+s, B_r) .

Using the definition of^∗, this last integral equation can be written as (9) wheres0 andx∈F₁. Hence, the lemma holds for any bounded, non-negative functionφ. By monotone convergence it also holds for anyφ∈B₊(E). If φ(·, x)= ˜φ(x), we get from (12) that

v_t(s, x)+Ex

∞ 0

d^∗_rv²_∗

r+t(r+s, B_r) =Ex

∞ 0

d^∗_rh(^∗_r +t )φ(B˜ _r) . (13)

In particularv_t^(s0) defined byv^(s_t ⁰⁾(s, x)=v_t(s₀+s, x)also solves (13). By uniqueness, we obtainv^(s_t ⁰⁾=v_t for anys₀0. Hence, we have that the functionv_t(s, x)does not depend ons, i.e.v_t(s, x)= ˜v_t(x)for anys0.

Following the arguments after (12), we deduce thatvdefined by (8) does not depend on time and solves (10). 2 Letη∈Mf(D)be a finite measure onD. Define νη∈Mf(R+×F1)to be the hitting distribution ofR+×F1

by(t, Bt), starting fromδ0⊗ηand killed onR+×F2: more preciselyνηis such that for anyψ∈B+(R+× D), we have

ν_η, ψ =

η(dx)Ex

1_{τ1<τ2}ψ (τ₁, B_τ1)

. (14)

Recall the definition of the densityρ from Lemma 2.1. We define, under P^Xν_η, the Mf(D)-valued process Z=(Z_t: t0)byZ₀:=ηand fort >0,

Z_t, ϕ = η, Q_tϕ +

Γ (dr,dx)1_{r<t}ρ(x)H^x

1_{{t−r<τ2<∞}}ϕ(e_t−r)

, (15)

whereϕ∈B₊(D) andQ_t denotes the semi group of the Brownian motionBkilled when it first hits∂D, i.e.

Q_tϕ(x)=Ex

ϕ(B_t)1_{{t <τ1∧τ2}} .

We writeP^Z_η the law of Z started atη. Let us give an intuitive interpretation of the measure valued processZ defined by (15). The measureZ_t describes a cloud of infinitesimal particles at timet. The first summand in (15) corresponds to those particles which have not reached the catalyst,F₁, at timetand which are distributed according to the starting measure ηat time 0. The second summand corresponds to the particles which have reached the catalyst before timet and perform a branching process. Particles are then released from the catalyst at time dr and location dx according to the random measureρ(x)Γ (dr,dx), and then they perform excursions outside the catalyst. As all these excursions are independent, a law of large numbers effect lets us only observe an average over all excursions.

LetC:=sup_x∈DEx[_τ2]<∞(see Lemma 6.3). The following proposition characterizes the finite dimensional marginals of the processZin terms of their Laplace transform.

Proposition 3.3. Let 0< t1· · ·tnandϕ1, . . . , ϕnelements ofB+(D), such that we have 2C n

i=1ϕi∞<1.

Then, E^Zη

exp−

n

i=1

Z_ti, ϕ_i =exp−

η, w(0,·) ,

where(w(s, x), s0, x∈ D)is the unique non-negative solution of w(s, x)+Ex

τ2

0

d_rw²(r+s, B_r) = n

i=1

1_{s<ti}Ex

1_{ti−s<τ2}ϕ_i(B_ti−s)

. (16)

Remark 3.4. From this proposition, it is easy to check thatZis a time-homogeneous Markov process. However, notice that the processZis not adapted to the filtration generated by the superprocessX.

Proof of Proposition 3.3. Usingφ(s, x):=n

i=11_{s<t_i}ρ(x)H^x[1_{t_i−s<τ₂<∞}ϕi(et_i−s)], we have E^Z_η

exp−

n

i=1

Z_ti, ϕ_i =exp− _n

i=1

η, Q_tiϕ_i + ν_η,w

, (17)

where, thanks to Lemma 3.1,(w(s, x), s 0, x∈F₁)is a non-negative solution of

w(s, x)+Ex

τ2

0

d_rw²(r+s, B_r) =Ex

τ2

0

d_rφ(r+s, B_r) . (18)

By Lemma 2.1, we have a.s. for allt0,

dL_t=ρ(B_t)1_F1(B_t)d_t. (19)

Using the definition ofφand the exit-formula (7) we obtain

Ex

τ₂ 0

drφ(r+s, Br) =Ex

τ₂ 0

dr

n

i=1

1_{r+s<t_i}ρ(Br)H^Br

1_{t_i−s−r<τ₂<∞}ϕi(et_i−s−r)

=Ex

τ2

0

dL_r n

i=1

1_{r+s<ti}H^Br

1_{{ti−s−r<τ2<∞}}ϕ_i(e_ti−s−r)

= n

i=1

1_{s<ti}Ex

1_{{τ1<ti−s<τ2}}ϕ_i(B_ti−s)

. (20)

We define fors0,x∈ D, w(s, x):=

n

i=1

1_{s<t_i}Qt_i−sϕi(x)+Ex

1_{τ₁<τ₂}w(s+τ₁, Bτ₁) .

Using the strong Markov property ofB at timeτ₁, (18) and (20), one check thatwsatisfies (16). Notice, that by construction, we have

η, w(0,·)

= n

i=1

η, Q_tiϕ_i +

η(dx)Ex

1_{τ1<τ2}w(τ ₁, B_τ1)

= n

i=1

η, Q_tiϕ_i + ν_η,w.

Thanks to (17), this implies the first equality of the lemma. To prove the uniqueness, letw₁andw₂be non-negative solutions of Eq. (16). Then both,w₁andw₂are bounded by_n

i=1ϕ_i∞. We have, w₁(s, x)−w₂(s, x)= −Ex

τ2

0

d_r

w₁²(s+r, B_r)−w₂²(s+r, B_r) .

Hence, we can deduce w₁−w_2∞ sup

x∈D,s0

Ex

τ2

0

d_rw₁²(s+r, B_r)−w²₂(s+r, B_r) 2C ∞

i=1

ϕ_i∞w₁−w_2∞.

As 2C_∞

i=1ϕ_i∞<1, we get thatw₁=w₂and (16) has a unique non-negative solution. 2 4. Dirichlet condition onF2

4.1. The exit measureZ^Dir

In this section, we define a measureZ^DironF₂and characterize it in terms of its Laplace functionals. According to Section 3, the measureZ^Dircan be seen as the exit-measure of theF₁-catalytic super-Brownian motion onF₂. Intuitively,Z^Dirdescribes the spatial distribution of the generic particles of aF₁-catalytic super-Brownian motion inD“frozen” when they first hitF₂.

Let us keep the same notation as in Section 3. In particular, forη∈Mf(D), the measure Γ is the total occupa- tion measure of the (non-catalytic) superprocessXstarting atX0=νη(see (14) for the definition ofνη).

Definition 4.1. We define the random measureZ^DironF2by: for allϕ∈B+(F2), Z^Dir, ϕ

=

η, Q¹(ϕ) +

Γ (dr,dx) ρ(x)H^x

ϕ(e_τ2)1_{τ2<∞}

,

withQ¹(ϕ)(r, x)=Ex[ϕ(B_τ2)1_{τ2τ1}]. We call the measure Z^Dir the exit measure of the F₁-catalytic super- Brownian motion onF₂, and writeP^Zη for its law.

Remark 4.2. To check thatZ^Diris finite, we compute its first moment. Thanks to (11), E^Zη

Z^Dir, ϕ

=

η, Q¹(ϕ)

+E^Xν_η Γ (dr,dx) ρ(x)H^x

ϕ(eτ₂)1_{τ₂<∞}

=

η, Q¹(ϕ) +

ν_η(ds,dx)Ex

τ2

0

d_rρ(B_r)H^Br

ϕ(e_τ2)1_{τ2<∞}

=

η, Q¹(ϕ) +

η(dx)Ex

1_{τ₁<τ₂}EB_τ1

τ₂ 0

dL_rH^Br

ϕ(e_τ2)1_{τ2<∞}

=

η(dx)Ex

ϕ(B_τ2)1_{τ2τ1} +

η(dx)Ex

τ₂ 0

dL_rH^Br

ϕ(e_τ2)1_{τ2<∞}

=

η(dx)Ex

ϕ(Bτ₂)1_{τ2τ1} +

η(dx)Ex

1_{τ₁<τ₂}ϕ(Bτ₂)

=

η(dx)Ex

ϕ(Bτ₂) ,

where we used Lemma 2.1 (or (19)) and the definition ofν_η, (14), for the third equality, the strong Markov property forBfor the fourth and (5) for the fifth.

Recall the definition of the constantC=sup_x∈DEx[τ₂]<∞. Lemma 4.3. For anyϕ∈B₊(F₂),

E^Z_η exp−

Z^Dir, ϕ

=exp−η, w,

where(w(x), x∈ D)is a non-negative solution of the integral equation onDgiven by

w(x)+Ex

τ₂ 0

drw²(Br) =Ex

ϕ(Bτ₂)

. (21)

If we additionally assume that 2Cϕ_∞<1, then the non-negative solutionwis also unique.

Proof. Usingφ(x, r):=ρ(x) H^x[ϕ(e_τ2)], we can compute E^Z_η

exp−

Z^Dir, ϕ

=E^X_νη exp−

η, Q¹(ϕ)

+ Γ, φ

=exp−

η, Q¹(ϕ)

+ ν_η, v ,

where, thanks to the second part of Lemma 3.1, the functionv is a non-negative solution on F₁ of the integral equation,

v(x)+Ex

τ2

0

d_rv²(B_r) =Ex

τ2

0

d_rρ(B_r)H^Br

ϕ(e_τ2)1_{τ2<∞}

=Ex

τ2

0

dL_rH^Br

ϕ(e_τ2)1_{τ2<∞}

=Ex

1_{τ₁<τ₂}ϕ(Bτ₂)

, (22)