Branching diffusion representation of semi-linear elliptic PDEs and estimation using Monte Carlo method

HAL Id: hal-01709033

https://hal-polytechnique.archives-ouvertes.fr/hal-01709033

Preprint submitted on 14 Feb 2018

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Branching diffusion representation of semi-linear elliptic PDEs and estimation using Monte Carlo method

Ankush Agarwal, Julien Claisse

To cite this version:

Ankush Agarwal, Julien Claisse. Branching diffusion representation of semi-linear elliptic PDEs and

estimation using Monte Carlo method. 2018. �hal-01709033�

Branching diusion representation of semi-linear elliptic PDEs and estimation using Monte Carlo method

Ankush Agarwal* ¹ and Julien Claisse ²

1 Adam Smith Business School, University of Glasgow, University Avenue, G128QQ Glasgow, United Kingdom. Email:ankush.agarwal@glasgow.ac.uk

2 Centre de Mathématiques Appliquées (CMAP), École Polytechnique and CNRS, Route de Saclay, 91128 Palaiseau Cedex, France. Email: julien.claisse@polytechnique.edu

February 14, 2018

Abstract

We study semi-linear elliptic PDEs with polynomial non-linearity and provide a probabilistic rep- resentation of their solution using branching diusion processes. When the non-linearity involves the unknown function but not its derivatives, we extend previous results in the literature by showing that our probabilistic representation provides a solution to the PDE without assuming its existence. In the general case, we derive a new representation of the solution by using marked branching diusion pro- cesses and automatic dierentiation formulas to account for the non-linear gradient term. In both cases, we develop new theoretical tools to provide explicit sucient conditions under which our probabilis- tic representations hold. As an application, we consider several examples including multi-dimensional semi-linear elliptic PDEs and estimate their solution by using the Monte Carlo method.

Key words: automatic dierentiation formula, branching diusion processes, elliptic, exit time, Monte Carlo method, partial dierential equation, semi-linear

AMS subject classications (2010): 35J61, 60H30, 60J85, 65C05

1 Introduction

In this paper, we are interested in the following class of semi-linear elliptic partial dierential equations (PDEs): given a bounded domain O ⊂ R ^d , f : O × R × R ^d → R, h : ∂O → R,

Lu + f u, Du

= 0 in O, u = h on ∂O, (1)

where L is the innitesimal generator of a diusion. This class of PDEs arises naturally in many elds of science (see, e.g., [2, 25] and references therein). In most cases, the related PDEs only involve non-linearity in the unknown function but not in its derivatives. For instance, the PoissonBoltzmann equation or the steady states in nonlinear equations of the KleinGordon or Schrödinger type. Besides, the stationary Burgers equation or Hamilton-Jacobi-Bellman equations when the diusion coecient is uncontrolled are examples of PDEs where the non-linearity involves both the unknown function and its rst order derivatives.

In this paper, we provide existence results for this class of PDEs when the generator f is polynomial and derive a new probabilistic representation of the solution which is well suited for numerical application, especially in high dimensions, using the Monte Carlo method.

The classical probabilistic approach for semi-linear PDEs relies on the theory of backward stochastic dierential equations (BSDEs) initiated by Pardoux and Peng [30]. The case of elliptic PDE with Dirichlet condition was rst studied by Darling and Pardoux [10] for Lipschitz generator and later more general results were provided by Pardoux [29] and Briand et al. [8]. These results rely on the so-called monotonicity assumption, which requires y 7→ f (x, y, z)−µy to be decreasing for some constant µ ∈ R, and the assumption that f has a linear or quadratic growth in z . We do not put such restrictions in our setting and propose a method to investigate a dierent class of elliptic PDEs with a generator which is a polynomial in y and z .

Our probabilistic representation is based on branching diusion processes. These processes describe the evolution of a population of independent and identical particles moving according to a diusion process.

They were rst introduced by Skorokhod [34] and later studied, more thoroughly and systematically, in a

series of papers by Ikeda et al. [20, 21, 22]. In particular, these authors, using branching diusion processes, established a probabilistic representation of semi-linear parabolic PDEs of the form

∂ t u + Lu + β X

l∈

N

p l u ^l − u

!

= 0 in [0, T ) × R ^d , u(T, ·) = g in R ^d , (2) where g : R ^d → R, β > 0 and (p l ) _l∈N is a probability mass function. More precisely, they consider a process where each particle moves according to a diusion with generator L and dies at an exponentially distributed random time with parameter β to give birth to l osprings with probability p l . Then, the solution of PDE (2) is expressed as E [ Q N

_T

i=1 g(X _T ⁱ )] where N T and (X _T ⁱ ) i=1,...,N

_T

denote the number of particles and their positions at time T respectively. The special case p

2

= 1 , which corresponds to the celebrated FisherKPP equation, has been particularly well-studied (see, e.g., McKean [26]).

In Henry-Labordère et al. [18], the authors extended the probabilistic representation of (path-dependent) PDE (2) where the probability mass function (p l ) l∈

N

is replaced by an arbitrary real-valued sequence of functions. See also Rasulov et al. [32]. This result was further extended in Henry-Labordère et al. [19] to parabolic semi-linear PDE of the form

∂ _t u + Lu + f (u, Du) = 0 in [0, T ) × R ^d , u(T, ·) = g in R ^d , (3) where f is a (d + 1) variate polynomial. The main idea consists of introducing marked particles which carry a weight function given by the Bismuth-Elworthy-Li formula to account for non-linearity in gradient of the solution. Under conditions of small non-linearity or small maturity T , the authors showed that their probabilistic representation provides a continuously dierentiable viscosity solution to PDE (3). They also performed numerical simulations to illustrate the accuracy of their method to solve PDEs using the Monte Carlo method.

The aim of this paper is to extend the results in Henry-Labordère et al. [18, 19] to the case of elliptic PDEs with Dirichlet condition. The rst result on the link between semi-linear elliptic PDE and branching diusion processes was obtained by Watanabe [36] who derived a criterion for extinction of branching Brownian motion absorbed at the boundary of a domain. Recently, Bossy et al. [4] extended this result to derive a probabilistic representation for PDE of the form

Lu + β X

l∈

N

c l u ^l − u

!

= 0 in O, u = h on ∂O, (4)

where (c l ) _l∈N is a sequence of real-valued functions. They used it to compute a Monte Carlo approximation of the solution to Poisson-Boltzmann equation in dimension three. One of the critical assumptions in their work is the existence of a smooth solution to PDE (4). In this paper, we do not require this assumption as we show directly that the probabilistic representation provides a continuous viscosity solution to PDE (4). The main diculty compared to [18] is to ensure the continuity and integrability of the probabilistic representation. In particular, the arguments of small maturity used in [18] cannot be exploited here and instead, we resolve this problem by developing new tools which ensure that our probabilistic representation holds in small domain.

In the second part of our paper, we perform a rigorous analysis for the case of semi-linear elliptic PDE involving non-linearity in gradient of the unknown function. To the best of our knowledge, this is the rst paper in the literature to provide a representation for this class of PDE using branching diusion processes. In contrast with Henry-Labordère et al. [19], Malliavin calculus cannot be used in this setting since the exit time of a diusion from a domain is not dierentiable in the Malliavin sense. However, Delarue [11] and Gobet [16] have established a suitable automatic dierentiation formula, based on the work of Thalmaier [35], which allows us to derive a probabilistic representation analogous to [19]. Under the assumption that the non-linear gradient term vanishes at the boundary, it provides a continuously dierentiable solution to PDE (1) when f is a multivariate polynomial.

Importantly, our probabilistic representation provides a means to evaluate solutions of semi-linear elliptic PDEs by using the Monte Carlo method whose accuracy relies on the dimensionless central limit theorem.

For this reason, it is particularly well suited for numerical applications in high dimension where deterministic methods usually fail to provide accurate estimates. Regarding the probabilistic approach, although dierent numerical methods for BSDEs have been introduced (see, e.g., [6, 37]), to the best of our knowledge, they have never been used in the literature for elliptic PDEs.

¹

In addition, the classical approach to solve BSDEs

1

A related problem has been studied by Bouchard and Menozzi [5] for the case of parabolic PDEs in a cylindrical domain

with nite time horizon.

induces a discretization bias and involves a sequence of conditional expectation estimators. The available methods to estimate conditional expectations, such as least-squares regression (see Gobet et al. [17]), induce an additional approximation bias and are computationally expensive in high dimension.

We consider several numerical examples, including multi-dimensional elliptic PDEs, to illustrate the accuracy of our method. One of the critical steps in the algorithm consists of simulating the rst exit time and position of a diusion from a domain. There are three classical approaches to perform this task walk on spheres scheme, walk on parallelepipeds scheme and Euler discretization method (see Bouchard et al. [7]

and references therein). For instance, the walk on spheres scheme was used by Bossy et al. [4] to estimate the solution of Poisson-Boltzmann equation in dimension three. In this paper, we consider numerical examples which involve Brownian motion over a rectangular domain and therefore, we generate unbiased samples for exit time and position using the walk on squares scheme as introduced in Faure [12] and in Milstein and Tretyakov [27], and implemented in the library developed by Lejay [24].

The rest of the paper is organized as follows. In the next section, we provide a precise formulation of the problem and introduce branching diusion processes used to derive our probabilistic representation.

Then, we consider the case of semi-linear PDE with linear gradient term in Section 3 and give explicit sucient conditions to ensure that the probabilistic representation provides a continuous viscosity solution to the PDE. In Section 4, we perform the analysis for semi-linear PDE with non-linear gradient term. In particular, we establish appropriate automatic dierentiation formulas. Finally, in Section 5, we present several numerical examples to illustrate the applicability of our results in dierent settings. We provide the proof of the automatic dierentiation formula in Appendix A.

1.1 Notations

Any element x ∈ R ^d , d ≥ 1, is a column vector with i th component x i and Euclidean norm |x|. x · y denotes the usual dot product and d(x, y) denotes the Euclidean distance for any x, y ∈ R ^d . Given an open set O ⊂ R ^d , O ¯ denotes its closure, ∂O its boundary and diam(O) its diameter. For any g : O → R, the supremum norm is dened as kgk ∞ = sup{|g(x)| : x ∈ O}. C(O) (resp. C( ¯ O) ) denotes the set of continuous functions on O (resp. O ¯ ). C ^k (O) (resp. C ^k ( ¯ O) ) denotes the set of functions with continuous derivatives of all orders less than or equal to k (resp. which further have continuous extensions to O ¯ ). C ^k,α (O) (resp.

C ^k,α ( ¯ O) ) are subspaces of C ^k (O) (resp. C ^k ( ¯ O) ) consisting of functions whose partial derivatives of order less than or equal to k are locally (resp. globally) Hölder continuous with exponent 0 < α < 1. Given T > 0 , C

^1,k

((0, T ] × O) denotes the set of functions with continuous rst derivative with respect to the rst component and continuous derivatives of all orders less than or equal to k with respect to the second component. For a smooth function u(t, x), Du and D

²

u stand for, respectively, its gradient (as a row vector) and Hessian matrix with respect to its second component. Furthermore, if g : R ^m → R ⁿ is a dierentiable function, its gradient Dg = (∂ x

₁

g(x), . . . , ∂ x

_m

g(x)) takes values in R ⁿ ⊗ R ^m . For d ≥ 1, M ^d denotes the set of all d × d matrices and I d ∈ M ^d the identity matrix.

2 Problem Denition

2.1 A Class of Semi-Linear Elliptic PDE

Let (µ, σ) : R ^d × R ^d → R ^d × M ^d denote the drift and diusion coecient. Then for a non-negative integer m, we consider a subset L ⊂ N ^m+1 and a generator function f : R ^d × R × R ^d → R dened as

f(x, y, z) := X

l=(l

₀

,l

₁

,...,l

_m)∈L

c l (x)y ^l

⁰

m

Y

i=1

(b i (x) · z) ^l

ⁱ

,

where (c _l ) _l∈L , c _l : R ^d → R and (b _i ) _i=1,...,m , b _i : R ^d → R ^d are sequences of functions. Furthermore, for every l = (l

₀

, l

₁

, . . . , l _m ), we denote |l| := P m

i=0 l _i . Given a bounded domain O ⊂ R ^d , we consider the following semi-linear elliptic PDE:

Lu + β f u, Du

− u

= 0 in O, u = h on ∂O, (5)

where β is a positive constant, h : ∂O → R is the Dirichlet boundary condition and L is the innitesimal

generator associated to a diusion process with parameters (µ, σ) . Under a set of general assumptions, we

provide a probabilistic representation of solution u using the theory of branching diusion processes. We

list the assumptions on parameters of PDE (5) which are needed for our results at the outset.

Assumption 2.1. (i) The functions b _i , c _l are continuous on O ¯ . (ii) The function h is continuous on ∂O .

(iii) The function (x, y, ζ) 7→ P

l∈L c l (x)y ^l

⁰

Q m

i=1 ζ _i ^l

ⁱ

is continuous on O × ¯ R × R ^m . Part (iii) above implies that the multivariate power series (y, ζ) 7→ P

l∈L c l (x)y ^l

⁰

Q m

i=1 ζ _i ^l

ⁱ

has an innite radius of convergence for all x ∈ O ¯ . It is a simplifying assumption which avoids the use of localization arguments in the rest of the paper.

2.2 Marked Branching Diusion Processes with Absorption

A (age-dependent) marked branching diusion process is characterized by diusion parameters (µ, σ) , a probability density function ρ : R

+

→ R

+

, and a probability mass function (p _l ) _l∈L . In this process, we start with one particle of mark 0 at position x ∈ R ^d which undergoes a diusion with parameters (µ, σ) during its lifetime distributed according to ρ. At the end of its lifetime (arrival time), the particle dies and gives rise to

|l| = P m

i=0 l _i osprings with probability p _l , among which l

₀

have mark 0, l

₁

have mark 1, and so on. After their birth, each ospring performs the same but an independent branching diusion process as the parent particle. Additionally, we consider that particles are absorbed at the boundary, i.e., they die without giving rise to any ospring when they leave the domain O . In order to construct the above process, we denote by K := {∅} ∪ S

n≥1 N ⁿ the set of labels and we consider the probability space (Ω, F, P ) equipped with

• a sequence of i.i.d. positive random variables (τ ^k ) _k∈

_K

distributed with density function ρ

• a sequence of i.i.d. random elements (I ^k ) _k∈

_K

with P (I ^k = l) = p _l , l ∈ L

• a sequence of independent d -dimensional Brownian motions (W ^k ) _k∈

_K

Furthermore, we consider the sequences (τ ^k ) _k∈

_K

, (I ^k ) _k∈

_K

and (W ^k ) _k∈

_K

to be mutually independent. The age-dependent branching process is constructed as follows:

1. Start from one particle at position x ∈ R ^d and index it by label ∅. This is the common ancestor of all particles, the only particle of generation 0. For notational convenience, we simply write τ , I and W instead of τ ^∅ , I ^∅ and W ^∅ . The dynamic of particle ∅ is given as follows:

• The position X ^∅ = X ^x of the particle during its lifetime is given by X _t ^x = x +

Z t

0

µ(X _s ^x ) ds + Z t

0

σ(X _s ^x ) dW s , P − a.s. ,

• The arrival time of the particle is given by

T ^∅ := τ ∧ η ^x where η ^x := inf n

t ≥ 0; X _t ^x ∈ O / o .

• At the arrival time, if η ^x ≤ τ , then the particle ∅ dies without giving rise to any ospring, else the particle ∅ dies and gives rise to |I| osprings which belong to the rst generation, and are indexed by label i for i = 0, . . . , |I| − 1.

• Given I = (I

0

, I

1

, . . . , I m ), we have |I| = P m

i=0 I i ospring particles, among which the rst I

0

have mark 0, I

1

have mark 1, and so on, so that each particle has mark i for i = 0, . . . , m.

2. For generation n ≥ 1, let the label for a particle be given as k = (k

1

, . . . , k _n−1 , k n ) ∈ N ⁿ . Furthermore, denote by k ⁻ := (k

1

, . . . , k _n−2 , k _n−1 ) the parent particle of k . The particle k starts from X _T ^k

⁻

k−

at time T _k

−

:

• The position X ^k of the particle during its lifetime is given by X _t ^k = X _T ^k

⁻

k−

+ Z t

T

_k−

µ(X _s ^k ) ds + Z t

T

_k−

σ(X _s ^k ) dW _s−T ^k

_k−

, P − a.s.

• The arrival time of the particle is given by T k := T _k

⁻

+ τ ^k

∧ inf n

t ≥ T _k

⁻

; X _t ^k ∈ O / o

.

• At the arrival time, if X _T ^k

k

∈ O / , then the particle k dies without giving rise to any ospring, else the particle k dies and gives rise to |I ^k | osprings which belong to the (n + 1) th generation, and are indexed by label (k

1

, . . . , k _n−1 , k n , i) for i = 0, . . . , |I ^k | − 1.

• Given I ^k = (I

₀

^k , I

₁

^k , . . . , I _m ^k ), we have |I ^k | = P m

i=0 I _i ^k ospring particles, among which the rst I

₀

^k have mark 0, I

₁

^k have mark 1, and so on, so that each particle has mark i for i = 0, . . . , m.

In addition, we denote by K ^x _n the collection of particles of the n th generation and by K ^x = S

n∈N K ^x _n the collection of all particles. Finally, we introduce the ltration

F n := σ τ ^k , I ^k , W ^k , k ∈ {∅} ∪

n

[

i=1

N ⁱ

!

, n ∈ N .

We make the following assumptions to ensure that the branching diusion process above is well-dened and for further developments.

Assumption 2.2. (i) The probability distribution (p l ) l∈L satises p l > 0 and P

l∈L |l|p l < ∞ . (ii) The probability density ρ is strictly positive.

(iii) The coecients (µ, σ) are Lipschitz on O ¯ .

Part (i) of the assumption above ensures that the number of particles remains nite in nite time in the underlying branching process, see, e.g., Athreya and Ney [1, Theorem 4.1.1]. In addition, under assertion (iii) , there exists a unique solution (up to the boundary) to the stochastic dierential equation (SDE) corresponding to (µ, σ) and so the branching diusion process is well-dened.

Assumption 2.3. The branching diusion process goes extinct almost surely.

It is clearly sucient to assume that P

l∈L |l|p l ≤ 1 for Assumption 2.3 to hold. However, since particles are absorbed at the boundary, we can derive much weaker conditions, especially if the domain is small. For instance, in the case of branching Brownian motion with exponential lifetime of parameter β , Assumption 2.3 is equivalent to

β X

l∈L

|l|p _l − 1

!

− λ

₁

2 ≤ 0, (6)

where λ

1

is the rst positive eigenvalue of the Laplacian operator in the domain O , see Sevast ⁰ yanov [33]

or Watanabe [36]. Also see Remark 3.3 below for further developments.

3 Semi-Linear PDEs with Linear Gradient Term

In this section, we are concerned with semi-linear PDEs with polynomial non-linearity involving the unknown function but not its derivatives, i.e., we assume that m = 0 in Section 2 so that all the particles have the same mark 0 and PDE (5) reads as

Lu + β (f (u) − u) = 0 in O, u = h on ∂O, (7)

where f (x, y) = P

l∈L c l (x)y ^l , L ⊂ N. Throughout this section, we suppose that Assumption 2.12.3 remain valid. Additionally, we set ρ as the density of exponential distribution with parameter β .

3.1 Probabilistic Representation

We consider a branching diusion process starting from x ∈ O as in Section 2.2 and we introduce the following random variable:

ψ ^x := Y

k∈K

^x

X

^k_Tk

∈O /

h(X _T ^k

k

) Y

k∈K

^x

X

_Tk^k

∈O

c _I

k

(X _T ^k

k

) p _I

k

.

Proposition 3.1. Suppose that PDE (7) has a solution u ∈ C

²

(O) ∩ C ( ¯ O) such that the sequence (ψ ^x _n ) _n∈

_N

given by

ψ ^x _n := Y

k∈∪

ⁿ_i=0

K

^x_i

X

_Tk^k

∈O /

h(X _T ^k

k

) Y

k∈∪

ⁿ_i=0

K

^x_i

X

_Tk^k

∈O

c _I

k

(X _T ^k

_k

) p _I

k

Y

k∈K

^x_n+1

u(X _T ^k

k−

),

is uniformly integrable, then we have u(x) = E [ψ ^x ] .

Proof. By applying Itô's formula to (e ^−βt u(X _t ^x )) _t≥0 , we obtain the following Feynman-Kac representation:

u(x) = E h

e ^−βη

^x

h(X _η ^x

x

) + Z η

^x

0

βe ^−βs f (·, u)(X _s ^x ) ds i .

See, e.g., Freidlin [13, Theorem 2.2.1] for a detailed proof. Then, we can write u(x) = E

h(X _η ^x

x

)1 _τ≥η

x

+ f (·, u)(X _τ ^x )1 _{τ <η}

x

(8)

= E

h(X _η ^x

x

)1 τ≥η

^x

+ c _I (X _τ ^x ) p I

u ^I (X _τ ^x )1 τ <η

^x

.

Since an empty product is equal to 1 by convention, it follows that u(x) = E [ψ ^x

₀

] . Next, since each ospring has the same dynamic as the parent particle, we can repeat the above calculations to write for k ∈ K

₁

^x ,

u(X _T ^k

k−

) = E

"

h(X _T ^k

_k

)1 _X

k

Tk

∈O / + c _I

k

(X _T ^k

k

) p _I

k

u ^I

^k

(X _T ^k

_k

)1 _X

k Tk

∈O

F

0

#

. (9)

We use the result in (9) and plug it back in (8) to obtain u(x) = E [ψ ^x

₁

] by conditional independence of particles in K ^x

₁

given F

0

. Similarly, we can show by iteration that for any n ∈ N , we have u(x) = E [ψ ^x _n ] . To conclude, it remains to observe that ψ _n ^x converges to ψ ^x almost surely in view of Assumption 2.3. Thus, if we suppose that (ψ _n ^x ) n∈

N

is uniformly integrable, as n → ∞, we get u(x) = E [ψ ^x ] .

Proposition 3.1 shows that there is at most one solution to PDE (7) satisfying an appropriate integrability condition and such a solution admits a probabilistic representation using branching diusion processes. It is not completely satisfactory since one needs to prove rst the existence of a classical solution and even then the uniform integrability condition is hard to derive as illustrated by the example of Section 5.1. In order to overcome these limitations, we next establish a result which shows directly that the probabilistic representation provides a (viscosity) solution of PDE (7). This is the main result of this section. It is stated under abstract assumptions for which we will provide explicit sucient conditions in the subsequent sections.

Theorem 3.1. Suppose u : x 7→ E [ψ ^x ] is well-dened and continuous on O ¯ . Then u is a viscosity solution of PDE (7).

Proof. We rst observe that by denition of u , it holds u(x) = E

"

h(X _η ^x

x

)1 _τ≥η

^x

+ c I (X _τ ^x ) p I

I−1

Y

i=0

ψ ^X

x τ

i 1 τ <η

^x

# .

where

ψ _i ^X

^xτ

:= Y

k=(i,...)∈K

^x

X

_Tk^k

∈O /

h(X _T ^k

k

) Y

k=(i,...)∈K

^x

X

_Tk^k

∈O

c _I

k

(X _T ^k

k

) p _I

k

.

Furthermore, the branching property, which says that each ospring starts the same but an independent branching diusion as the parent particle, yields that conditioned on X _τ ^x and I , (ψ ^X

x τ

0

, . . . , ψ ^X

x τ

I−1 ) are inde- pendent random variables identical in law to ψ ^X

⁰

where X

₀

is distributed as X _τ ^x and independent of F n for all n ∈ N. Through this argument, we deduce that

E h ^I Y ⁻¹

i=0

ψ ^X

x τ

i

X _τ ^x , I i

1 τ <η

^x

= u ^I (X _τ ^x ) 1 τ <η

^x

.

Working backward along the lines of the proof of Proposition 3.1, we deduce that u(x) = E

h e ^−βη

^x

h(X _η ^x

x

) + Z η

^x

0

βe ^−βs f (·, u)(X _s ^x ) ds i

. (10)

Next, for any δ > 0, it follows from Markov property that u(x) = E

h e ^−β(η

^x

^∧δ) u(X _η ^x

x

∧δ ) + Z η

^x

∧δ

0

βe ^−βs f (·, u)(X _s ^x ) ds i

. (11)

The fact that u is a viscosity solution of PDE (7) now follows from classical arguments. For the sake of completeness, let us show that u is a viscosity subsolution. Consider ϕ ∈ C _b

²

(O) such that x ∈ O is a maximum point of u − ϕ and u(x) = ϕ(x) . First, we observe by applying Itô's formula that

E h

e ^−β(η

^x

^∧δ) ϕ(X _η ^x

x

∧δ ) i

= ϕ(x) + E

"

Z η

^x

∧δ

0

e ^−βs (Lϕ − βϕ) (X _s ^x ) ds

# .

Since u(x) = ϕ(x) and u ≤ ϕ otherwise, we deduce by using (11) that

E

"

1 δ

Z η

^x

∧δ

0

e ^−βs (Lϕ − βϕ + βf (·, u)) (X _s ^x ) ds

#

≥ 0.

Since y 7→ f(y, u(y)) is continuous and bounded, it follows from the mean value theorem and the dominated convergence theorem that

Lϕ(x) + β (f (x, ϕ(x)) − ϕ(x)) ≥ 0.

Thus u is a viscosity subsolution of PDE (7). The fact that u is a viscosity supersolution results from similar arguments.

Remark 3.1. It will turn out from the study of the next sections that the explicit conditions we provide to ensure that x 7→ E [ψ ^x ] is a solution to PDE (7) also entail that x 7→ E [|ψ ^x |] is a solution to

Lv + β X

l∈L

|c l | v ^l − v

!

= 0 in O, v = |h| on ∂O.

In particular, we observe that the monotonicity assumption is not relevant in our setting. The limiting assumption in our approach is the integrability condition on (ψ ^x ) _x∈O which ensures that u : x 7→ E [ψ ^x ] is well-dened. See Section 3.3 for more details.

Remark 3.2. We can also work with a general lifetime distribution ρ and extend the arguments above, as done in Section 4, to derive another probabilistic representation which is given as

u(x) = E









 Y

k∈K

^x

X

_Tk^k

∈O /

e ^−β∆T

^k

h(X _T ^k

k

) F ¯ (∆T _k )

Y

k∈K

^x

X

_Tk^k

∈O

βe ^−β∆T

^k

c _I

k

(X _T ^k

k

) p _I

k

ρ(∆T k )









 ,

where ∆T _k := T _k − T _k

−

is the lifetime of particle k and F ¯ (t) := R ∞

t ρ(s) ds , t ≥ 0 . However, this leads to more stringent assumptions when studying the integrability condition as in Section 3.3.

3.2 Continuity Assumption

Let us give explicit sucient conditions for the continuity assumption in Theorem 3.1 to hold. We use two dierent approaches leading to slightly dierent conditions. Both approaches rely on appropriate integra- bility conditions on (ψ ^x ) _x∈O which we discuss in the next section.

3.2.1 PDE Approach

Assumption 3.1. (i) The diusion coecient σ is uniformly elliptic.

²

(ii)The boundary ∂O is of class C

^1,α

.

Proposition 3.2. Suppose Assumption 3.1 holds and (ψ ^x ) _x∈O is uniformly bounded in L

¹

, then the map u : x 7→ E [ψ ^x ] is continuous.

The proof follows immediately from (10) and the following lemma.

Lemma 3.1. Suppose Assumption 3.1 holds, then the following statements are satised:

(i) The map O 3 ¯ x 7→ E [e ^−βη

^x

h(X _η ^x

x

)] is continuous.

(ii) For any g : O → R bounded measurable, the map O 3 ¯ x 7→ E [ R η

^x

0

e ^−βs g(X _s ^x ) ds] is continuous.

2

There exists λ > 0 such that σσ

^∗

(x) ≥ λI

d

for all x ∈ O ¯ .

Proof. (i) Let us rst study the continuity of x 7→ E [e ^−βη

^x

h(X _η ^x

x

)] . Under the assumptions above, it is well-known that there exists a smooth solution ϕ ∈ C

²

(O) ∩ C( ¯ O) to the following PDE:

Lϕ − βϕ = 0 in O, ϕ = h on ∂O.

See, e.g., Gilbarg and Trudinger [15, Theorem 6.13]. By Itô's formula, we deduce that ϕ(x) = E [e ^−βη

^x

h(X _η ^x

x

)] and the conclusion follows.

(ii)Let us now turn to the continuity of x 7→ E [ R η

^x

0

e ^−βs g(X _s ^x ) ds] . We start by showing that x 7→

E [g(X _s ^x )1 s<η

^x

] is continuous for s > 0 . If g is continuous and g = 0 on ∂O , it is known that the unique smooth solution χ ∈ C

^1,2

((0, T ] × O) ∩ C([0, T ] × O) ¯ of

∂ _t χ − Lχ = 0, in (0, T ] × O, χ(0, ·) = g, on O,

χ = 0, on (0, T ] × ∂O, (12)

is of the form

χ(s, x) = Z

O

G(s, x; 0, y)g(y) dy, 0 < s ≤ T,

where G is the so-called Green function of PDE (12) (see, e.g., Ladyºenskaja et al. [23, Theorem 4.16.2]).

Then, it follows from Itô's formula that

E [g(X _s ^x )1 s<η

^x

] = Z

O

G(s, x; 0, y)g(y) dy.

In particular, y 7→ G(s, x; 0, y) appears as the density of X _s ^x on the event {s < η ^x } , and so the identity above remains valid for any g bounded measurable. Furthermore, x 7→ G(s, x; 0, y) is continuous and satises

|G(s, x; 0, y)| ≤ Cs ⁻

^d2

e ^−C

^|x−y|

2 s

,

for some constant C > 0 depending on T , see [23, Equation 4.16.16]. Hence the desired result follows from the dominated convergence theorem. To conclude, it remains to observe that

E

"

Z η

^x 0

e ^−βs g (X _s ^x ) ds

#

= Z

+∞

0

e ^−βs E [g (X _s ^x ) 1 s<η

^x

] ds

and to apply once again the dominated convergence theorem.

3.2.2 Probabilistic Approach

Assumption 3.2. (i)The diusion coecient σ is uniformly elliptic on ∂O .

³

(ii)The boundary ∂O satises an exterior cone condition.

⁴

(iii)The stopping time η ^x is nite almost surely.

Sucient conditions for Part (iii) to hold are provided in Freidlin [13, Lemma 3.3.1]. For instance, it suces to assume that there exists 1 ≤ i ≤ d such that P d

j=1 σ _ij

²

(x) > 0 or |µ i (x)| > 0 for all x ∈ O . Proposition 3.3. Suppose Assumption 3.2 holds and (ψ ^x ) x∈O is uniformly integrable, then u : x 7→ E [ψ ^x ] is continuous.

Proof. Clearly it suces to prove that O 3 ¯ x 7→ ψ ^x is almost surely continuous, in the sense that for all sequence (x _n ) _n∈

_N

converging to x , it holds

P

n→∞ lim ψ ^x

ⁿ

= ψ ^x

= 1.

This essentially follows from the almost sure continuity of x 7→ η ^x stated in Lemma 3.2 below.

3

There exists λ > 0 such that σσ

^∗

(x) ≥ λI

_d

for all x ∈ ∂O .

4

See, e.g., Gilbarg and Trudinger [15, Problem 2.12] for a denition.

(i) Let us rst show that the contribution of the rst particle to ψ ^x is almost surely continuous, i.e.,

n→∞ lim h(X _η ^x

xnⁿ

)1 _τ≥η

xn

= h(X _η ^x

x

)1 _τ≥η

^x

, P − a.s. , (13)

n→∞ lim

c I (X _τ ^x

ⁿ

) p I

1 τ <η

^xn

= c I (X _τ ^x ) p I

1 τ <η

^x

, P − a.s. (14)

To achieve this, let us consider the set Ω ^∅ := {η ^x 6= τ } ∩ n

n→∞ lim η ^x

ⁿ

= η ^x o

∩ n

n→∞ lim X ^x

ⁿ

= X ^x o .

In view of Lemma 3.2 below, it is clear that P (Ω ^∅ ) = 1 . Furthermore, one easily checks that, for every ω ∈ Ω ^∅ , both (13)(14) hold.

(ii) Let us show next that the contribution of the particles of the rst generation to ψ ^x is almost surely continuous. For every i ∈ N, we denote by (X _s ^i,x ) s≥0 the unique solution of

X _t ^i,x = x + Z t

0

µ(X _s ^i,x ) ds + Z t

0

σ(X _s ^i,x ) dB _s ⁱ , P − a.s. , where the Brownian motion B ⁱ is dened by

B _t ⁱ := W _t∧τ + W _t−τ ⁱ 1 _t≥τ .

Clearly, (X _s ^i,x ) _s≥0 coincides with the trajectory X ⁱ of particle i during its lifetime. Let η ^i,x be the rst exit time of X ^i,x from O and consider

Ω ⁱ :=

η ^i,x 6= ¯ τ ⁱ ∩ n

n→∞ lim η ^i,x

ⁿ

= η ^i,x o

∩ n

n→∞ lim X ^i,x

ⁿ

= X ^i,x o .

where τ ¯ ⁱ := τ + τ ⁱ . Once again, P (Ω ⁱ ) = 1 and

n→∞ lim h X _η ^i,x

_i,xnⁿ

1 _τ

_¯i

≥η

^i,xn

= h X _η ^i,x

i,x

1 _τ

_¯i

≥η

^i,x

, on Ω ⁱ ,

n→∞ lim c I

ⁱ

X

_¯

_τ ^i,x

iⁿ

p _I

i

1 _τ

_¯i

<η

^i,xn

= c I

ⁱ

X _τ ^i,x

_¯i

p _I

i

1

_¯

_τ

i

<η

^i,x

, on Ω ⁱ . Thus, we conclude that on the set Ω ^∅ ∩ T

i∈N Ω ⁱ of probability 1 , the contributions of the particles of generations 0 and 1 to ψ ^x is almost surely continuous.

(iii) The desired result follows by iteration. We denote for all n ≥ 2 , k ∈ N ⁿ , Ω ^k :=

η ^k,x 6= ¯ τ ^k ∩ n

n→∞ lim η ^k,x

ⁿ

= η ^k,x o

∩ n

n→∞ lim X ^k,x

ⁿ

= X ^k,x o ,

where τ ¯ ^k := ¯ τ ^k

⁻

+ τ and (X _s ^k,x ) s≥0 is the unique solution of X _t ^k,x = x +

Z t

0

µ(X _s ^k,x ) ds + Z t

0

σ(X _s ^k,x ) dB _s ^k , P − a.s. , with

B _t ^k := B ^k

⁻

t∧¯ τ

^k−

+ W ^k

t−¯ τ

^k−

1 _{t≥¯ τ}

k−

. Then on the set T

k∈K Ω ^k of probability 1 , it holds that lim n→∞ ψ ^x

ⁿ

= ψ ^x .

Lemma 3.2. Suppose Assumption 3.2 holds, then the map O 3 ¯ x 7→ η ^x is almost surely continuous, in the sense that for all sequence (x n ) n∈

N

converging to x , it holds

P

n→∞ lim η ^x

ⁿ

= η ^x

= 1.

Proof. The proof is due to Darling and Pardoux [10], see also [28, Proposition 4.4]. More precisely, the authors show that if the stopping time

τ ^x :=

s ≥ 0, X _s ^x ∈ / O ¯ ,

is nite almost surely and P (τ ^x > 0) = 0 for all x ∈ ∂O , then x 7→ τ ^x is almost surely continuous. It

remains to observe that, under Assumption 3.2 (i)(ii), P (τ ^x = 0) = 1 for all x ∈ ∂O (see Bass [3, Corollary

3.3.2] or Pinsky [31, Theorem 2.3.3]) and thus τ ^x = η ^x for all x ∈ O ¯ .

3.3 Integrability Condition

In this section, we provide explicit sucient conditions to verify the integrability conditions on (ψ x ) _x∈O required for Propositions 3.2 and 3.3. Actually we study boundedness of (ψ x ) x∈O in L ^q for q ≥ 1 . In particular, the case q = 2 ensures that the corresponding Monte Carlo estimator has nite variance. Let us introduce the constant

C

₀

:= max

khk ∞ , sup

l∈L

kc l k ∞

p l

.

Clearly, it holds |ψ ^x | ≤ C

₀

^|K

^x

^| where |K ^x | denotes the cardinality of the set K ^x , i.e., the total number of particles. In particular, if C

₀

≤ 1 , then |ψ ^x | ≤ 1 . To the best of our knowledge, this is the only condition that has been used so far in the literature to ensure the integrability of ψ ^x .

In the rest of this section, we provide more technical but weaker conditions. First, we establish the desired result under minimal assumptions.

Proposition 3.4. Suppose that there exists a non-negative function v ∈ C

²

(O) ∩ C( ¯ O) satisfying Lv + β X

l∈L

|c l | ^q p ^q−1 _l v ^l − v

!

≤ 0 in O, v ≥ |h| ^q on ∂O, then we have E [|ψ ^x | ^q ] ≤ v(x) . In particular, (ψ ^x ) x∈O is uniformly bounded in L ^q . Proof. First, we observe that Itô's formula yields

v(x) ≥ E h

e ^−βη

^x

h(X _η ^x

x

)

q + Z η

^x

0

βe ^−βs X

l∈L

|c l (X _s ^x )| ^q

p ^q−1 _l v ^l (X _s ^x ) ds i .

Next, by repeating the arguments of Proposition 3.1, we get

v(x) ≥ E









 Y

k∈∪

ⁿ_i=0

K

^x_i

X

_Tk^k

∈O /

h(X _T ^k

_k

)

q Y

k∈∪

ⁿ_i=0

K

^x_i

X

_Tk^k

∈O

c _I

k

(X _T ^k

k

)

q

p ^q _I

_k

Y

k∈K

^x_n+1

v(X _T ^k

k−

)









 .

Then the conclusion follows easily from Fatou's lemma.

In practice, one can look for a constant supersolution in order to apply Proposition 3.4. For instance, for the case q = 1 , such a solution exists if and only if P

l∈L |c l |khk ^l _∞ ≤ khk _∞ . Otherwise, nding a supersolution has to be done on a case by case basis and might turn out to be a dicult task. For this reason, we provide an alternative result for which conditions are easier to verify than Proposition 3.4 and less stringent than assuming C

0

≤ 1 . Essentially, it signies that the integrability condition is satised if the domain is suciently small.

The idea is to dispose of the spatial parameter x by introducing a branching process which stochastically dominates the branching diusion process. Denote

δ := 1 − inf

x∈O

n E

h

e ^−βη

^x

io .

Let us introduce a new probability mass function (˜ p l ) _l∈ L

˜

with L ˜ := L ∪ {0} as follows:

˜

p

0

:= 1 − δ + δp

0

and p ˜ l := δp l for all l ≥ 1, and the corresponding transition matrix P ˜ = ( ˜ P i,j ) _i,j≥0 given by

P ˜

_0,0

= 1 and P ˜ _i,i+l−1 = ˜ p _l for all i ≥ 1, l ∈ L. ˜

Proposition 3.5. Denote by R the common radius of convergence of f(s) := P

l∈L p l s ^l and f ˜ (s) :=

P

l∈ L

˜

p ˜ _l s ^l . If R > 1 and P

l∈ L

˜

l p ˜ _l < 1 , then (ψ ^x ) _x∈O is uniformly bounded in L ^q provided that C

₀

^q ≤ γ where

γ := s ^∗

f ˜ (s ^∗ ) = s ^∗

1 − δ + δf(s ^∗ ) , (15)

where s ^∗ is the solution (if any) to s f ˜ ⁰ (s) = ˜ f (s) and s ^∗ = R otherwise. In addition, if R = ∞ , then γ goes

to innity as diam(O) goes to zero.

Proof. (i) Denote by (N _n ^x ) _n∈

_N

and ( ˜ N _n ) _n∈

_N

the number of particles at the n th jump time (arrival time) in the branching diusion process and in a branching process with ospring distribution (˜ p l ) _l∈ L

˜

respectively.

In particular, ( ˜ N n ) n∈

N

is a Markov chain with transition matrix P ˜ . Let us rst show that (N _n ^x ) n∈

N

is stochastically dominated by ( ˜ N n ) , i.e., P (N _n ^x ≥ l) ≤ P ( ˜ N n ≥ l) for all n, l ≥ 1 . We observe that

˜ p

0

= inf

x∈O { P (N

₁

^x = 0)} and p ˜ l = sup

x∈O

{ P (N

₁

^x = l)} for l ∈ L ˜ \ {0}.

It follows that P (N

₁

^x ≥ l) ≤ P ( ˜ N

1

≥ l) for all l ≥ 1 . For the incremental step, we rst observe that for all l ≥ 1 ,

P N _n+1 ^x ≥ l, N _n ^x ≥ l

≤ P (N _n ^x ≥ l) − P N _n ^x = l, N _n+1 ^x = l − 1

≤ P (N _n ^x ≥ l) − p ˜

₀

P (N _n ^x = l)

≤ (1 − p ˜

0

) P (N _n ^x ≥ l) + ˜ p

0

P (N _n ^x ≥ l + 1) , where the second inequality follows from Markov property. In addition, it holds

P N _n+1 ^x ≥ l, N _n ^x ≤ l − 1

≤

l−1

X

j=1

P N _n ^x = j, N _n+1 ^x − N _n ^x ≥ l − j

≤

l−1

X

j=1

∞

X

i=l−j+1

˜

p i P (N _n ^x = j)

≤

∞

X

i=l

˜

p i P (N _n ^x ≥ 1) −

∞

X

i=2

˜

p i P (N _n ^x ≥ l) +

l−1

X

j=2

˜

p _l−j+1 P (N _n ^x ≥ j) ,

where the second inequality follows once again from Markov property. Combining both computations above, we deduce that

P N _n+1 ^x ≥ l

≤

∞

X

i=l

˜

p i P (N _n ^x ≥ 1) +

l−1

X

j=2

˜

p _l−j+1 P (N _n ^x ≥ j) + ˜ p

1

P (N _n ^x ≥ l) + ˜ p

0

P (N _n ^x ≥ l + 1) . Hence the desired result follows easily by induction and Markov property.

(ii) Let us show next that E [|ψ ^x |] ≤ E [C

₀

^ζ ] where ζ the extinction time of ( ˜ N n ) _n∈N . First we recall that

|ψ ^x | ≤ C

₀

^|K

^x

^| where |K ^x | denotes the cardinality of the set K ^x . We observe further that |K ^x | coincides with the extinction time of (N _n ^x ) _n∈N . Clearly, Step (i) above yields that P (|K ^x | ≥ n) ≤ P (ζ ≥ n) for all n ≥ 1 , and thus

E [|ψ ^x |] ≤ E [C

₀

^|K

^x

^| ] ≤ E [C

₀

^ζ ].

(iii) We are now in a position to conclude the proof. It follows from Daley [9, Theorem 2] that the power series E [s ^ζ ] converges on its radius of convergence γ =

_˜

^s

^∗

f(s

^∗)

. In addition, if R = ∞ , then there exists a solution s ^∗ to s f ˜ ⁰ (s) = ˜ f (s) . Using further f ˜ (s) = 1 − δ + δf(s) , we deduce that

s ^∗ f ⁰ (s ^∗ ) − f (s ^∗ ) = 1 − δ δ .

It follows that s ^∗ goes to innity as δ goes to zero, or equivalently, diam(O) goes to zero. Then we have γ = 1

f ˜ ⁰ (s ^∗ ) = 1

δf ⁰ (s ^∗ ) = s ^∗ 1 − δ

1 − f (s ^∗ ) s ^∗ f ⁰ (s ^∗ )

.

To conclude, it remains to observe that _s

∗

^f(s f

⁰(s^∗)∗)

is bounded away from 1 since s 7→ _sf ^f(s)

0(s)

is decreasing.

Remark 3.3. It follows immediately from Step (i) of the proof of Proposition 3.5 that if P

l∈ L

˜

l p ˜ _l ≤ 1 , then the branching diusion process goes extinct almost surely, i.e., Assumption 2.3 holds.

4 Semi-Linear PDEs with Non-Linear Gradient Term

In this section we study the case of semi-linear PDE with non-linearity in gradient of the solution, i.e., we

assume that m ≥ 1 in Section 2 so that the particles in the branching diusion process carry dierent marks

to account for it. Throughout this section, we suppose that Assumption 2.12.3 remain valid.

4.1 Probabilistic Representation

Our next assumption is the key automatic dierentiation condition on the underlying diusion process X ^x . We will provide explicit conditions and formulas for it in the next sections.

Assumption 4.1. (i) The map x 7→ E [e ^−βη

^x

h(X _η ^x

x

)] belongs to C

¹

(O) ∩ C( ¯ O) and there exists a measurable function W ∂O (x, W ) = W ∂O (x, (W _r ) _r∈[0,η

x]

) such that

D E h

e ^−βη

^x

h(X _η ^x

x

) i

= E h

e ^−βη

^x

h(X _η ^x

x

)W ∂O (x, W ) i .

(ii) For any g : O → R bounded measurable, the map x 7→ E [ R η

^x

0

e ^−βs g(X _s ^x ) ds] belongs to C

¹

(O) ∩ C( ¯ O) and there exists a measurable function W _O (s, x, W ) = W _O (s, x, (W r ) _r∈[0,s] ) such that

D E

"

Z η

^x 0

e ^−βs g(X _s ^x ) ds

#

= E

"

Z η

^x 0

e ^−βs g(X _s ^x )W _O (s, x, W ) ds

# .

Let us dene W(s, x, W ) = W(s, x, (W r ) _r∈[0,s] ) as follows:

W(s, x, W ) := W ∂O (x, W )1 X

_s^x

∈O / + W O (s, x, W )1 X

_s^x

∈O .

We consider a marked branching diusion process starting from x ∈ O as in Section 2.2 and denote W k := 1 m

_k=0

+ 1 m

_k

6=0 b m

_k

(X _T ^k

k−

) · W (∆T k , X _T ^k

k−

, W ^k ),

where m k and ∆T k := T k − T _k

−

stand for the mark and the lifetime of particle k respectively. We next introduce the following random variable:

ψ ^x := Y

k∈K

^x

X

_Tk^k

∈O /

e ^−β∆T

^k

h(X _T ^k

k

) F ¯ (∆T _k ) W k

Y

k∈K

^x

X

_Tk^k

∈O

βe ^−β∆T

^k

c _I

k

(X _T ^k

k

) p _I

k

ρ(∆T k ) W k .

where F ¯ (t) := R ∞

t ρ(s) ds , t ≥ 0 .

Proposition 4.1. Suppose Assumption 4.1 holds. Assume further that PDE (5) has a solution u ∈ C

²

(O) ∩ C( ¯ O) such that the functions (b _i · Du) _{i=1,··· ,m} are bounded and the sequence (ψ ^x _n ) _n∈

_N

dened by

ψ _n ^x := Y

k∈∪

ⁿ_i=1

K

^x_i

X

_Tk^k

∈O /

e ^−β∆T

^k

h(X _T ^k

k

) F ¯ (∆T _k ) W k

Y

k∈∪

ⁿ_i=1

K

^x_i

X

_Tk^k

∈O

βe ^−β∆T

^k

c _I

k

(X _T ^k

k

) p _I

k

ρ(∆T k ) W k

Y

k∈K

^x_n+1

m

k=0

u(X _T ^k

k−

) Y

k∈K

^x_n+1

m

_k

6=0

(b m

_k

· Du)(X _T ^k

k−

),

is uniformly integrable. Then, we have u(x) = E [ψ ^x ] .

Proof. The proof follows similar arguments as the proof of Proposition 3.1. Using Itô's formula, we have the following Feynman-Kac representation:

u(x) = E h

e ^−βη

^x

h(X _η ^x

x

) + Z η

^x

0

βe ^−βs f (·, u, Du)(X _s ^x ) ds i .

Then, we can write u(x) = E

"

e ^−βη

^x

h(X _η ^x

x

)

F(η ¯ ^x ) 1 _τ≥η

x

+ βe ^−βτ f (·, u, Du)(X _τ ^x ) ρ(τ) 1 τ <η

^x

#

(16)

= E

"

e ^−βη

^x

h(X _η ^x

x

)

F(η ¯ ^x ) 1 _τ≥η

^x

+ βe ^−βτ c I (X _τ ^x )

p _I ρ(τ) u ^I

⁰

(X _τ ^x )

m

Y

i=1

(b i · Du) ^I

ⁱ

(X _τ ^x )1 τ <η

^x

# .

In other words, we have u(x) = E [ψ ^x

₀

] . Furthermore, in the original Feynman-Kac formula, we obtain by dierentiating and using Assumption 4.1,

Du(x) = E

"

e ^−βη

^x

h(X _η ^x

x

)W ∂O (x, W ) + Z η

^x

0

βe ^−βs f (·, u, Du)(X _s ^x )W O (s, x, W ) ds

#

= E

ψ ^x

₀

W(T _∅ , x, W )

.

Since each ospring has the same dynamic as the parent particle, we can repeat the above calculations for k ∈ K ^x

₁

and plug the results back in (16) to obtain u(x) = E [ψ

₁

^x ] by conditional independence of particles in K

₁

^x given F

0

. We conclude by iteration that, for any n ∈ N, u(x) = E [ψ ^x _n ] and, as n → ∞, u(x) = E [ψ ^x ] .

Similar to Section 3.1, Proposition 4.1 provides a result of uniqueness for a class of semi-linear PDEs given an appropriate uniform integrability condition is satised. Next we establish a result of existence by showing that the probabilistic representation is a viscosity solution of PDE (5). This is the main result of this section.

Theorem 4.1. Suppose Assumption 4.1 holds. If we further assume that (ψ ^x ) x∈O and, for i = 1, . . . , m , (ψ ^x b i (x) · W(T _∅ , x, W )) x∈O are uniformly bounded in L

¹

, then u : x 7→ E [ψ ^x ] belongs to C

¹

(O) ∩ C( ¯ O) and solves PDE (5) in the viscosity sense.

Proof. The proof follows similar arguments as the proof of Theorem 3.1. We observe rst that, by denition of u , it holds

u(x) = E





e ^−βη

^x

h(X _η ^x

x

)

F(η ¯ ^x ) 1 _τ≥η

x

+ βe ^−βτ c _I (X _τ ^x ) p I ρ(τ)

|I|−1

Y

i=0

ψ ^X _i

^τx

1 _{τ <η}

x



 .

where

ψ ^X

x τ

i := Y

k=(i,...)∈K

^x

X

_Tk^k

∈O /

e ^−β∆T

^k

h(X _T ^k

k

) F(∆T ¯ _k ) W k

Y

k=(i,...)∈K

^x

X

_Tk^k

∈O

βe ^−β∆T

^k

c _I

k

(X _T ^k

k

) p _I

k

ρ(∆T k ) W k .

It further follows from the branching property that, conditioned on F

0

, (ψ ^X

x τ

i ) i=0,...,|I|−1 are independent random variables, among which the rst I

0

are identical in law to ψ ^X

⁰

, the next I

1

are identical in law to ψ ^X

⁰

b

₁

(x) · W(T _∅ , x, W ) and so on, where X

₀

is distributed as X _τ ^x and independent of F n for all n ∈ N.

Through this argument, we deduce that

E h ^|I Y ^|−1

i=0

ψ ^X _i

^τx

F

0

i

1 τ <η

^x

= u ^I

⁰

(X _τ ^x )

m

Y

i=1

v _i ^I

ⁱ

(X _τ ^x )1 τ <η

^x

.

where (v i ) i=1,...,m , v i : O 7→ R are dened as

v i (x) := E [ψ ^x b i (x) · W (T _∅ , x, W )].

Working backward along the lines of the proof of Proposition 4.1, we deduce that u(x) = E

h

e ^−βη

^x

h(X _η ^x

x

) + Z η

^x

0

βe ^−βs X

l∈L

c l u ^l

⁰

m

Y

i=1

v ^l _i

ⁱ

(X _s ^x ) ds i

. (17)

In particular, since u and (v i ) i=1,...,m are bounded by assumption, it follows from Assumption 4.1 that u belongs to C

¹

(O) ∩ C( ¯ O) and

Du(x) = E h

e ^−βη

^x

h(X _η ^x

x

)W _∂O (x, W ) + Z η

^x

0

βe ^−βs X

l∈L

c _l u ^l

⁰

m

Y

i=1

v _i ^l

ⁱ

(X _s ^x )W _O (s, x, W ) ds i

= E [ψ ^x W(T _∅ , x, W )].

Thus, for all i = 1, . . . , m , v i coincides with b i · Du and (17) reads as u(x) = E

h

e ^−βη

^x

h(X _η ^x

x

) + Z η

^x

0

βe ^−βs f (·, u, Du)(X _s ^x ) ds i .

The fact that u is a viscosity solution of PDE (5) now follows by classical arguments.

4.2 Automatic Dierentiation Formula: the General Case

The aim of this section is to provide sucient conditions to ensure that Assumption 4.1 holds and to derive explicit formula for W . The automatic dierentiation formula discussed in the following originates from Thalmaier [35] and was subsequently developed by Delarue [11] and Gobet [16].

Assumption 4.2. (i) The coecients (µ, σ) belong to C

^1,α

( ¯ O) . (ii) The diusion coecient σ is uniformly elliptic.

(iii) The boundary ∂O is of class C

²

.

(iv) The function h can be extended to a function of class C

^1,α

on O ¯ .

We start by establishing a technical lemma. Fix a nite horizon T > 0 and denote for any s > 0 , θ _s (r, y) := 1

d (y, ∂ O)

²

(s − r) , for all y ∈ O, r ∈ [0, s).

Lemma 4.1. Under Assumption 4.2, it holds for all x ∈ O and s > 0 , Z η

^x

∧s

0

θ s (r, X _r ^x ) dr = ∞, P − a.s. (18)

In addition, if we denote

ζ s := inf

t > 0 : Z t

0

θ s (r, X _r ^x ) dr = 1

,

then there exists t < s such that ζ s ≤ η ^x ∧ t and for all q ≥ 1 ,

E

"

Z ζ

s∧T

0

θ _s∧T

²

(r, X _r ^x ) dr

! q #

≤ C

d (x, ∂O)

^4q−2

(s ∧ T ) ^q , (19) where C > 0 depends on q and T but not on x or s .

Proof. The proof essentially follows from Delarue [11]. Indeed, if O is a ball, both identities (18) and (19) are easily obtained by repeating the arguments of Propositions 2.3 and 2.4 in [11] while working with (r, y) 7→ d(y, ∂ O) √

s − r instead of (r, y) 7→ d(y, ∂ O)(s − r) . For an arbitrary domain, it suces to work with a C

²

extension of the distance to the boundary (see, e.g., Gilbarg and Trudinger [15, Lemma 14.16]).

Additionally, it follows from (18) that ζ _s ≤ η ^x ∧ s . Furthermore, it holds for all t < s , Z t

0

θ _s (r, X _r ^x ) dr 1 _t≤η

x

≥ −C ⁻¹ log

1 − t s

1 _t≤η

x

,

where C := diam(O)

²

/4 . Thus for t = (1 − e ^−C )s , we have ζ s ≤ η ^x ∧ t .

Proposition 4.2. Under Assumption 4.2, the assertions of Assumption 4.1 are satised with W _∂O ^> (x, W ) =

Z ζ

_T 0

θ T (r, X _r ^x ) σ ⁻¹ (X _r ^x )Y _r ^{x >}

dW r ,

W _O ^> (s, x, W ) = Z ζ

s∧T

0

θ _s∧T (r, X _r ^x ) σ ⁻¹ (X _r ^x )Y _r ^x >

dW _r ,

where Y ^x is the tangent process given by

Y _s ^x = I _d + Z s

0

Dµ(X _r ^x )Y _r ^x dr +

d

X

i=1

Z s

0

Dσ _i (X _r ^x )Y _r ^x dW _r ⁱ .

and σ _i denotes the i th column of σ . In addition, it holds for all q ≥ 1 ,

E

"

Z ζ

s∧T

0

θ s∧T (r, X _r ^x ) σ ⁻¹ (X _r ^x )Y _r ^x >

dW r

q #

≤ C

d(x, ∂O)

^2q−1

(s ∧ T )

^q2

. (20)