Noufel Frikha
email: frikha@math.univ-paris-diderot.fr

Arturo Kohatsu-Higa

Libo Li
email: libo.li@unsw.edu.au

ON THE FIRST HITTING TIMES FOR ONE-DIMENSIONAL ELLIPTIC DIFFUSIONS

Keywords: 2000 Mathematics Subject Classification. Primary 60H10, 60G46, Secondary 60H30, 35K65 Killed diffusion, parametrix expansion, first hitting time, density estimates

In this article, we obtain properties of the law associated to the first hitting time of a threshold by a one-dimensional uniformly elliptic diffusion process and to the associated process stopped at the threshold. Our methodology relies on the parametrix method that we apply to the associated Markov semigroup. It allows to obtain explicit expressions for the corresponding transition densities and to study its regularity properties up to the boundary under mild assumptions on the coefficients. As a by product, we also provide Gaussian upper estimates for these laws and derive a probabilistic representation that may be useful for the construction of an unbiased Monte Carlo path simulation method, among other applications.

Introduction

In this article, we consider the following one-dimensional stochastic differential equation (SDE in short)

(1.1) X u,x t " x `ż t u bpX u,x s qds `ż t u σpX u,x s qdW s , t ě u ě 0, x P R where pW t q tě0 stands for a one-dimensional Brownian motion on a given filtered probability space pΩ, F, pF t q tě0 , Pq.

Our main interest is to study the law of the first hitting time of the level L (or equivalently the exit time of the open set p´8, Lq) by the one-dimensional process X defined by τ u,x " inf tv ě 0, X u,x u`v ě Lu and the associated killed diffusion process pX x τ u,x t q tě0 . Here, we write τ u,x t :" τ u,x ^pt ´uq. In various applications, such as ruin probability, mathematical finance [START_REF] Musiela | Martingale methods in financial modelling[END_REF] or neurosciences [START_REF] Delarue | Global solvability of a networked integrate-and-fire model of McKean-Vlasov type[END_REF], one is interested in results related to the existence of a density for τ u,x t or the vector pτ u,x t , X x u`τ u,x t q and if it exists its regularity properties as well as sharp upper-bounds for the density and its derivatives.

Several results concerning existence and smoothness properties (as well as some Gaussian bounds) for the densities of the exit time of one-dimensional diffusion processes and the associated killed diffusions have been established in the literature. When the coefficients are smooth, we refer e.g. to [START_REF] Pauwels | Smooth first-passage densities for one-dimensional diffusions[END_REF] for the existence and smoothness of a first-passage density using a Lamperti transformation technique combined with Girsanov theorem. We also refer to the recent unpublished note [START_REF] Delarue | First hitting times for general non-homogeneous 1d diffusion processes: density estimates in small time[END_REF] for some Gaussian upper-bounds in the case of a non-homogeneous smooth drift coefficient and a constant diffusion coefficient using a PDE point of view of the parametrix method.

On the other hand, in a multi-dimensional setting and for a domain D such that BD is smooth and noncharacteristic, Cattiaux [START_REF] Cattiaux | Calcul stochastique et opérateurs dégénérés du second ordre. II. Problème de Dirichlet[END_REF] developed a Malliavin's calculus approach to prove that the semigroup associated to a process killed when it hits the boundary BD admits an infinitely differentiable kernel under a restricted Hörmander condition on the vector fields. Gaussian bounds on this kernel are also established in small time. We also refer the reader to Ladyzenskaja and al. [START_REF] Ladyženskaja | Linear and quasilinear equations of parabolic type[END_REF], Friedman [START_REF] Friedman | Partial differential equations of parabolic type[END_REF] and Garroni and Menaldi [START_REF] Garroni | Green functions for second order parabolic integro-differential problems[END_REF] for constructions of Green functions related to a class of Cauchy-Dirichlet value problems in a uniformly elliptic setting using a partial differential equation framework.

In order to study this problem, one is naturally led to define the collection of linear maps pP t q tě0 , acting on B b pRq, as follows (1.2) @pu, xq P R `ˆR, P t hpu, xq " E " hpu `τ u,x u`t , X u,x u`τ u,x u`t q ı .

From the above definition, one realizes that the main problem to analyse the above quantity is that the probability measure generated by the couple pτ u,x u`t , X u,x u`τ u,x u`t q is singular. Indeed, if the process X does not reach the boundary in the interval ru, u `ts, the law of τ u,x u`t is concentrated on time t. Conversely, if the process exits the domain before time u `t, the law of X u,x u`τ u,x u`t will have a point mass at L.

To investigate this problem, we rely on a perturbation technique, known as the parametrix method, that we apply to the process pu `τ u,x u`t , X u,x u`τ u,x u`t q tě0 in order to obtain an expansion of P t hpu, xq as an infinite series. The main interest for introducing the linear maps pP t q tě0 is that it allows to include the first exit time and the related killed diffusion in the same analysis. From this representation, we obtain the existence of a transition density for pP t q tě0 on R `ˆp´8, Lq under mild smoothness assumption on the coefficients. As a by product, we study its regularity properties and derive Gaussian upper-bounds.

The parametrix method is a classical perturbation technique used in partial differential equation (PDE in short) theory that allows to give an expansion in infinite series of iterated kernels of the fundamental solution of an elliptic or parabolic PDE as exposed e.g. in Friedman [START_REF] Friedman | Partial differential equations of parabolic type[END_REF] or McKean and Singer [START_REF] Mckean | Curvature and the eigenvalues of the Laplacian[END_REF]. Its success is due to its robustness and flexibility as it can be invoked for a wide variety of PDEs both for theoretical goals such as density estimates, see e.g. Delarue and Menozzi [START_REF] Delarue | Density estimates for a random noise propagating through a chain of differential equations[END_REF], Kohatsu-Higa & al. [START_REF] Kohatsu-Higa | The parametrix method for skew diffusions[END_REF] and for numerical approximations see e.g. Konakov and Mammen [START_REF] Konakov | Local limit theorems for transition densities of Markov chains converging to diffusions[END_REF] and Frikha and Huang [START_REF] Frikha | A multi-step Richardson-Romberg extrapolation method for stochastic approximation[END_REF] among others. Though its application seems to be restricted to Markov processes, it notably allows for coefficients to be less regular than in the Malliavin calculus approach for the study of transition densities.

Recently, Bally and Kohatsu-Higa [START_REF] Bally | A probabilistic interpretation of the parametrix method[END_REF] used a semigroup approach to the parametrix method in order to obtain a probabilistic representation for the transition density of the solution to elliptic diffusion processes and some Lévy driven SDEs. Let us note that the case of stable-like driven SDE with Hölder continuous coefficients has been handled in [START_REF] Kohatsu-Higa | Regularity of the density of a stable-like driven SDE with Hölder continuous coefficients[END_REF]. Although a difficult aspect of the problem lays in the singular behavior of the joint law of pu `τ u,x u`t , X u,x u`τ u,x u`t q tě0 , we try to follow the approach initiated in [START_REF] Bally | A probabilistic interpretation of the parametrix method[END_REF] by considering two kinds of techniques, namely the forward parametrix method and the backward parametrix method which require different smoothness assumptions on the coefficients and provide different properties on the underlying density.

Roughly speaking, the forward parametrix method consists in approximating the process pu `τ u,x u`t , X u,x u`τ u,x u`t q tě0

by the proxy process pu `τ u,x u`t , Xu,x u`τ u,x u`t q tě0 where p Xu,x u`t q tě0 has dynamics given by (1.1) with diffusion coefficient frozen at the initial point x and with zero drift, τ u,x being its associated exit time. In order to make the argument of the forward parametrix approach works properly, one has to assume that the drift coefficient1 is C 1 b pRq and that the diffusion coefficient is bounded, uniformly elliptic and C 2 b pRq. Then conclusions on the regularity of the density with respect to the terminal point are obtained.

On the other hand, a backward parametrix expansion, usually uses an Euler scheme with coefficients frozen at the terminal point of the density as proxy process. For this reason, the method is called backward. This method can be applied if the drift coefficient is measurable and bounded and the diffusion coefficient is bounded, uniformly elliptic and Hölder-continuous. Regularity properties with respect to the starting point x can be established. Under the mild smoothness assumptions of the backward parametrix framework (see assumption (H2) in Section 4), we were unable to find references on the existence, regularity properties and Gaussian estimates for the density of the couple pτ u,x u`t , X u,x τ u,x u`t q up to the boundary value L. Although it may be possible to link pt, u, xq Þ Ñ P t hpu, xq to the unique classical solution of a Cauchy-Dirichlet PDE thanks to a Feynman-Kac representation formula, this will require additional smoothness on the coefficients that we do not want to impose here.

As explained before, in our case, the situation is more challenging than in the standard diffusion or Lévy driven setting studied in [START_REF] Bally | A probabilistic interpretation of the parametrix method[END_REF], since we have to deal with two processes which have a singular behavior with respect to each other. Another technical difficulty (compared to the standard diffusion setting investigated in [START_REF] Bally | A probabilistic interpretation of the parametrix method[END_REF]) that appears in the backward setting lies in the proof of the convergence of the parametrix series corresponding to the first hitting time since the singularity in time induced by the exit time distribution of the frozen process is of higher order. As it will become clear later on, when dealing with the part corresponding to the exit time, the key idea is to use an Euler scheme with coefficients frozen at the barrier L whereas, when using this approach for the stopped process, as in the case studied in [START_REF] Bally | A probabilistic interpretation of the parametrix method[END_REF], one has to use a standard Euler scheme with coefficients frozen at the terminal point of the density.

In conclusion, the main advantage of the parametrix expansion is that it allows to prove the existence of the density for the couple pu `τ u,x u`t , X u,x u`τ u,x u`t q and to study its regularity properties under such rather mild assumptions on the coefficients. We believe that the methodology and the collection of results established here may be extended to certain type of multi-dimensional smooth domains. This will be taken up in future works.

One of the main advantages for considering the forward parametrix expansion and not only the backward method is that it allows to obtain regularity of the transition density with respect to the terminal point and also it leads to a more natural probabilistic representation for the density of the process in consideration and therefore also provides a representation for P t hpu, xq that can be used for an unbiased Monte Carlo numerical simulation or as an alternative to Malliavin calculus. We refer to [START_REF] Anderson | Unbiased simulation of stochastic differential equations using parametrix expansions[END_REF] for a comprehensive insight on unbiased simulation of SDEs. As far as numerical approximations of (1.2) are concerned, the standard methodology to evaluate such expectation is to discretize the dynamics (1.1) using an Euler scheme and to consider its discrete hitting time. A more sophisticated procedure consists in interpolating the standard approximation scheme into a continuous Euler scheme and then using the law of some Brownian bridge in order to take into account the probability that the process has left the domain between two discretization times or not. For a rigorous treatment of the weak discretization error for the evaluation of Erf pX x T q1 tτ x ąT u s and some implementable simulation schemes, we refer to [START_REF] Gobet | Weak approximation of killed diffusion using Euler schemes[END_REF] and [START_REF] Gobet | Exact approximation rate of killed hypoelliptic diffusions using the discrete Euler scheme[END_REF] and the references therein. The two probabilistic representations obtained in the paper notably allow to remove the discretization error that appears in the numerical evaluation of P T hpu, xq.

This article is divided as follows: in Section 2, we will discuss some properties of the process in consideration. Notably, we will see that the collection of positive linear maps given by (1.2) defines a Markov semigroup and characterize its infinitesimal generator. In Section 3, we introduce the forward parametrix method and its application to the current problem. The main results of this section are given in Theorem 3.1 where an expansion is obtained for the semigroup, which is then used to prove in Theorem 3.2 the existence of the transition density function and also to obtain Gaussian upper estimates. Some regularity properties are also studied in Theorem 3.3, namely we consider the differentiability of kernel related to killed diffusion with respect to its terminal point and also the Hölder regularity in time of the kernel related to the exit time. We then conclude the section by providing some applications such as the probabilistic presentation (see Theorem 3.5) that leads to an unbiased Monte Carlo path simulation for Erhpτ x t , X x τ x t qs, an integration by parts formula with respect to the killed process or bounds on Erhpτ x t , X x τ x t qs and ErB 2 hpτ x t , X x τ x t qs under weak conditions on the test function h. In Section 4, we introduce and establish the backward parametrix expansion for the semigroup under mild regularity assumptions on the coefficients, namely, the drift coefficient is measurable and bounded and the diffusion coefficient is bounded, uniformly elliptic and Hölder-continuous. Similarly to the forward parametrix method, an expansion of the semigroup is obtained in Theorem 4.1, then the existence of the transition density and its regularity properties with respect to the initial point are discussed in Theorem 4.2 and Theorem 4.5 respectively. Finally, we discuss some applications such as a probabilistic representation for the semigroup or the transition density (see Theorem 4.8), a Bismut type formula with respect to the killed process or bounds on Erhpτ x t , X x τ x t qs and B x Erhpτ x t , X x τ x t qs under weak conditions on the test function h and the coefficients. Finally, in a short appendix, we provide some useful key estimates in order to construct our parametrix expansions.

Preliminaries

Notations.

We first give some basic notations and definitions used throughout this paper. For a sequence of linear operators pS i q 1ďiďn , we define ś n i"1 S i " S 1 ¨¨¨S n and ś 1 i"n S i " S n ¨¨¨S 1 . We will often use the convention ś H " 1 which appears when we have for example ś ´1 i"0 . Furthermore we will use the following notation for time and space variables s p " ps 1 , ¨¨¨, s p q, z p " pz 1 , ¨¨¨, z p q, the differentials ds p " ds 1 ¨¨¨ds p , dz p " dz 1 ¨¨¨dz p and for a fixed time t ě 0, we denote by ∆ p ptq " ts p P r0, ts p : s p`1 :" 0 ď s p ď s p´1 ď ¨¨¨ď s 1 ď t ": s 0 u and ∆ p ptq " ts p P r0, ts p : s 0 :" 0 ď s 1 ď s 2 ď ¨¨¨ď s p ď t ": s p`1 u. For a multi-index α " pα 1 , ¨¨¨, α q of length , we sometimes write B α f pxq " B xα 1 ¨¨¨B xα f pxq, for a vector x. For a real valued function f defined on R, we will also use the notation |f | 8,L :" sup xPp´8,Ls |f pxq| whenever this quantity is finite. We denote by y Þ Ñ gpct, yq the transition density function of the standard Brownian motion with variance c, i.e. gpct, yq " p2πtcq ´1{2 expp´y 2 {p2tcqq, y P R. The associated Hermite polynomials are defined respectively as H i pct, yq " gpct, yq ´1B i y gpct, yq for i P N. We write Φpxq " ş x ´8 gp1, yqdy for the cumulative distribution function of the standard normal law. Sometimes, we also use the alternative notation erfcpxq " 2p1 ´Φp ? 2xqq. For a fixed given point z P R, the Dirac measure is denoted by δ z pdxq.

For any function h with domain D Ď R, we denote its support by suppphq Ď D. We follow the common practice of denoting by C k b pEq the collection of all real-valued bounded continuous functions defined on E which have continuous and bounded derivatives of every order up to k. The set B b pEq is the collection of real-valued bounded measurable maps defined on E. If pu, xq Þ Ñ hpu, xq P B b pR `ˆRq is a continuous function on R `ˆp´8, Ls with partial derivatives B 1 hpu, xq, B 2 hpu, xq and B 2,2 hpu, xq " B 2 2 hpu, xq, being continuous and bounded on R `ˆp´8, Ls (continuity and derivatives at x " L are always understood as left-continuity and left-derivatives at x " L), we write h P C 1,2 b pR `ˆp´8, Lsq and similar notation will be used when the domain is a general product space. The reader is warned that the latter space is not standard but is introduced here in order to reduce the amount of notation. We finally introduce the space C 0 pR `ˆRq of continuous function defined on R `ˆR that vanishes at infinity. 2.2. Markov semigroup, Itô's formula and related infinitesimal generator.

The aim of this section is to study the collection of positive linear maps defined by (1.2). We assume that there exists a unique weak solution to (1.1) that satisfies the strong Markov property. We first emphasize that since the process pX u,x u`t q tě0 given in (1.1) is time-homogeneous it may be understood as a shifted version of pX 0,x t q tě0 . Specifically, we can choose the canonical Wiener space for pΩ, pF t q tě0 , Pq and thus introduce the shift operator pθ u : w Þ Ñ θ u pwq " wpu `.q ´wpuqq uě0 . Then, pX u,x u`t q tě0 " pX 0,x t ˝θu q tě0 , or we will simply write pX x t ˝θu q tě0 , with the convention X x " X 0,x . Notably, one has @pu, xq P R `ˆR, P t hpu, xq " Erhpu `τ x t , X x τ x t qs.

To apply the parametrix method, we claim that the process pu `τ x t , X x τ x t q tě0 is a Markov process, and for the readers convenience, the proof of this fact is provided in Proposition 5.1 whose statement and proof is postponed in Appendix 5.3. Under additional smoothness assumptions on the coefficients, namely that b and σ are bounded Lipschitz continuous functions and that Ppτ x " tq " 0, t ą 0 and x ă L, we prove that pP t q tě0 is a strongly continuous Feller semigroup but we will not need this property for the analysis developed below. Note that the property Ppτ x " tq " 0 has been proven in [START_REF] Hayashi | Smoothness of the distribution of the supremum of a multi-dimensional diffusion process[END_REF] under enough regularity of the coefficients b and σ. If one is interested in establishing the strong Feller property, one may assume for the moment that coefficients here satisfy the assumptions in [START_REF] Hayashi | Smoothness of the distribution of the supremum of a multi-dimensional diffusion process[END_REF] which guarantee Ppτ x " tq " 0. Later we will see that the absolute continuity property of the law of τ x only depends on (H1) or (H2) (see Section 3 and Section 4 below). Therefore, a limit procedure will finish the argument. Also note that the following relation is satisfied: Ppmax 0ďsďt X x s ă Lq " Ppτ x ą tq, therefore showing the duality between stopped process and its associated exit time.

We also consider the following proxy process Xy,u,x u`t with coefficients frozen at a fixed point y P R and with dynamics given by Xy,u,x u`t " x `σpyqpW t`u ´Wu q and its corresponding exit time τ y,u,x :" inf tv ě 0, Xy,u,x u`v ě Lu, τ y,u,x t :" τ y,u,x ^pt ´uq. From now on, pu τ y,u,x t

, Xy,u,x u`τ u,x t q tě0 and p P y t q tě0 will be referred as the frozen process and its associated semigroup defined for h P B b pR `ˆRq by P y t hpu, xq " Erhpu `τ y,u,x t , Xy,u,x u`τ y,u,x t qs " Erhpu `τ y,x t , Xy,x τ y,x t qs with τ y,x t :" τ y,0,x ^t. Note that we removed the drift part in the dynamics of Xy,u,x since it plays no role in the analysis below. In order to simplify the notations, we will remove the superscript y and write p Pt q tě0 and pτ u,x t , Xu,x u,u`τ u,x t q tě0 when there is no confusion. Similarly, by time-homogeneity, we work with the process pu `τ x t , Xx τ x t q tě0 and follow the same notation as for the original process.

We now characterize the infinitesimal generators L and Ly of respectively pP t q tě0 and p Pt q tě0 . Lemma 2.1.

"ż t 0 1 tsďτ x u B 2 hpu `s, X x s qbpX x s qds `ż t 0 1 tsďτ x u B 2 hpu `s, X x s qσpX x s q dW s  `1txăLu "ż t 0 1 tsďτ x u B 1 hpu `s, X x s qds `1 2 ż t 0 1 tsďτ x u B 2 2 hpu `s, X x s qapX x s qds  .
Hence, we get , Lq| 8 and the coefficients b, σ. Now using that τ x ą 0 a.s. for x ă L and the right-continuity of t Þ Ñ Ppτ x ď tq, one gets Ppτ x ď tq Ñ 0 as t Ó 0. Now the continuity of the paths of the process pu `τ x t , X x τ x t q tě0 and the continuity of the coefficients b, σ on p´8, Ls finally yield

Erhpu `τ x ^t, X x τ x ^tqs ´hpu, xq t " 1 t ż t 0 1 txăLu pErbpX x s^τ x qB 2 hpu `s ^τ x , X x s^τ x q `1 2 apX x s^τ x qB 2 2 hpu `s ^τ x , X x s^τ x qs `ErB 1 hpu `s ^τ x , X x s^τ x q sqds ´1txăLu 1 t ż t 0 ErpbpLqB 2 hpu `τ x , Lq `1 2 apLqB 2 2 hpu `τ x , Lq `B1 hpu `τ x , Lqq1
P t hpu, xq ´hpu, xq t Ñ Lhpu, xq as t Ó 0.
The same line of reasoning gives the result for the infinitesimal generator Ly of the proxy process so that we omit its proof.

Remark 2.1. From the proof of Lemma 2.1, we also get that P t h P dompLq for h P C 1,2 b pR `ˆp´8, Lsq. Indeed, since 1 ε pP t`ε h ´Pt hq " pP ε ´Iq ε P t h " P t pP ε ´Iq ε h for all ε ą 0 and that ε ´1pP ε ´Iqhpu, xq Ñ Lhpu, xq as ε Ñ 0, by dominated convergence theorem, one gets P t h P dompLq and also ε ´1pP t`ε h ´Pt hq Ñ LP t h " P t Lh as ε Ñ 0. For other properties on semigroups and related results on their infinitesimal generators, we refer to Ethier and Kurtz [START_REF] Ethier | Markov processes[END_REF].

Forward parametrix expansion

In this section we apply the forward parametrix expansion using a semigroup approach. Section 3.1 is devoted to the expansion of the semigroup pP t q tě0 . In Section 3.2, the existence and an expansion of the transition density function are derived as a by product of the semigroup expansion. Some regularity estimates and Gaussian upperbounds are also obtained. Finally, in Section 3.3, several applications are discussed. In particular, a probabilistic representation is provided.

Through this section, we will make the following assumptions on the coefficients b, σ : R Ñ R:

Assumptions (H1).

(i) σ is bounded. Moreover, a " σ 2 is uniformly elliptic, that is there exist a, a ą 0 s.t. for any x P R, a ď apxq ď a. Before performing the forward parametrix expansion, we first need to study the transition density of the proxy semigroup and to obtain some key estimates. Hence, we remark that the density of the proxy process is composed of two singular measures. As it will appear clearly in the following analysis, this fact raises difficulties in establishing a parametrix expansion of the semigroup.

In order to simplify the expressions appearing in the parametrix series, we define the following two kernels Observe here the double use of K and S as an operator and a kernel. Moreover, let us note that the operator Kt is not standard compared to the diffusion setting since it does not involve the integral of a kernel. This operator comes from the very nature of the forward parametrix method used in this section which require doing integration by parts and dealing with such boundary terms. In particular, let us remark that under (H1) using the space-time inequality2 : @x P R, |x| p e ´qx 2 ď ppp{p2qeqq p{2 , valid for any p, q ą 0, one easily gets gpct, z ´xq (3.5)

(
In this section, in order to simplify the notations, we will write p Pt q tě0 for the semigroup with frozen coefficients at the starting point x that is Pt hpu, xq " P x t hpu, xq and also write S t hpu, xq " S x t hpu, xq, K t hpu, xq " K x t hpu, xq when there is no confusion.

The following proposition corresponds to a first order expansion of pP t q tě0 around p Pt q tě0 and is the keystone to build the forward parametrix expansion for the semigroup pP t q tě0 . Proposition 3.1. where we performed an integration by parts formula in the last equality and used the fact that qx T ´spx, Lq " 0 for all x P p´8, Ls. We now let η goes to zero in the previous result. By dominated convergence theorem and the continuity of z Þ Ñ Lhpu, zq, one gets ż R pErLhpu `τ x s , X x τ x s qs ´ErLhpu `τ x s , z `Xx

τ x s qsqgpη, zqdz Ñ 0, η Ó 0.
Under (H1), from Theorem 3.1 in Pauwels [START_REF] Pauwels | Smooth first-passage densities for one-dimensional diffusions[END_REF] (see also Theorem 4.2 in Section 4), τ x admits a positive density for x ă L so that in particular Ppτ x " tq " 0. By Proposition 5.1, it follows that z Þ Ñ P s hpu `T ´s, zq and z Þ Ñ B 1 P s hpu `T ´s, zq " P s B 1 hpu `T ´s, zq are continuous on R. This in turn yields

ż L ´8 ż R dzdy qx T ´spx
P T hpu, xq ´P T hpu, xq " ż T 0 ds `K T ´shpu, xq `S T ´sP s hpu, xq ˘.
We also remark that for any bounded Borel function h, one has Ks hpu, Lq " Ss hpu, Lq " 0 and therefore

KT ´s1 Ks1´s2 hpu, xq " 1 txăLu papLq ´apxqq apxq f x τ px, T ´s1 q Ks1´s2 hpu `T ´s1 , Lq " 0 KT ´s1 Ss1´s2 hpu, xq " 1 txăLu papLq ´apxqq apxq f x τ px, T ´sq Ss1´s2 hpu `T ´s1 , Lq " 0 (3.7)
so that, by induction, we get

P T hpu, xq " PT hpu, xq `ż T 0 ds 1 KT ´s1 hpu, xq `S T ´s1 P s1 hpu, xq ( " PT hpu, xq `ż T 0 ds 1 KT ´s1 hpu, xq `S T ´s1 Ps1 hpu, xq ( `ż T 0 ds 1 ż s1 0 ds 2 KT ´s1 Ks1´s2 P s2 hpu, xq `K T ´s1 Ss1´s2 P s2 hpu, xq (
`ż T 0 ds 1 ż s1 0 ds 2 ST ´s1 Ks1´s2 P s2 f px, uq `S T ´s1 Ss1´s2 P s2 hpu, xq (
" PT hpu, xq `N ÿ r"1 ż ∆rpT q ds r ST ´s1 Ss1´s2 ¨¨¨S sr´2´sr´1 Ksr´1´sr Psr hpu, xq `N ÿ r"1 ż ∆rpT q ds r ST ´s1 Ss1´s2 ¨¨¨S sr´1´sr Psr hpu, xq `ż∆ N `1pT q ds N `1 ST ´s1 Ss1´s2 ¨¨¨S s N ´1 ´sN Ks N ´sN`1 P s N `1 hpu, xq (
`ż∆ N `1pT q ds N `1 ST ´s1 Ss1´s2 ¨¨¨S s N ´sN`1 P s N `1 hpu, xq ( .
The idea now is to let N Ñ `8 in order to obtain an expansion of the semigroup P T as infinite series.

Using repeatedly (3.4) and (3.5) with Lemma 5.4 as well as the asymptotic of the Gamma function at infinity, one gets ˇˇˇˇż

∆ N `1pT q ds N `1 ST ´s1 Ss1´s2 ¨¨¨S s N ´1´s N Ks N ´sN`1 P s N `1 hpu, xq ˇˇˇď ż ∆ N `1pT q ds N `1 ż p´8,Ls N dz N 1 ?
T ´s1 gpcpT ´s1 q, z 1 ´xq ¨¨¨1 ? s N ´1 ´sN gpcps N ´1 ´sN q, z N ´zN´1 q ˆgpcps N ´sN`1 q, L ´zN q|hpu `T ´sN`1 , Lq|

ď |h| 8 C N `1 T ż ∆ N `1pT q ds N `1 1 ? T ´s1 ¨¨¨1 ? s N ´1 ´sN gpcpT ´sN`1 q, L ´xq ď |h| 8 C N `1 T ż ∆ N `1pT q ds N `1 1 ? T ´s1 ¨¨¨1 ? s N ´sN`1 gpcT, L ´xq " |h| 8 C N `1 T T pN `1q{2 pΓp1{2qq N `1 Γp1 `pN `1q{2q gpcT, L ´xq Ñ 0, as N Ñ `8.
Now we investigate the second remainder term. Similarly to the previous term, we get ˇˇˇˇż

∆ N `1pT q ds N `1 ST ´s1 Ss1´s2 ¨¨¨S s N ´sN`1 P s N `1 hpu, xq ˇˇˇď ż ∆ N `1pT q ds N `1 ż p´8,Ls N `1 dz N `1 1 ? T ´s1 gpcpT ´s1 q, z 1 ´xq ¨¨1 ? s N ´sN`1 gpcps N ´sN`1 q, z N `1 ´zN q ˇˇP s N `1 hpu `T ´sN`1 , z N `1q ˇď |h| 8 C N `1 T ż ∆ N `1pT q ds N `1 1 ? T ´s1 ¨¨¨1 ? s N ´sN`1 ď |h| 8 C N `1 T T pN `1q{2 pΓp1{2qq N `1 Γp1 `pN `1q{2q Ñ 0, as N Ñ `8.
Hence, the two series converge absolutely and uniformly for pt, u, xq P r0, T s ˆR`ˆR .

In order to state the forward parametrix expansion for the semigroup pP t q tě0 , we define for ps 0 , u, xq P r0, T s R`ˆp ´8, Ls and h P B b pR `ˆRq, the following family of operators

(3.8) I n s0 hpu, xq " # ş ∆nps0q ds n !´ś n´1 i"0 Ssi´si`1 ¯P sn hpu, xq `´ś n´2 i"0 Ssi´si`1 ¯K sn´1´sn hpu, xq ) if n ě 1, Ps0 hpu, xq if n " 0,
where we recall that we use the convention ś H " 1, and the operators St hpu, xq, Kt hpu, xq and Pt hpu, xq have been defined in (3.3), (3.2) and (3.1) respectively. As seen from the above discussion, we obtain the following expansion in infinite series of the Markov semigroup pP t q tě0 around p Pt q tě0 . The transition density and the probabilistic representation will be obtained from this result in the following sections.

Theorem 3.1. Let T ą 0. Assume that (H1) holds. Then, for every h P C 1,2 b pR `ˆp´8, Lsq, the series ř ně0 I n T hpu, xq converges absolutely and uniformly for pu, xq P R `ˆR and one has (3.9) P T hpu, xq "

ÿ ně0 I n T hpu, xq.
3.2. Existence of a transition density, its expansion and related properties.

In the previous section, we obtained an expansion in infinite series of the semigroup pP t q tě0 on smooth test functions. In this section, we retrieve from (3.9) the expansion of the transition density function. With the convention that t 0 " 0, z 0 " x, we introduce the following kernels

I K,n`1 px, tq :" ż ∆ n ptq dt n ż p´8,Ls n dz n ˜n´1 ź i"0
Sti`1´ti pz i , z i`1 q ¸K t´tn pz n , Lq, (3.10)

J K,n`1 px, tq :" ż ∆ n`1 ptq dt n`1 ż p´8,Ls n`1 dz n`1 ˜n ź i"0 Sti`1´ti pz i , z i`1 q ¸f zn`1 τ pz n`1 , t ´tn`1 q, (3.11) I D,n`1 s0 px, zq :" ż ∆n`1ps0q ds n`1 ż p´8,Ls n`1 dz n`1 n ź i"0 Ssi´si`1 pz i , z i`1 qq zn`1 sn`1 pz n`1 , zq (3.12)
where the terms St hpu, xq and Kt hpu, xq are given in (3.3), (3.2) and (3.1) respectively. As will become clear below, the two sequence of kernels pI K,n q ně1 and pJ K,n q ně1 are related to the exit time whereas pI D,n q ně1 correspond to the killed process. Using the change of variable t i " s 0 ´si , i " 0, ¨¨¨, n, (t 0 " 0 and z 0 " x) and Fubini's theorem, we write

I K,n`1 s0 hpu, xq :" ż ∆n`1ps0q ds n`1 ˜n´1 ź i"0 Ssi´si`1 ¸K sn´sn`1 hpu, xq " ż s0 0 ż s0 t1 ¨¨¨ż s0 tn dt n`1 ż p´8,Ls n dz n ˜n´1 ź i"0 Sti`1´ti pz i , z i`1 q ¸K tn`1´tn pz n , Lqhpu `tn`1 , Lq " ż s0 0 ż tn`1 0 ¨¨¨ż t2 0 dt n`1 ż p´8,Ls n dz n ˜n´1 ź i"0 Sti`1´ti pz i , z i`1 q ¸K tn`1´tn pz n , Lqhpu `tn`1 , Lq " ż s0 0 dt n`1 « ż ∆ n ptq dt n ż p´8,Ls n dz n ˜n´1 ź i"0 Sti`1´ti pz i , z i`1 q ¸K tn`1´tn pz n , Lq ff hpu `tn`1 , Lq " ż s0 0 I K,n`1 px, tqhpu `t, Lqdt.
Similarly, one gets

J K,n`1 s0 hpu, xq :" ż ∆n`1ps0q ds n`1 ˜n ź i"0 Ssi´si`1 ¸Ksn`1 hpu, xq " ż s0 0 ż s0 t1 ¨¨¨ż s0 tn`1 dt n`2 ż p´8,Ls n`1 dz n`1 ˜n ź i"0 Sti`1´ti pz i , z i`1 q ¸f zn`1 τ pz n`1 , t n`2 ´tn`1 qhpu `tn`2 , Lq " ż s0 0 dt n`2 « ż ∆ n`1 ptn`2q dt n`1 ż p´8,Ls n`1 dz n`1 ˜n ź i"0 Sti`1´ti pz i , z i`1 q ¸f zn`1 τ pz n`1 , t n`2 ´tn`1 q ff ˆhpu `tn`2 , Lq " ż s0 0 J K,n`1 px, tqhpu `t, Lqdt.
Inside the domain, one has

I D,n`1 s0 hpu, xq :" ż ∆n`1ps0q ds n`1 ˜n ź i"0 Ssi´si`1 ¸Ssn`1 hpu, xq " ż ∆n`1ps0q ds n`1 ż p´8,Ls n`1 dz n`1 n ź i"0 Ssi´si`1 pz i , z i`1 q « ż p´8,Ls hpu `s0 , z n`2 qq zn`1 sn`1 pz n`1 , z n`2 qdz n`2 ff " ż L ´8 dz n`2 « ż ∆n`1ps0q ds n`1 ż p´8,Ls n`1 dz n`1 n ź i"0 Ssi´si`1 pz i , z i`1 qq zn`1 sn`1 pz n`1 , z n`2 q ff hpu `s0 , z n`2 q " ż L ´8 I D,n`1 s0 px, zqhpu `s0 , zqdz.
Hence, we are naturally led to define the following kernels for ps 0 , x, zq P p0, T s ˆp´8, Ls 2

(3.13) p K,n px, s 0 q "

" I K,n px, s 0 q `JK,n px, s 0 q if n ě 1, f x τ px, s 0 q if n " 0, and
(3.14) p D,n s0 px, zq " " I D,n s0 px, zq if n ě 1, qx s0 px, zq if n " 0.
where the terms I K,n px, s 0 q, J K,n px, s 0 q and I D,n s0 px, zq are defined in (3.10), (3.11) and (3.12) respectively. As one of the main results of this section, we present the forward parametrix expansion of the transition density of the process pu `τ x t , X x τ x t q tě0 . Theorem 3.2. Let T ą 0. Assume that (H1) holds. For all pu, xq P R `ˆp´8, Ls, define the measure @pt, zq P pu, u `T s ˆp´8, Ls, p T pu, x, dt, dzq :" p K px, t ´uqδ L pdzqdt `pD T px, zqδ u`T pdtqdz with p K px, tq :"

ÿ ně0 p K,n px, tq and p D T px, zq " ÿ ně0 p D,n T px, zq.
Then, the series defining p K px, tq and p D T px, zq converge absolutely for px, t, zq P R ˆR˚ˆR and uniformly for px, t, zq P R ˆˆK T ˆR, where K T is any compact subset of p0, T s. Moreover, for h P B b pR `ˆRq, one has

(3.15) @pu, xq P R `ˆR, P T hpu, xq " hpu, xq1 txěLu `1txăLu ż u`T u ż L
´8 hpt, zqp T pu, x, dt, dzq and, for some positive C, c ą 1, for all pt, zq P p0, T s ˆp´8, Ls, the following Gaussian upper-bounds hold (3.16) p K px, tq ď Ct ´1{2 gpct, L ´xq and p D T px, zq ď CgpcT, z ´xq. Therefore, for all pu, xq P R `ˆp´8, Lq, p T pu, x, ., .q is the probability density function of the random vector pu `τ x T , X x τ x T q. More precisely, the first hitting time τ x T has a mixed type law. That is for t P ru, u `T q, τ x T has the density p K px, tq and at t " u `T , Ppu `τ x T " u `T q " ş L ´8 dz p D T px, zq. Similarly, the stopped process

X x τ x
T also has a mixed type law. That is, for z P p´8, Lq, X x τ x T has the density p D T px, zq and at z " L,

PpX x τ x T " Lq " ş u`T u dt p K px, t ´uq. Finally, px, zq Þ Ñ p D T px, zq (resp. px, tq Þ Ñ p K px, tq) is continuous on p´8, Ls 2 (resp. on p´8, Ls ˆp0, T s) and satisfies lim zÒL p D T px, zq " lim xÒL p D T px, zq " lim xÒL p K px, tq " 0.
Proof. We first remark that from (3.8) and (3.1), one has

I n T hpu, xq " ż ∆npT q ds n #˜n ´1 ź i"0 Ssi´si`1 ¸P sn hpu, xq `˜n´2 ź i"0 Ssi´si`1 ¸K sn´1´sn hpu, xq + " ż T 0 pI K,n px, tq `JK,n px, tqqhpu `t, Lqdt `ż L ´8 I D,n T px, zqhpu `T, zqdz " ż u`T u hpt, zqp K,n px, t ´uqδ L pzqdtdz `ż L ´8 hpt, zqp D,n
T px, zqδ u`T ptqdtdz.

for n ě 1. Moreover, for n ě 0, from the semigroup property and Lemma 5.4, one easily gets the following estimates

|I D,n`1 s0 px, zq| ď 2 ż ∆n`1ps0q ds n`1 ż p´8,Ls n`1 dz n`1 p n ź i"0 C ? s i ´si`1 gpcps i ´si`1 q, z i`1 ´zi qqgpcs n`1 , z ´zn`1 q ď 2C n`1 ż ∆n`1ps0q ds n`1 n ź i"0 1 ? s i ´si`1 gpcs 0 , z ´xq ď 2pCs 1 2 0 q n`1 pΓp 1 2 qq n`1 Γp1 `n`1 2 q
gpcs 0 , z ´xq.

Similar arguments yield

|I K,n`1 px, tq| ď C n`1 ż ∆ n ptq dt n ż ´s8,Ls n dz n p n´1 ź i"0 1 ? t i`1 ´ti gpcpt i`1 ´ti q, z i`1 ´zi qq ˆgpcpt ´tn q, L ´zn q " C n`1 t n`1 2 pΓp1{2qq n Γpn{2qp1 `n{2q
gpct, L ´xq and finally

|J K,n`1 px, tq| ď C n`2 ż ∆ n`1 ptq dt n`1 ż ´s8,Ls n`1 dz n`1 p n ź i"0 1 ? t i`1 ´ti gpcpt i`1 ´ti q, z i`1 ´zi qq ˆ1 ? t ´tn`1 gpcpt ´tn`1 q, L ´zn`1 q " C n`2 t ´1 2 `n`1 2 pΓp 1 2 qq n`2 Γp1 `n`2 2 q gpct, L ´xq.
From the asymptotics of the Gamma function at infinity, we deduce that both series ř ně0 p K,n px, tq and ř ně0 p D,n T px, zq converge absolutely and uniformly for pt, x, zq P R ˚ˆR 2 . From equation (3.9), we easily deduce (3.15) and the Gaussian upper-bound (3.16) follows from the preceding computations. Now, from Theorem 3.1 and the above discussion, for all h P C 1,2 b pR `ˆp´8, Lsq and all pu, xq P R `ˆp´8, Lq, one has

P T hpu, xq " ż u`T u ż L
´8 hpt, zqp T pu, x, dt, dzq so that p T pu, x, ., .q is the probability density function of the random vector pu `τ x T , X x τ x T q. As px, zq Þ Ñ p D,n T px, zq (resp. px, tq Þ Ñ p K,n px, t´uq) is continuous on p´8, Ls 2 (resp. on p´8, Lsˆpu, u`T s) and satisfies lim zÒL p D,n T px, zq " lim xÒL p D,n T px, zq " 0 (resp. lim xÒL p K,n px, t ´uq " 0), then px, zq Þ Ñ p D T px, zq (resp. px, tq Þ Ñ p K px, t ´uq) is also continuous and satisfies lim zÒL p D T px, zq " 0 (resp. lim xÒL p K px, t ´uq " 0). Now that we have obtained the parametrix expansion for the transition density, we will discuss its regularity properties.

Theorem 3.3 (Differentiability of the density). Following the notations introduced in Theorem 3.2, let T ą 0 be fixed and assume that (H1) holds. Then for any x P p´8, Ls and any α P r0, 1q, z Þ Ñ p D T px, zq P C 1`α pp´8, Lsq. In particular, one has

@z P p´8, Ls, B 2 p D T px, zq " ÿ ně0 B 2 p D,n T px, zq
with the following bound (3.17)

|B 2 p D T px, zq| ď C T 1{2
gpcT, z ´xq. Moreover, for any α P r0, 1q, for any pz, z 1 q P p´8, Ls 2 , one has

(3.18) |B 2 p D T px, zq ´B2 p D T px, z 1 q| ď C|z ´z1 | α T 1´γ 2
`gpcT, z ´xq `gpcT, z 1 ´xq with γ " 1 ´α. Finally, for all η P r0, 1{2q, for all pu, xq P R `ˆp´8, Ls, t Þ Ñ p K px, t ´uq is η-Hölder continuous on pu, u `T s. In particular, for all pt, t 1 q P pu, u `T s, one has

|p K px, t ´uq ´pK px, t 1 ´uq| ď C|t ´t1 | η ˆ1 pt ´uq 1 2 `η gpcpt ´uq, L ´xq `1 pt 1 ´uq 1 2 `η gpcpt 1 ´uq, L ´xq ˙.
Proof. We first remark that by Fubini's theorem and the change of variable t i " T ´si , one has

p D,n T px, zq " I D,n T px, zq " ż ∆ n pT q dt n ż p´8,Ls n dz n n´1 ź i"0 Sti`1´ti pz i , z i`1 qq zn T ´tn pz n , zq, n ě 1.
Denote by Ψ s px, zq the solution to the Volterra integral equation

Ψ s px, zq " Ss px, zq `ż s 0 ż L ´8 Ψ t1 px, z 1 q Ss´t1 pz 1 , zqdz 1 dt 1 .
From estimate (3.5), we see that the kernel Ss´t1 pz 1 , zq leads to an integrable singularity (in time) in the above space time integral so that the solution exists and is given by the (uniform) convergent series

Ψ s px, zq " Ss px, zq `ÿ ně1 ż ∆npsq ds n ż p´8,Ls n dz n n ź i"1
Ssi´si`1 pz i , z i`1 q Ss´s1 pz, z 1 q with the convention z n`1 " x, s n`1 " 0. Furthermore, the inequality (3.19) @ps, x, zq P p0, T s ˆp´8, Ls 2 , |Ψ s px, zq| ď C s 1{2 gpcs, z ´xq is easily obtained. Moreover, plugging this expansion in the following equality, we observe that @px, zq P p´8, Ls 2 , p D T px, zq " qx T px, zq

`ż T 0 ż L ´8 Ψ t1 px, z 1 qq z1 T ´t1 pz 1 , zqdt 1 dz 1 .
From the Lebesgue differentiation theorem, we get

B 2 p D T px, zq " B 2 qx T px, zq `ż T 0 ż L ´8 Ψ t1 px, z 1 qB 2 qz1 T ´t1 pz 1 , zqdt 1 dz 1
and estimate (3.17) follows from (3.19) and |B 2 qz1 T ´t1 pz 1 , zq| ď CpT ´t1 q ´1{2 gpcpT ´t1 q, z ´z1 q. It remains to prove (3.18). First, let us assume that |z 1 ´z| 2 ă T ´t1 . Using the mean value theorem, the bound |B 2 2 qz1 T ´t1 pz 1 , zq| ď CpT ´t1 q ´1gpcpT ´t1 q, z ´z1 q and noting that for any point ζ in the interval pz, z 1 q, one has

exp ˆ´|ζ ´z1 | 2 cpT ´t1 q ˙ď C " exp ˆ´|z 1 ´z1 | 2 cpT ´t1 q ˙`exp ˆ´|z ´z1 | 2 cpT ´t1 q ˙* we get |B 2 qz1 T ´t1 pz 1 , zq ´B2 qz1 T ´t1 pz 1 , z 1 q| ď C|z 1 ´z| T ´t1 " exp ˆ´|z 1 ´z1 | 2 cpT ´t1 q ˙`exp ˆ´|z ´z1 | 2 cpT ´t1 q ˙* ď C|z 1 ´z| α pT ´t1 q 1´γ{2 " exp ˆ´|z 1 ´z1 | 2 cpT ´t1 q ˙`exp ˆ´|z ´z1 | 2 cpT ´t1 q ˙*
for |z 1 ´z| 2 ă T ´t1 . Otherwise, one gets

|B 2 qz1 T ´t1 pz 1 , zq| ď CpT ´t1 q α 2
pT ´t1 q 1´γ 2 gpcpT ´t1 q, z ´z1 q ď C|z ´z1 | α pT ´t1 q 1´γ 2 gpcpT ´t1 q, z ´z1 q and similarly,

|B 2 qz1 T ´t1 pz 1 , z 1 q| ď CpT ´t1 q α 2
pT ´t1 q 1´γ 2 gpcpT ´t1 q, z 1 ´z1 q ď C|z ´z1 | α pT ´t1 q 1´γ 2 gpcpT ´t1 q, z 1 ´z1 q when |z 1 ´z| 2 ě T ´t1 . Combining these estimates, (3.19) and the equality

B 2 p D T px, zq ´B2 p D T px, z 1 q " B 2 qx T px, zq ´B2 qx T px, z 1 q `ż T 0 ż L ´8 Ψ t1 px, z 1 qpB 2 qz1 T ´t1 pz 1 , zq ´B2 qz1 T ´t1 pz 1 , z 1 qqdt 1 dz 1
we obtain (3.18). We now prove the second part of the theorem. We first remark that for 0 ă t ď T p K px, tq " f x τ px, tq `ÿ ně0 pI K,n`1 `JK,n`1 qpx, tq

" f x τ px, tq `K t px, Lq `ÿ ně0 ż ∆ n`1 ptq dt n`1 ż p´8,Ls n`1 dz n`1 n ź i"0 Sti`1´ti pz i , z i`1 q ˆ Kt´tn`1 pz n`1 , Lq `f zn`1 τ pz n`1 , t ´tn`1 q ( " f x τ px, tq `K t px, Lq `ż t 0 ż L ´8 Ψ s px, z 1 q Kt´s pz 1 , Lq `f z1 τ pz 1 , t ´sq ( dsdz 1 . (3.20)
Let pt, t 1 q Ps0, T s and 0 ă η ă 1{2. We first prove the following bound

(3.21) |f x τ px, t 1 q ´f x τ px, tq| ď C|t 1 ´t| η ˆ1 t 1p1`2ηq{2 gpct 1 , L ´xq `1 t p1`2ηq{2 gpct, L ´xq ˙.
Assume first that |t 1 ´t| ă pL ´xq 2 . By Lemma 5.2 and the mean value theorem, one gets

|f x τ px, t 1 q ´f x τ px, tq| ď C ż 1 0 |t 1 ´t| |λt `p1 ´λqt 1 | 3{2 gpcpλt `p1 ´λqt 1 q, L ´xqdλ ď C|t ´t1 | η ż 1 0 1 |λt `p1 ´λqt 1 | p1`2ηq{2 gpcpλt `p1 ´λqt 1 q, L ´xqdλ.
Now noting that for any point t P pt, t 1 q which satisfies |t 1 ´t| ď pL ´xq 2 , we deduce the inequality ˆ1 t 1p1`2ηq{2 gpct 1 , L ´xq `1 t p1`2ηq{2 gpct, L ´xq ḟor all pt, t 1 q Ps0, T s and 0 ă η ă 1{2. Let 0 ă t ď t 1 ď T and x P p´8, Ls. From (3.20), we now write

1 t1`η expp´p L ´xq 2 c t q ď C ˆ1 t 1 1`η expp´p L ´xq 2 c 1 t 1 q `1 t 1`η expp´p L ´xq 2 c 1 t q ḟor some constants C, c 1 ą 1,
p K px, t 1 q ´pK px, tq " `f x τ px, t 1 q ´f x τ px, tq ˘`p Kt 1 px, Lq ´K t px, Lqq `ż t 1 t ż p´8,Ls Ψ s px, z 1 q Kt 1 ´spz 1 , Lq `f z1 τ pz 1 , t 1 ´sq ( dsdz 1 `ż t 0 ż p´8,Ls Ψ s px, z 1 q Kt 1 ´spz 1 , Lq ´K t´s pz 1 , Lq `f z1 τ pz 1 , t 1 ´sq ´f z1 τ pz 1 , t ´sq ( dsdz 1
and bound the first two terms of the above equality using (3.21) and (3.22). From (3.19), (3.4), (3.5), Lemma 5.2 and the semigroup property, we obtain

ˇˇˇˇż t 1 t ż p´8,Ls Ψ s px, z 1 q Kt 1 ´spz 1 , Lq `f z1 τ pz 1 , t 1 ´sq ( dsdz 1 ˇˇˇˇď C ż t 1 t ˆ1 s 1{2 `1 s 1{2 1 pt 1 ´sq 1{2 ˙gpct 1 , L ´xqds ď C pt 1 ´tq η t 1η gpct 1 , L ´xq.
Similarly, from Lemma 5.2, (3.21) and (3.22), we also get ˇˇˇˇż

t 0 ż p´8,Ls Ψ s px, z 1 q Kt 1 ´spz 1 , Lq ´K t´s pz 1 , Lq `f z1 τ pz 1 , t 1 ´sq ´f z1 τ pz 1 , t ´sq ( dsdz 1 ˇˇˇď Cppt 1 ´tq 1 2 `η `pt 1 ´tq η q ż t 0 ˆ1 s 1 2 1 pt 1 ´sq 1 2 `η gpct 1 , L ´xq `1 s 1 2 1 pt ´sq 1 2 `η gpct, L ´xq ˙ds ď C T pt 1 ´tq η ˆ1 t 1η gpct 1 , L ´xq `1 t η gpct, L ´xq
ḟor some positive constant C T (non-decreasing with respect to T ). This completes the proof.

Remark 3.4. In order to investigate the differentiability of t Þ Ñ p K px, tq, one is naturally led to differentiate the representation formula (3.20) with respect to t. The first two terms appearing in the right-hand side of this formula can be readily differentiated. The difficulty comes when one tries to differentiate the time-space convolution with respect to t. Actually the singularity in time appearing in the density f z τ pz, t ´sq prevents us to do so unless additional smoothness assumptions on the coefficients b and σ are provided. This phenomenon does not appear in the standard diffusion framework because the density f z τ pz, t ´sq is replaced by a Gaussian density. 3.3. Applications.

In this section we collect some applications of the results established in Theorem 3.2 and Theorem 3.3. From the Gaussian upper bounds satisfied by p K px, tq, p D T px, zq and their derivatives with respect to x, we claim: Corollary 3.2. Let T ą 0 and x P p´8, Lq be fixed. Then, the following bound

|Erhpτ x T , X x τ x T qs| ď ż T 0 ds|hps, Lq| 1 ? s gpcs, x ´Lq `ż L ´8 dz |hpT, zq|gpcT,
x ´zq is valid for any Borel function function h defined on R `ˆp´8, Ls as soon as the above integrals are finite. Moreover, if h P C 1 pp´8, Lsq, the following bound is valid

|ErB 2 hpτ x T , X x τ x T qs| ď ż T 0 ds|B 2 hps, Lq| 1 s gpcs, x ´Lq `ż L ´8 dz |hpT, zq| 1 ? T gpcT, x ´zq,
as soon as the above integrals are finite.

The above bounds may be useful since combined with Theorem 3.2 and Theorem 3.3 they allow to establish the continuity of the maps

x Þ Ñ Erhpτ x T , X x τ x T qs and x Þ Ñ ErB 2 hpτ x T , X x τ x
T qs on p´8, Lq for a large class of test function. We omit its proof.

We now aim at giving a probabilistic representation of the transition density of the process pu `τ x t , X x τ x t q tě0 and pP t hpu, xqq tě0 that may be useful for unbiased Monte Carlo path simulation or probabilistic infinite dimensional analysis. First, for z P p´8, Lq, we write T q θT px, Xx T qs. From (3.5), we note that Er| θt px, Xx t q|s ď C T t ´1{2 for t P p0, T s, which in particular implies that θt px, X x t q P L 1 pPq. We also remark that for 0 ď s 1 ď s 2 ď T , one has

Ss1 Ss2´s1 hpu, xq " Erhpu `s2 , Xs1, Xx s 1 s2 qθ s2´s1 p Xx s1 , Xs1, Xx s 1 s2 qθ s1 px, Xx s1 q1 tτ s 1 , Xx s 1 ěs2´s1u 1 tτ x ěs1u s " Erhpu `s2 , Xs1, Xx s 1 s2 q θs2´s1 p Xx s1 , Xs1, Xx s 1 s2
q θs1 px, Xx s1 qs. In order to extend this probabilistic representation to the semigroup expansion obtained in (3.9), we first apply Fubini's theorem and the change of variable t i " T ´si , i " 0, ¨¨¨, n, in order to write

ż ∆npT q ds n ˜n´1 ź i"0 Ssi´si`1 ¸P sn hpu, xq " ż ∆ n pT q dt n ˜n´1 ź i"0 Sti`1´ti ¸P T ´tn hpu, xq.
For a given time partition π : 0 " t 0 ă t 1 ă ¨¨¨ă t N ă t N `1 " T , we introduce the Euler scheme Xπ " p Xπ ti q 0ďiďN `1 with the following dynamics Xπ ti`1 " Xπ ti `σp Xπ ti qpW ti`1 ´Wti q Xπ t0 " x which in turn allows us to write the following (partial) probabilistic representation

˜n´1 ź i"0 Sti`1´ti ¸P T ´tn hpu, xq " Erhpu `T,
ds n ˜n´2 ź i"0 Ssi´si`1 ¸K sn´1´sn hpu, xq " ż ∆n´1pT q ds n´1 ˜n´2 ź i"0 Ssi´si`1 ¸q K sn´1 hpu, xq " ż ∆ n´1 pT q dt n´1 ˜n´2 ź i"0 Sti`1´ti ¸q K T ´tn´1 hpu, xq.
Similarly to the previous term, one gets ˜n´2 ź

i"0 Sti`1´ti ¸q K T ´tn´1 hpu, xq " Erhpu `tn´1 `τ tn´1, Xπ t n´1 , Lq1 tτ t n´1 , Xπ t n´1 ďT ´tn´1u
papLq ´ap Xπ tn´1 qq ap Xπ tn´1 q ˆθ tn´1´tn´2 p Xπ tn´2 , Xπ tn´1 q ¨¨¨θ t1 px, Xπ t1 qs. Now, in order to give a probabilistic representation of the time integral, we let pN ptqq tě0 be a simple Poisson process with intensity parameter λ ą 0 and define N " N pT q. Let ζ 1 ă ζ 2 ă ¨¨¨ă ζ N be the jump times of the Poisson process and set ζ 0 " 0, ζ N `1 " T . We know that conditional on N , the event times are distributed as the uniform order statistics associated to a sequence of i.i.d. uniform r0, T s-valued random variables satisfying PpN " n, ζ 1 P dt 1 , ¨¨¨, ζ n P dt n q " λ n e ´λT dt n , on the set ∆ npT q " tt n P r0, T s n : 0 ă t 1 ă t 2 ă ¨¨¨ă t n ă T u.

We still denote by π the random time partition π : ζ 0 " 0 ă ζ 1 ă ¨¨¨ă ζ N `1 " T and denote by Xπ " p Xπ ζi q 0ďiďN `1 its associated Euler scheme. As a consequence, we may rewrite the time integral appearing in the above expressions in a probabilistic way as follows for n ě 0, where we use the convention ś H " 1. Given the above discussion, we obtain the final result of this section. Theorem 3.5. Let T ą 0 and assume that (H1) holds. Define the two sequences pΓ N pxqq N ě0 and p ΓN pxqq N ě0 as follows

ż ∆npT q ds n ˜n´1 ź i"0 Ssi´si`
Γ N pxq " " ś N ´1 j"0 λ ´1 θζj`1´ζj p Xπ ζj , Xπ ζj`1 q if N ě 1, 1 if N " 0, and ΓN pxq " $ & % papLq´ap Xπ ζ N ´1 qq ap Xπ ζ N ´1 q ś N ´2 j"0 λ ´1 θζj`1´ζj p Xπ ζj , Xπ ζj`1 q if N ě 1, 0 if N " 0.
Then, for all h P B b pR `ˆRq, for all pu, xq P R `ˆR, the following probabilistic representation holds

Erhpτ x T , X x τ x T qs " e λT E « hppζ N pT q `τ ζ N pT q , Xπ ζ N pT q q ^T, Xπ pζ N pT q `τ ζ N pT q , Xπ ζ N pT q q^T qΓ N pT q pxq ff `eλT E " hpζ N pT q´1 `τ ζ N pT q´1 , Xπ ζ N pT q´1 , Lq1 tτ ζ N pT q´1 , Xπ ζ N pT q´1 ďT ´ζNpT q´1 u
ΓNpT q pxq  .

Similarly, the following probabilistic representation for the density is satisfied @pt, x, zq P p0, T s ˆp´8, Ls 2 , p T p0, x, dt, dzq " δ T pdtqp D T px, zq `δL pdzqp K px, tq with, for all pt, zq P p0, T s ˆp´8, Ls,

p D T px, zq " e λT E " q Xπ ζ N pT q T ´ζNpT q p Xπ ζ N pT q , zqΓ N pT q pxq  , p K px, tq " e λT E " f Xπ ζ N pT q τ p Xπ ζ N pT q , t ´ζNpT q q1 ttěζ N pT q u Γ N pT q pxq `f Xπ ζ N pT q´1 τ p Xπ ζ N pT q´1 , t ´ζNpT q´1 q1 ttěζ N pT q´1 u ΓNpT q pxq  .
Remark 3.6. We observe that the probabilistic representation of P T hp0, xq has a natural interpretation. The first term can be decomposed into two expectations. The first one involves paths of the Euler scheme Xπ that do not exit the domain p´8, Lq (note that Λ t is a factor in the definition of θt so that θt px, zq " 0 for px, zq R p´8, Lq) on the interval r0, T s whereas the second term involves paths of the Euler scheme that exit the domain on the last time interval of the Poisson process rζ N pT q , T s by sampling according to the law of the exit time τ ζ N pT q , Xπ ζ N pT q on the last interval. The last term appearing in the probabilistic representation is an additional correction term which is due to the very nature of the forward parametrix method and comes from the integration by parts formula used in the proof of Proposition 3.1. It also involves paths of the Euler scheme that exit the domain on the last time interval rζ N pT q´1 , T s.

Remark 3.7. An unbiased Monte Carlo method for evaluating P T hp0, xq or p T p0, x, dt, dzq stems from the probabilistic representations obtained in Theorem 3.5. The explosion of the variance may be an important issue that can induce poor convergence rate of the method as pointed out in [START_REF] Anderson | Unbiased simulation of stochastic differential equations using parametrix expansions[END_REF] for unbiased simulation of multi-dimensional diffusions. In these situations, an importance sampling method on the time steps using a Beta or Gamma distribution may be used. In short, it would seem that this approximation will work well in the case of small parameters. Although a very close analysis could be carried here, we do not intend to develop importance sampling schemes and refer the interested reader to [START_REF] Anderson | Unbiased simulation of stochastic differential equations using parametrix expansions[END_REF] for some developments in the diffusion case. From the above probabilistic representation, one may also infer the possibility of infinite-dimensional analysis based on the analysis of the corresponding approximation or the possibility of density expansions with respect to a small parameter as investigated in [START_REF] Frikha | A parametrix approach for asymptotic expansion of markov semigroups with applications to multi-dimensional diffusion processes[END_REF]. These issues will be developed in a future work.

We conclude this section by one simple corollary that provides a kind of integration by parts formula for the killed process.

Corollary 3.3. Let T ą 0 and assume that (H1) holds. Let h P C 1 pp´8, Lsq satisfying: there exist C, c ą 0, such that for all z P p´8, Ls, |hpzq| `|h 1 pzq| ď C exppc|z|q. Then, for all x P p´8, Lq, one has

Erh 1 pX x T q1 tτ x ąT u s " ´eλT E " hp Xπ T qΛ T ´ζNpT q p Xπ ζ N pT q , Xπ T qµ 1 T ´ζNpT q p Xπ ζ N pT q , Xπ T qΓ N pT q pxq ı .
Proof. Combining theorems 3.2 and 3.3 with an integration by parts formula yield

Erh 1 pX x T q1 tτ x ąT u s " ż L ´8 h 1 pzqp D T px, zqdz " ´ż L ´8 hpzqB 2 p D T px, zqdz,
where we used the fact that lim zÒL p D T px, zq " 0. From Theorem 3.5 and Lebesgue differentiation theorem, one obtains the following probabilistic representation formula

B 2 p D T px, zq " e λT E " B 2 q Xπ ζ N pT q T ´ζNpT q p Xπ ζ N pT q , zqΓ N pT q pxq  " e λT E " q Xπ ζ N pT q
T ´ζNpT q p Xπ ζ N pT q , zqµ 1 T ´ζNpT q p Xπ ζ N pT q , zqΓ N pT q pxq  which with the previous computation readily concludes the proof.

Backward parametrix expansion

In this section we apply the backward parametrix expansion using a semigroup approach in order to study the law of pu`τ x t , X x τ x t q with respect to x under Hölder continuity assumptions on the coefficients. Through this section, we will make the following assumptions on the coefficients b and σ: Assumption (H2).

(i) σ : R ÝÑ R is bounded on R and a " σ 2 is uniformly elliptic. That is there exist a, a ą 0 such that for any x P R, a ď apxq ď a. In the forward case the kernel K is never differentiated because of the cancelling property (3.7). In the backward setting this is not the case. The differentiation with respect to the time variable of the kernel associated to the density f τ gives a degeneration which does not appear in the usual case.

Therefore, we first introduce a regularizing parameter r ą 0 in Lemma 4.1 and 4.2 to avoid the singularity in time when deriving the first order expansion of the semigroup associated with the process pτ x t , X x τt q with respect to the parametrix process, whose coefficients are frozen at some point y P p´8, Ls. The strategy to deal with the time singularity is to take the limit as r goes to zero, using the boundary conditions on the approximation processes as given in Lemma 3.1 and by choosing h from an appropriate class of test functions, we show that the limits are well defined and the first order backward parametrix expansion is achieved in Lemma 4.3.

To avoid confusion, we point out that in the rest of the paper, the support of a function f : X Ñ R refers to the subset of its domain X, for which the function f is non-zero and we do not take the topological closure, although X is often a subset of a topological space.

Lemma 4.1. Let y P p´8, Ls and r ą 0 with apyq ą 0. Suppose that either h P C 2,0 b pR `ˆp´8, Lsq and suppphq Ď R `ˆp´8, Lq is satisfied or h P C By dominated convergence theorem, we deduce that pt, u, xq Þ Ñ P y T ´t`r hpu, xq is jointly continuous on r0, T s ˆR`p ´8, Lq and that the left limit as x Ò L is given by hpu, Lq for any pt, uq P r0, T s ˆR`. Similar arguments show that u Þ Ñ P y T ´t`r hpu, xq is continuously differentiable on R `, for pt, xq P r0, T s ˆp´8, Lq and that the left-limit as x Ò L is equal to B 1 hpu, Lq. Moreover, each term appearing in the right-hand side of the above equality is bounded uniformly on r0, T s ˆR`ˆp ´8, Ls.

Similarly, by dominated convergence theorem and integration by parts, one has for x P p´8, Lq where we used the relation B x f y τ px, sq " 2B s gpapyqs, L ´xq (see Lemma 3.1). Moreover, the two terms appearing in the right-hand side of the last equality are continuous and uniformly bounded on r0, T s ˆR`ˆp ´8, Ls when seen as functions of pt, u, xq. Similarly, for the second derivatives w.r. We consider now the function pt, u, xq Þ Ñ ş p´8,Lq hpu `T ´t `r, zqq y T ´t`r px, zqdz, which is the integral against the difference of two Gaussian densities. By standard arguments for the Gaussian densities and the fact that h P C 2,0 b pR `ˆp´8, Lsq, we can show that the first partial derivatives in u and t, and the first and second partial derivative in x can be taken under the integral and are continuous on r0, T s ˆR`ˆp ´8, Lq with finite left limit at L and uniformly bounded for pt, u, xq P r0, T s ˆR`ˆp ´8, Ls. We omit the remaining technical details.

From the proof we see that x Þ Ñ P y T ´t`r hpu, xq is continuous, but not differentiable at L. However, since lim xÒL B r x P y T ´t`r hpu, xq is finite for r " 1, 2, we can set the left derivatives of P y T ´t`r hpu, xq with respect to x at L to be equal to their respective left limits. We then work with this modification of the function P y T ´t`r hpu, xq, which belongs to C 1,1,2 b pr0, T s ˆR`ˆp ´8, Lsq.

For the next result, we introduce the two following kernels ). We present first some auxiliary estimates and results on the above kernels and integral operators which can be useful later in proving the convergence of the backward parametrix expansion. Under (H2), by using Lemma 5.2 and Hölder continuity of a " σ 2 , we have for any β P r0, 1s and any pt, x, zq P p0, T s ˆp´8, Ls 2

| Ŝz t px, zq| ď C ˆ1 t 1´η 2 ^|L ´z| β t 2`β´η 2
˙gpct, x ´zq, (4.12)

| KL t px, sq| ď C 1 s 3´η 2
gpcs, L ´xq1 tsătu . (4.13)

The exponent β is appropriately chosen later on in Theorem 4.2, so that the asymptotic expansion of the transition density of pu `τ x t , X x τ x t q converges. For h P C 2,0 b pR `ˆRq, the estimate (4.14) below is obtained directly from (4.12)

| Ŝt hpu, xq| ď C T |h| 8 1 t 1´η 2 , (4.14) | Kt hpu, xq| ď C T p|h| 8 , |B 1 h| 8 q 1 t 1´η 2
, (4.15) while (4.15) can be obtained by applying Lemma 5.2 and the inequality gps, x ´yq ď C ? s to (4.6). Moreover, by combining (4.6), (4.12), (4.14) and (4.15), we see that if h P C 8,0 b pR `ˆRq then Ŝt h and Kt h belongs to C 8,0 b pR `ˆp´8, Lsq and their support are contained in R `ˆp´8, Lq. We point out to the reader that in order to obtain a convergent expansion of the semigroup pP t q tě0 , the above mentioned support property of Ŝt h and Kt h or more specifically the fact that Ŝt hpu, Lq " Kt hpu, Lq " 0 is crucial, and the non-zero left limit Kt hpu, L´q does not play a role. For any fixed t ą 0, it is clear from (4.12) and dominated convergence theorem that lim rÓ0 Ŝt`r hpu, xq " Ŝt hpu, xq. From (4.6) and the fact that h P C 2,0 b pR `ˆRq, we have lim rÓ0 Kt`r hpu, xq " Kt hpu, xq. Finally, by applying Lemma 5.2, equation (4.2) and dominated convergence theorem we have lim rÓ0 S t`r hpu, xq " S t hpu, xq and lim rÓ0 K t`r hpu, xq " K t hpu, xq.

We are now in position to prove the first order expansion of the semigroup pP t q tě0 . Notice that we have proved two different expansions given in (4.16) and (4.17) respectively. The difference in the two expansions is due to the assumption on the support of h, and the reason that KT ´sh term does not appear in (4.17) is precisely due to the fact that hpu, Lq " 0. Lemma 4.3. Assume that (H2) holds and that b is continuous on p´8, Ls. For h P C 2,0 b pR `ˆRq, the following first order expansion for the semigroup pP t q tě0 holds (4. Proof. The result is straightforward for x ě L, so from now on we assume that x ă L. We will do the proof for the first case only. For y P R, we apply Lemma 4.2 to the function pu, xq Þ Ñ gpε, y ´xqhpu, xq P C 2,0 b pR `ˆRq and integrate both hand sides of (4.8) with respect to dy. The goal now is to prove that we can take the limit as ε Ó 0 first and then r Ó 0. To do this, each term in the expansion given in Lemma 4.2 is analyzed. We first remark that for x P p´8, Lq one has ż R dy ż P T pu, x, du 1 , dx 1 q P y r pgpε, that the order of integration in the above integral can be freely interchanged using Fubini's theorem.

y ´¨qhqpu 1 , x 1 q " ż R dy ż P T pu, x, du 1 , dx 1 q ! ż L ´8 dz gpε,
To take the limit as ε Ó 0, we see that by using Lemma 5.2, (4.18) and ş R dy gpε, y ´zqhpu 1 `r, zq ď |h| 8 to obtain ż R dy gpε, y ´zq|hpu 1 `r, zq|q y r px 1 , zq ď C|h| 8 gpār, z ´x1 q ż R dy gpε, L ´yq|hpu 1 `s, Lq|f y τ px 1 , sq ď C|h| 8 B x gpās, L ´x1 q, which are ş P T pu, x, du 1 , dx 1 q ş L ´8 dz and ş P T pu, x, du 1 , dx 1 q ş r 0 ds integrable respectively. Therefore, by dominated convergence theorem, this shows that

ż P T pu, x, du 1 , dx 1 q ż L ´8 dz lim εÓ0 ż R
dy gpε, y ´zqhpu 1 `r, zqq y r px 1 , zq "

ż P T pu, x, du 1 , dx 1 q ż L ´8 dz hpu 1 `r, zqq z r px 1 , zq ż P T pu, x, du 1 , dx 1 q ż r 0 ds lim εÓ0 ż R
dy gpε, L ´yqhpu 1 `s, Lqf y τ px 1 , sq " ż P T pu, x, du 1 , dx 1 q ż r 0 dshpu 1 `s, Lqf L τ px 1 , sq.

To take the limit as r Ó 0, we apply dominated convergence theorem by noticing that both inner integrals on the right hand of the above expressions are bounded by |h| 8 . Then by Lemma 5. To compute the right hand side of (4.8), we note that the strategy is also to first apply Fubini's theorem and then dominated convergence theorem. By using again Lemma 5.2 to estimate B 2 x qy T ´s`r px, zq and B x qy T ´s`r px, zq, the term

ż R dy ż T 0 ds ż P s pu, x, du 1 , dx 1 q Ŝy T ´s`r hpu 1 , x 1 q " ż R dy ż T 0 ds ż P s pu, x, du 1 , dx 1 q ż L ´8 dz hpu 1 `T ´s `r, zq ˆgpε, z ´yq " 1 2 rapx 1 q ´apyqsB 2 x 1 qy T ´s`r px 1 , zq `bpx 1 qB x 1 qy T ´s`r px 1 , zq * 1 tx 1 ăLu
is absolutely integrable and one can apply the Fubini's theorem to interchange the order of integration. By using Lemma 5.2 we see that

ż L ´8 dz ż R dy gpε, z ´yqhpu 1 `T ´s `r, zq " 1 2 rapx 1 q ´apyqsB 2 x 1 qy T ´s`r px 1 , zq `bpxqB x 1 qy T ´s`r px 1 , zq * ď |h| 8 T ´s `r
which is independent of ε and ş T 0 ds ş P s pu, x, du 1 , dx 1 q integrable. Therefore, one can take the limit in ε by dominated convergence theorem and it is sufficient to compute

lim εÓ0 ż L ´8 dz ż R dy gpε, z ´yqhpu 1 `T ´s `r, zq " 1 2 rapx 1 q ´apyqsB 2
x 1 qy T ´s`r px 1 , zq `bpx 1 qB x 1 qy T ´s`r px 1 , zq

* 1 tx 1 ăLu " ż L ´8 dzhpu 1 `T ´s `r, zq " 1 2 rapx 1 q ´apzqsB 2 x 1 qz T ´s`r px 1 , zq `bpx 1 qB x 1 qz T ´s`r px 1 , zq * 1 tx 1 ăLu
" ŜT ´s`r hpu 1 , x 1 q.

To take the limit as r Ó 0, we see that by (4.14), | ŜT ´s`r hpu 1 , x 1 q| ď C T |h| 8

1 pT ´sq 1´η 2
, which is is independent of r and ş T 0 ds ş P s pu, x, du 1 , dx 1 q integrable. The arguments to prove that the limit as ε Ó 0 for the term associated with Ky pgpε, y ´¨qhq in Lemma 4.2 are more involved. In fact, in order to apply Fubini's theorem and take the limit as ε Ó 0, we need to apply inequality (4.7) and Lemma 5.2,

| Ky

T ´s`r pgpε, y ´¨qhqpu 1 , x 1 q| " Our aim now is to iterate the first order expansion formula (4.16) and (4.17) in order to obtain an expansion in infinite series of the Markov semigroup pP t q tě0 in the spirit of Theorem 3.1. For h P C 8,0 b pR `ˆRq, we recall that the terms S t hpu, xq, K t hpu, xq, Ŝt hpu, xq and Kt hpu, xq are given in (4.10) and (4.11), and we set

ż T ´s`r 0 dv gpε, L ´yq|hpu 1 `v, Lq| ˇˇ1 2 rapx 1 q ´apyqsB 2 x 1 f y τ px 1 , vq `bpx 1 qB x 1 f y τ px 1 ,
I n T hpu, xq :" # ş ∆npT q ds n S sn Ŝsn´1´sn . . . Ŝs1´s2 p ŜT ´s1 `K T ´s1 qhpu, xq n ě 1, pS T `KT qhpu, xq n " 0.
We point out that the operator K only appears once in the above, because to study the transition density functions, one must take test functions h with domain R `ˆR or in particular, test functions which belong to C 8,0 b pR `ˆRq. It is only after the first iteration, we notice that Ŝt h and Kt h belongs to C 8,0 b pR `ˆp´8, Lsq and their support are contained in R `ˆp´8, Lq, and (4.17) is used to obtain the expansion after the first iteration. We present in the following, one of the main results of this section. Proof. We know that for all h P C 8,0 b pR `ˆRq and t P p0, T s, Ŝt h and Kt h belongs to C 8,0 b pR `ˆp´8, Lsq and has support contained in R `ˆp´8, Lq (see the discussion after (4.14)). Therefore, by replacing h by ŜT ´th `K T ´th in (4.16) of Lemma 4.3 and iterating using (4.17), we obtain

P T h " pS T `KT qh `N´1 ÿ n"1 ż ∆npT q ds n S sn Ŝsn´1´sn . . . Ŝs1´s2 p ŜT ´s1 `K T ´s1 qh `RN T h
where the remainder term is given by

R N T hpu, xq :" ż ∆ N pT q ds n P s N Ŝs N ´1´s N . . . Ŝs1´s2 p ŜT ´s1 `K T ´s1 qhpu, xq.
We first show that the remainder term converges to zero as N Ñ 8. From estimates (4.14) and (4.15), for any pu, xq P R `ˆR, the remainder term is bounded by

|R N T hpu, xq| ď C T p|hp., Lq| 8 , |B 1 hp., Lq| 8 q ż ∆ N pT q ds N N ´1 ź n"0 C T ps n ´sn`1 q ´p1´η 2 q " C T p|hp., Lq| 8 , |B 1 hp., Lq| 8 qC N T T N η{2 Γpη{2q N Γp1 `N η{2q ,
where we used Lemma 5.4 with b " 0, a " 1 ´η{2 and t 0 " T for the last equality. Hence, from the asymptotics of the Gamma function at infinity, we clearly see that the remainder goes to zero uniformly in pu, xq P R `ˆR as n Ò 8. Similar estimates also give the absolute and uniform convergence of the infinite sum.

Existence of a transition density, its expansion and related properties.

In this section, we retrieve from Theorem 4.1 the existence and an expansion of the transition density function. In order to do this, one needs estimates on the series obtained in the previous theorem which do not involve the regularity of h. In particular, we have used |B 1 hp¨, Lq| ă 8 in the previous proof (see also Section 3.2 in the forward case). Therefore we start by an examination of the n-th term of the series expansion in Theorem 4.1 related to the killed diffusion process:

S sn Ŝsn´1´sn . . . Ŝs1´s2 ŜT ´s1 hpu, xq " ż L ´8 dz 0 hpu `T, z 0 q ˜żp´8,Ls n dz n qzn sn px, z n q Ŝz0 T ´s1 pz 1 , z 0 q n´1 ź i"1 Ŝzi si´si`1 pz i`1 , z i q ¸.
Similarly, for the term associated to the exit time,

S sn Ŝsn´1´sn . . . Ŝs1´s2 KT ´s1 hpu, xq " ż p´8,Ls n dz n qzn sn px, z n q ż ds 1 p0,T ´s1s psq hpu `s1 `s, Lq KL T ´s1 pz 1 , sq n´1 ź i"1 Ŝzi si´si`1 pz i`1 , z i q " ż T 0 dt 1 ts1ătu hpu `t, Lq ˜żp´8,Ls n dz n qzn sn px, z n q KL T ´s1 pz 1 , t ´s1 q n´1 ź i"1 Ŝzi si´si`1 pz i`1 , z i q
where in the last equality, we have made the change of variable t " s 1 `s and use the fact that 1 ps1,T q ptq " 1 t0ătăT u 1 ts1ătu . In the following, we write z i`1 " x and z 0 " z which represents the initial point and terminal point respectively. To obtain a representation in terms of infinite series for the transition density, we apply Fubini's theorem to obtain

I D,n T hpu, xq :" ż ∆npT q ds n S sn Ŝsn´1´sn . . . Ŝs1´s2 ŜT ´s1 hpu, xq " ż L ´8 dz 0 hpu `T, z 0 q ˜ż∆npT q ds n ż p´8,Ls n dz n qzn sn px, z n q Ŝz0 T ´s1 pz 1 , z 0 q n´1 ź i"1 Ŝzi si´si`1 pz i`1 , z i q " ż L ´8 dz 0 hpu `T, z 0 q I D,n T px, z 0 q (4.19)
for the first term and the second term is given by

I K,n T hpu, xq :" ż ∆npT q ds n S sn Ŝsn´1´sn . . . Ŝs1´s2 KT ´t1 hpu, xq " ż T 0 dt hpu `t, Lq ˜ż∆nptq ds n ż p´8,Ls n dz n qzn tn px, z n q KL T ´s1 pz 1 , t ´s1 q n´1 ź i"1 Ŝzi si´si`1 pz i`1 , z i q " ż T 0 dt hpu `t, Lq I K,n px, tq, (4.20)
where for an integer n ě 1, we introduced the two kernels

I D,n T px, zq :" ż ∆npT q ds n ż p´8,Ls n dz n qzn sn px, z n q Ŝz T ´s1 pz 1 , zq n´1 ź i"1 Ŝzi si´si`1 pz i`1 , z i q, (4.21) I K,n px, tq :" ż ∆nptq ds n ż p´8,Ls n dz n qzn sn px, z n q KL T ´s1 pz 1 , t ´s1 q n´1 ź i"1 Ŝzi si´si`1 pz i`1 , z i q. (4.22)
In (4.22), the dependence of the term KL T ´s1 pz 1 , t ´s1 q with respect to T is only in the indicator function 1 tt´s1ăT ´s1u " 1 ttăT u . Therefore, we omit writing the dependence of T in I K,n px, tq as it is always understood that t ă T .

From the above computations, we are naturally led to define for pt, x, zq P p0, T s ˆp´8, Ls 2 the following kernels p D,n T px, zq :"

# I D,n T px, zq if n ě 1 qz T px, zq
if n " 0 and p K,n px, tq :"

# I K,n px, tq if n ě 1 f L τ px, tq if n " 0.
We are ready to give the backward parametrix representation of the transition density of the process pu τ x t , X x τ x t q tě0 . One must point out that the proof of the convergence of the asymptotic expansion of the transition density is not trivial in the current setting. In the standard diffusion setting, the parametrix expansion of the transition density converges since the order of singularity in time of the space integrals in (4.21) is 1 ´η 2 ă 1, where η is the Hölder exponent of the diffusion coefficient. The situation here is much more delicate. At first glance, the order of the singularity in I K,n px, tq due to the kernel p K L (the third derivative of a Gaussian density) is of order 3´η 2 ě 1 which can not be made smaller than one by using the Hölder continuity of the diffusion coefficient. Therefore the classical argument does not guarantee the convergence of the integral. To overcome this difficulty and show that the parametrix expansion for the transition density converges, one has to make use of the estimate of the function qzn tn px, z n q for z n close to L in order to improve the order of the singularity in time.

Theorem 4.2. Let T ą 0. Assume that (H2) holds and that b is continuous on p´8, Ls. For pu, xq P R `ˆp´8, Ls, define the measure p T pu, x, dt, dzq :" p K px, t ´uqδ L pdzqdt `pD T px, zqδ u`T pdtqdz with p K px, tq :" ÿ ně0 p K,n px, tq and p D T px, zq :"

ÿ ně0 p D,n T px, zq.
Then, both series defining p K px, tq and p D T px, zq converge absolutely for px, t, zq P R ˆR˚ˆR and uniformly for px, t, zq P R ˆKT ˆR, where K T is any compact subset of p0, T s. Moreover for h P C 8,0 b pR `ˆRq, the following representation for the semigroup holds,

P T hpu, xq " hpu, xq1 txěLu `1txăLu ż u`T u ż L
´8 hpt, zq p T pu, x, dt, dzq. Finally, for some positive C, c ą 1, for all pt, zq P p0, T s ˆp´8, Ls, the following Gaussian upper-bounds hold (4.23) p K px, tq ď Ct ´1{2 gpct, L ´xq and p D T px, zq ď CgpcT, z ´xq.

Therefore, for all pu, xq P R `ˆp´8, Lq, p T pu, x, ., .q is the probability density function of the random vector pu `τ x T , X x τ x T q. More precisely, the first hitting time τ x T has a mixed type law. That is, for t P ru, u `T q, τ x T has the density p K px, t ´uq and at t " u `T , Ppu `τ x T " u `T q " ş L ´8 dz p D T px, zq and p K pL´, tq " 0. Similarly, the stopped process X x τ x T also has a mixed type law. That is, for z P p´8, Lq, X x τ x T the density p D T px, zq exists and at the boundary, we have PpX

x τ x T " Lq " ş u`T u dt p K px, t ´uq, p D T pL´, zq " p D T px, L´q " 0.
Proof. To show the convergence of ř ně0 |p D,n T px, zq|, it is sufficient to apply estimates (4.12) and (5.3) together with the semigroup property to obtain

|I D,n T px, zq| ď C n T # ż ∆npT q ds n n´1 ź i"0 ps i ´si`1 q ´p1´η 2 q + gpcT, x ´zq " C n T T nη{2 Γpη{2q n Γp1 `nη{2q
gpcT, x ´zq so that we see that the series p D T px, zq "

ř ně0 p D,n
T px, zq is uniformly convergent for px, zq P R 2 and satisfies the mentioned Gaussian upper-bound. Proving the convergence of the series ř ně0 |p K,n px, tq| requires greater effort. We proceed by induction. For n " 2, we apply estimate (5.3) for any β P r0, 1s and any px, z 1 , z 2 q P p´8, Ls 3 ,

|q z2 s2 px, z 2 q Ŝz1 s1´s2 pz 2 , z 1 q KL T ´s1 pz 1 , t ´s1 q| ď Cs ´β 2 2 |L ´z2 | β gpcs 2 , x ´z2 q| Ŝz1 s1´s2 pz 2 , z 1 q|| KL T ´s1 pz 1 , t ´s1 q|.
The key idea of the above inequality is to use the regularity of qz2 s2 in order to remove the singularity appearing in the kernel KL T ´s1 . We now proceed by writing

|L ´z2 | β ď |L ´z1 | β `|z 2 ´z1 | β . For n " 2, s 0 " t and z 0 " L,
we can bound the term with L ´z1 by using (4.13), (4.12) and the space-time inequality

|L ´z1 | β | Ŝz1 s1´s2 pz 2 , z 1 q|| KL T ´s1 pz 1 , t ´s1 q| ď C 2 T ps 1 ´s2 q ´p1´η 2 q pt ´s1 q ´3´pη`βq 2 ˆgpcpt ´s1 q, L ´z1 qgp2cps 1 ´s2 q, z 1 ´z2 q ď C 2 T n´1 ź i"0 ps i ´si`1 q ´3´pη`βq 2 gp2cps i ´si`1 q, z i ´zi`1 q
where we require 3´pη`βq 2 ă 1 and β P r0, 1s. To satisfy these conditions, β is chosen such that 1 ´η ă β ď 1. For the term involving |z 2 ´z1 | β , one first apply (4.12) and the order of the singularity for s 1 ´s2 can be improved using |z 2 ´z1 | β and the space-time inequality. Secondly, by using the |L ´z1 | β term and space-time inequality, we improve the order of the singularity in the estimate of p K L given in (4.13). That is

|z 2 ´z1 | β | Ŝz1 s1´s2 pz 2 , z 1 q|| KL T ´s1 pz 1 , t ´s1 q| ď C T |L ´z1 | β ps 1 ´s2 q ´p1´η 2 q ˆgp2cps 1 ´s2 q, z 2 ´z1 q| KL T ´s1 pz 1 , t ´s1 q| ď C 2 T n´1 ź i"0 ps i ´si`1 q ´3´pη`βq 2 gp2cps i ´si`1 q, z i ´zi`1 q.
Hence we have shown the following estimates for n " 2

|L ´z2 | β | Ŝz1 s1´s2 pz 2 , z 1 q|| KL T ´s1 pz 1 , t ´s1 q| ď 2C 2 T n´1 ź i"0 ps i ´si`1 q ´3´pη`βq 2 gp2cps i ´si`1 q, z i ´zi`1 q.
In general, suppose the following induction hypothesis holds for n ´1, that is

|L ´zn´1 | β | KL T ´s1 pz 1 , t ´s1 q| n´2 ź i"1 | Ŝzi si´si`1 pz i`1 , z i q| ď p2C T q n´1 n´2 ź i"0 ps i ´si`1 q ´3´pη`βq 2 gp2cps i ´si`1 q, z i ´zi`1 q. (4.24)
To show that the above inequality holds for n and obtain the estimate for I K,n , we use the inequality |L ´zn | β ď |L ´zn´1 | β `|z n ´zn´1 | β valid for β P r0, 1s and from the induction hypothesis (4.24) and estimate (4.13), we have

|L ´zn | β | KL T ´s1 pz 1 , t ´s1 q| n´1 ź i"1 | Ŝzi si´si`1 pz i`1 , z i q| ď 2 n´1 pC T q n n´1 ź i"0 pt i ´ti`1 q ´3´pη`βq 2 gp2cps i ´si`1 q, z i ´zi`1 q `|z n ´zn´1 | β | KL T ´t1 pz 1 , t ´t1 q| ˆn´1 ź i"1 | Ŝzi si´si`1 pz i`1 , z i q|.
For the term associated with |z n ´zn´1 | β , one applies (4.12) to | Ŝzn´1 sn´1´sn pz n , z n´1 q| and use the induction hypothesis in (4.24) to obtain

|z n ´zn´1 | β | KL T ´s1 pz 1 , t ´s1 q| " n´1 ź i"1 | Ŝzi si´si`1 pz i`1 , z i q| ı ď C T ps n´1 ´sn q ´3´pη`βq 2 gpcps n´1 ´sn q, z n´1 ´zn q|L ´zn´1 | β | KL T ´s1 pz 1 , t ´s1 q " n´2 ź i"1 | Ŝzi si´si`1 pz i`1 , z i q| ı ď 2 n´1 pC T q n n´1 ź i"0 ps i ´si`1 q ´3´pη`βq 2 gp2cps i ´si`1 q, z i ´zi`1 q.
Therefore by combining the two terms we have shown that (4.24) holds for n, that is

|L ´zn | β | KL T ´s1 pz 1 , t ´s1 q| " n´1 ź i"1 Ŝzi si´si`1 pz i`1 , z i q| ı ď p2C T q n n´1 ź i"0
ps i ´si`1 q ´3´pη`βq 2 gp2cps i ´si`1 q, z i ´zi`1 q. (4.25)

We consider the integrand in I K,n and by applying (5.3) to qzn sn px, z n q and (4.25)

|q zn sn px, z n q KL T ´s1 pz 1 , t ´s1 q

n´1 ź i"1 Ŝzi si´si`1 pz i`1 , z i q| ď Cs ´β 2 n gpcs n , x ´zn q|L ´zn | β | KL T ´s1 pz 1 , t ´s1 q| n´1 ź i"1 | Ŝzi si´si`1 pz i`1 , z i q| ď Cp2C T q n gpcs n , x ´zn qs ´β 2 n n´1 ź i"0 ps i ´si`1 q ´3´pη`βq 2 gp2cps i ´si`1 q, z i ´zi`1 q.
From the semigroup property and Lemma 5.4, we derive

|I K,n px, tq| ď Cp2C T q n # ż ∆nptq ds n s ´β 2 n n´1 ź j"0 ps i ´si`1 q ´3´pη`βq 2 + gp2ct, L ´xq " Cp2C T q n t ´β 2 `np η`β 2 ´1 2 q Γ n p η`β 2 ´1 2 qΓp1 ´β 2 q Γp1 ´β 2 `np η`β 2 ´1 2 qq gp2ct, L ´xq. (4.26)
The above shows that the n-th term is finite and the series ř n |p K,n px, tq| converges absolutely for every px, tq P R ˆR˚a nd uniformly in px, tq P R ˆKT where K T is any compact set of p0, T s. The Gaussian upper-bound (4.23) also follows (4.26).

To show that the infinite sum and the integral can be interchanged, we apply Fubini-Tonelli's theorem. Using the fact that h is bounded and the series ř ně0 |p D,n T px, yq| is convergent and satisfies the Gaussian upper bounded given in (4.23). To show that the infinite sum in the right-hand side above is finite, we use the estimate of |I K,n px, tq| in (4.26) to show that for n ě 1,

ż T 0 dt |I K,n px, tq| ď ż T 0 dt t ´β 2 ´1 2 `np η`β´1 2 q Γp1 ´β 2 qΓp η`β´1 2 q n Γp1 ´β 2 `np η`β´1q 2 qq " » -T 1´β 2 `np η`β´1 2 q 1´β 2 `n ´η`β´1 2 ¯fi fl Γp1 ´β 2 qΓp η`β´1 2 q n Γp1 ´β 2 `np η`β´1
2 qq which forms a convergent series since β P p1 ´η, 1s.

By using an appropriate approximation argument, we can extend the statement of Theorem 4.2 for bounded measurable drift coefficients. That is, we remove the continuity hypothesis of b on p´8, Ls. Proof. The proof is given in subsection 5.4 of the appendix.

Remark 4.4. A careful reading of the proofs of the main results obtained in this section show that we do not have to impose regularity assumptions of the coefficients b and σ on the whole real line but only on the interval p´8, Ls. In particular, one may obtain similar results by only assuming that b is bounded and continuous on p´8, Ls and that a " σ 2 is uniformly elliptic and η-Hölder continuous on p´8, Ls. We introduced assumption (H2) in order to make the approximation argument of Theorem 4.3 work properly, that is in order to construct a sequence of probability measure pP N q N ě1 (on the path space) that converges to the probability measure P induced by X the unique weak solution to (1.1). We do not know if such argument works if one only assumes that b is bounded measurable on p´8, Ls and a " σ 2 is uniformly elliptic and η-Hölder continuous on p´8, Ls. Now that we have obtained the parametrix expansion for the density, we study the differentiability of the functions x Þ Ñ p D T px, zq and x Þ Ñ p K T px, tq, as well as Gaussian bounds for their first partial derivatives. Theorem 4.5. Let T ą 0. Assume that (H2) holds. For any pz, tq P p´8, Ls ˆp0, T s, the functions p D T px, zq and p K px, tq given in Theorem 4.2 are differentiable with respect to x P p´8, Ls. Moreover, for some positive C, c ą 1, for all pt, zq P p0, T s ˆp´8, Ls, the following Gaussian upper-bounds hold Similarly, one has B x Ppu `τ x T " u `T q "

ş L ´8 dz B x p D T px, zq, B x PpX x τ x
T " Lq " ş T 0 dt B x p K px, tq and the following bounds hold

|B x Ppu `τ x T " u `T q| ď CgpcT, L ´xq, |B x PpX x τ x T " Lq| ď C T 1{2 gpcT, L ´xq.
Proof. By dominated convergence theorem, for x P p´8, Ls, one has

B x I D,n T px, zq " ż ∆npT q ds n ż p´8,Ls n dz n B x qzn tn px, z n q Ŝz T ´s1 pz 1 , zq n´1 ź i"1 Ŝzi si´si`1 pz i`1 , z i q
where we used the following estimate @px, zq P p´8, Ls 2 , |B x qz t px, zq| ď

C T t 1 2
gpct, x ´zq and similarly to the proof of Theorem 4.2, using Lemma 5.4, we obtain the bound

|B x I D,n T px, zq| ď C n T # ż ∆npT q ds n s ´1 2 n n´1 ź i"0 ps i ´si`1 q ´p1´η 2 q + gpcT, z ´xq " C n T T ´1 2 `n η 2 Γ n p η 2 qΓp 1 2 q Γp 1 2 `n η 2 q
gpcT, z ´xq (4.28) and, from the asymptotics of the Gamma function, the series ř ně0 B x I D,n T px, zq converges absolutely and uniformly for px, zq P R 2 and one has B x p D T px, zq "

ř ně0 B x p D,n
T px, zq. The Gaussian bound (4.27) also follows from (4.28). Similarly, we have

B x I K,n px, tq " ż ∆nptq ds n ż p´8,Ls n dz n B x qzn sn px, z n q KL T ´s1 pz 1 , t ´s1 q n´1 ź i"1 Ŝzi si´si`1 pz i`1 , z i q.
For β P r0, 1s, we use the following estimate

|B x qz t px, zq| ď C |L ´z| β t 1`β 2
gp2āt, x ´zq and select β P p1 ´η, 1q. Now using the same proof as in Theorem 4.2 (we omit the induction argument), one gets

B x I K,n px, tq ď C n T # ż ∆nptq ds n t ´1`β 2 n´1 ź i"0 ps i ´si`1 q ´3´pη`βq 2 + gpct, L ´xq " C n T t ´1`β 2 `n η`β´1 2 Γ n p η`β´1 2 qΓp 1´β 2 q Γp 1´β 2 `np η`β´1
2 qq gpct, L ´xq (4.29) which shows that the series ř ně0 B x I K,n px, tq converges absolutely for px, tq P R ˆR˚a nd uniformly for px, tq P R ˆKT , K T being any compact set of p0, T s, and that one has B x p K px, tq " ř ně0 B x I K,n px, tq. The Gaussian upper-bound (4.27) follows from (4.29). The bounds for the derivatives of the probabilities are also obtained from (4.27).

Remark 4.6. Similarly to the forward method, in order to investigate the differentiability of t Þ Ñ p K px, tq, one is naturally led to differentiate the representation (4.22) with respect to t. The difficulty comes when one tries to differentiate the KL T ´s1 pz 1 , t ´s1 q term with respect to t which involves the derivatives of t Þ Ñ B 2 z1 f z τ pz 1 , t ´sq. The singularity in time then prevents us to do so unless additional smoothness assumptions on the coefficients b and σ are provided. Again, this phenomenon does not appear in the standard diffusion framework because the density f z τ pz, t ´sq is replaced by a Gaussian density.

4.3. Applications.

We conclude this section, by giving some applications of the results established in Theorem 4.2 and Theorem 4.5. From the Gaussian upper bounds satisfied by p K px, tq, p D T px, zq and their derivatives with respect to x, we claim: Corollary 4.1. Let T ą 0 and x P p´8, Lq be fixed. Then, the following upper bounds Similarly to the forward case, the above bounds may be useful since combined with Theorem 4.2 and Theorem 4.5 they allow to establish the continuity of the maps

|Erhpτ x T , X x τ x T qs| ď ż T 0 ds|hps,
x Þ Ñ Erhpτ x T , X x τ x T qs and x Þ Ñ B x Erhpτ x T , X x τ x
T qs on p´8, Lq for a large class of test function. Although an approximation argument on the function h is needed, we omit the proof of Corollary 4.1.

Corollary 4.2. For x ă L, the first hitting time τ x has a probability density function given by t Þ Ñ p K px, tq defined on p0, 8q and an atom of size lim T Ò8 ş L ´8 dz p D T px, zq at infinity.

Proof. For every T ą 0, note that the law of τ x restricted to r0, T q is equal to the law of τ x ^T restricted to r0, T q, since for any borel set A P BpR `q Ppτ x P A, 0 ď τ x ă T q " Ppτ x ^T P A, 0 ď τ x ^T ă T q.

From Theorem 4.2, we have, ż R`1 r0,T q ptq1 A ptqPpτ x P dtq " ż R`1 r0,T q ptq1 A ptqp K px, tqdt which shows that p K px, tq1 r0,T q ptq is non-negative almost everywhere with respect to the Lebesgue measure. This implies that t Þ Ñ p K px, tq is non-negative almost everywhere on r0, 8q. Therefore, letting T Ò 8 by monotone convergence theorem, we have

ż R`1 r0,8q ptq1 A ptqPpτ x P dtq " ż R`1 r0,8q ptq1 A ptqp K px, tqdt.
To compute the atom at infinity, we see that Ppτ x ě T q " Ppτ x ^T ě T q and by Theorem 4.2

Ppτ x ě T q " ż L ´8 dz p D T px, zq.
The left hand side in the above is non-negative and decreasing with respect to T , therefore the limit as T Ò 8 exists and

Ppτ x " 8q " lim T Ò8 ż L ´8 dz p D T px, zq " 1 ´ż 8 0 dt p K px, tq.
Remark 4.7. We make the two following remarks. Firstly, in general given the solution X to a SDE with inital condition x and the corresponding hitting time τ x of a level L, the existence/size of the atom Ppτ x " 8q depends on the form of the drift and is usually a non-trivial quantity. For example, in the case of Brownian motion with a negative constant linear drift, that is X t " bt `Bt for b ă 0, the atom is of size 1 ´e2|b|L . Secondly, we see that for T ą 0, Ppmax 0ďsďT X s ă Lq " Ppτ x ą T q and therefore ż T 0 p K px, tqdt " Pp0 ď τ x ă T q " 1 ´ż L ´8 dz p D T px, zq by differentiating with respect to T , we observe that for t P p0, 8q, p K px, tq " ´Bt ş L ´8 dz p D t px, zq. We now aim at providing a probabilistic representation of the transition density using the backward parametrix method. For x ă L, we use the change of variables s i " T ´ti and s i " t ´ti for i " 1, . . . , n for (4.19) and (4.20) respectively to obtain for any bounded measurable test function h,

I D,n T hpu, xq " ż dz hpu `T, zq ż ∆ n pT q dt n ż p´8,Ls n dz n qzn T ´tn px, z n q Ŝz t1 pz 1 , zq n´1 ź i"1 Ŝzi ti`1´ti pz i`1 , z i q, I K,n hpu, xq " ż ds hpu `s, Lq ż ∆ n ptq dt n ż p´8,Ls n dz n qzn t´tn px, z n q KL t1`T ´tpz 1 , t 1 q n´1 ź i"1 Ŝzi ti`1´ti pz i`1 , z i q.
We first notice that qz t px, zq " qz t pz, xq and proceed similarly to the forward method except the role of z and x is reversed. We set From (4.12), we note that Er| θt pz, Xz t q|s ď C T t ´p1´η{2q for t P p0, T s, which in particular implies that θt pz, Xz t q P L 1 pPq. For a given time partition π : 0 " t 0 ă t 1 ă ¨¨¨ă t n ă t n`1 " T and z P R, we define Xπ,z " p Xπ,z ti q 0ďiďn`1 to be the following Euler scheme Xπ,z ti`1 " Xπ,z ti `σp Xπ,z ti qpW ti`1 ´Wti q (4.31) Xπ,z t0 " z. Hence, by induction pt 0 " 0q, we obtain ż p´8,Ls n dz n qzn T ´tn px, z n q Ŝz t1 pz 1 , zq

n´1 ź i"1 Ŝzi ti`1´ti pz i`1 , z i q " E " q Xπ,z tn T ´tn px, Xπ,z tn q n´1 ź i"0 θti`1´ti p Xπ,z tn , Xπ,z tn`1 q ı .
Let pN ptqq tě0 be a Poisson process with intensity parameter λ ą 0 independent from Xπ,z , and define N :" N pT q. Let ζ 1 ă ζ 2 ă . . . ζ N be the event times of the Poisson process and we set ζ 0 " 0, ζ N `1 " T . We know that conditional on N " n, the distribution of the event times follows a uniform order statistic given by PpN T " n, ζ 1 P dt 1 , . . . , ζ n P dt n q " λ n e ´λT dt i on the set ∆ i pT q " tt n P r0, T s n : 0 ă t 1 ă t 2 ă ¨¨¨ă t n ă T u. We still denote by π the random time partition π : ζ 0 " 0 ă ζ 1 ă ¨¨¨ă ζ n`1 " T and Xπ,z " p Xπ,z ζi q 0ďiďn`1 and its associated Euler scheme defined in (4.31). As a consequence, given a random variable Z independent from Xπ,z and the Poisson process N with density function g, we may rewrite I D,n T hpu, xq in a probabilistic way as follows

I D,n T hpu, zq " E " ż R dz hpu `T, zqe λT q Xπ,z ζ N T ´ζN px, Xπ,z ζ N qλ ´N N ´1 ź i"0 θζi`1´ζi p Xπ,z ζi , Xπ,z ζi`1 q1 tN "nu ı , n ě 0. " E " e λT hpu `T, ZqgpZq ´1 q Xπ,Z ζ N T ´ζN px, Xπ,Z ζ N qλ ´N N ´1 ź i"0 θζi`1´ζi p Xπ,Z ζi , Xπ,Z ζi`1 q1 tN "nu ı , n ě 0.
We now consider I K,n hpu, xq. We note that the derivatives of f y τ px, tq are given by

B x f y τ px, tq " " L ´x apyqt ´1 L ´x  f y τ px
ź i"1 ϑ si`1´si py i`1 , y i qq yi`1 si`1´si py i , y i`1 q
where, as convention, we set s 0 " 0, s n`1 " t, y 0 " x and y n`1 " L. We use the idea of importance sampling and rewrite the integrand in the above as the following, # θL py n , t ´sn qf L τ py n , t ´sn q

qy1 s1 px, y 1 q qx s1 px, y 1 q n´1 ź i"1 ϑ si`1´si py i`1 , y i q qyi`1 si`1´si py i , y i`1 q qyi si`1´si py i , y i`1 q + n´1 ź i"0 qyi si`1´si py i , y i`
ź i"0 θi si`1´si p Xπ,x si`1 , Xπ,x si q ‰ " e λT E " hpu `ζN `τ Xπ,x ζ N , Lqλ ´N 1 t0ăτ Xπ,x ζ N ăT ´ζN u θL p Xπ,x ζ N , τ Xπ,x ζ N q N ´1 ź i"0 θi ζi`1´ζi p Xπ,x ζi`1 , Xπ,x ζi q1 tN "nu ı .
We point out that the form of the above probabilistic representation for the I K,n hpu, xq is different from the one introduced in Bally and Kohatsu-Higa [START_REF] Bally | A probabilistic interpretation of the parametrix method[END_REF], where the Euler scheme therein has initial value z, which represents the terminal value of the process X. In the current case, the change of variable and the use of importance sampling, effectively reversed the direction of the Euler scheme and similarly to the forward method, the initial value now is x, which represents the initial value of the process X. We believe that the final representation derived here is more intuitive from a simulation point of view.

Theorem 4.8. Let T ą 0. Assume that (H2) holds. Define the two sequences p ΓD N pzqq N ě0 and p ΓK N pxqq N ě0 as follows

ΓD N pzq " # λ ´N ś N ´1 i"0 θζi`1´ζi p Xπ,z ζi , Xπ,z ζi`1 q if N ě 1, 1 if N " 0 and ΓK N pxq :" $ & % λ ´N 1 t0ăτ Xπ,x ζ N ăT ´ζN u θL p Xπ,x ζ N , τ Xπ,x ζ N q ś N ´1 i"0 θi ζi`1´ζi p Xπ,x ζi`1 , Xπ,x ζi q N ě 1 1 N " 0.
Then, the following probabilistic representation holds. Let Z be a random variable independent from Xπ,x and the Poisson process N with positive density function g. Then, for any test function h P B b pR `ˆRq, for all x P p´8, Ls, one has

Erhpτ x T , X x τ x T qs " e λT E " hpT, ZqgpZq ´1 q Xπ,Z ζ N T ´ζN px, Xπ,Z ζ N q ΓD N pZq ‰ `eλT E " hpζ N `τ Xπ,x ζ N , Lq ΓK N pxq ‰ .
Moreover, a probabilistic representation for the transition density holds, namely @x P p´8, Ls, p T p0, x, dt, dzq " δ T pdtqp D T px, zq `δL pdzqp K px, tq with for all pt, xq P p0, T s ˆp´8, Ls,

p D T px, zq " e λT E " q Xπ,z ζ N T ´ζN px, Xπ,z ζ N q ΓD N pzq ı , p K px, tq " e λT E " f L τ p Xπ,x ζ N , t ´ζN q ΓK N pxq ı .
Corollary 4.3. From Theorem 4.8, for any h P B b pRq, one has

B x ErhpX x T q1 tτ x ąT u s " e λT E " hpZqgpZq ´1 μ1 T ´ζN px, Xπ,Z ζ N qq Xπ,Z ζ N T ´ζN px, Xπ,Z ζ N q ΓD N pZq ‰ ,
where the density of Z is given by the positive function g.

Appendix

5.1. On some useful technical results. and the last line we have used Lemma 5.3 with y ˚" 2L ´z and y " z.

For the case that 4|L ´z| 2 ą t, by using triangular inequality we have 4|L ´z| 2 ą t

|q z t px, zq| ď C |L ´z| β t β 2
gp4āt, x ´zq.

The proof of the first and second derivatives of qz t px, zq as well as the estimates on B r x f y τ px, sq " apyqB r`1 x gpapyqs, L ´xq1 txăLu follow similar arguments and details are omitted. Here C ε :" Cp1 `1 ε q.

Proof. Using Young's inequality, we have that for any ε ą 0, |x| 2 ´p1 `εq|y| 2 ď p1 `1 ε q|x ´y| 2 , we obtain that p1 `1 ε q|x ´y˚´p ȳ ´y˚q | 2 ě |x ´y˚|2 ´p1 `εq|ȳ ´y˚|2 ě |x ´y˚|2 ´p1 `εq|y ´y˚|2

On the set |y ˚´y| 2 ď v, we have that p1 `1 ε q|x ´y˚´p ȳ ´y˚q | 2 ě |x ´y˚|2 ´p1 `εqv and therefore e where Bpx, yq " ş 1 0 t x´1 p1 ´tq y´1 dt stands for the standard Beta function. Using this equality repeatedly, we obtain the statement.

Markov semigroup property.

We will assume that there exists a unique weak solution to (1.1) for all x P R that satisfies the strong Markov property and our goal is to prove that pτ x t , X x τ x t q tě0 is a Markov process. The main result is given in Proposition 5.1. We first need the following preparative lemma.

Lemma 5.5. On the set tτ x s ě su, one has τ x s`t " s `τ s,X x s s`t . Proof. We just have to notice that on the set tτ x ^s ě su " tτ x ě su, the process pX x t q tě0 never crosses the level L before time s. Therefore, on the set tτ x s ě su, one has τ x " inf tv ě 0, X x v ě Lu " s `inf tv ě 0, X s,X x s s`v ě Lu " s `τ s,X x s which in turn implies τ x s`t " ps `τ s,X x s q ^ps `tq " s `pτ s,X x s ^pt `s ´sqq " s `τ s,X x s t .

We are now in position to prove the Markov property.

Proposition 5.1. The collection of positive linear maps pP t q tě0 given by (1.2) defines a Markov semigroup. Assume that b, σ are bounded and Lipschitz continuous functions on R and that Ppτ x " tq " 0 for all t ą 0 and x ă L. Then, pP t q tě0 is a strongly continuous Feller semigroup.

Proof.

Step 1: Semigroup property: Let h be a bounded continuous function. We first prove the semigroup property: P t`s hpu, xq " P s P t hpu, xq. For x ě L, one has τ x " 0 so that the semigroup property reduces to P t`s hpu, xq " hpu, xq " P s P t hpu, xq. For now on, we assume that x ă L. By the tower property of conditional expectation, it is sufficient to show that E " hpu `τ x s`t , X x τ x s`t q ˇˇF τ x s ‰ " P t hpu `τ x s , X x τ x s q. (5.4)

The computation is done on the sets tτ x s ă su and tτ x s ě su separately. Firstly, on the set tτ x s ă su " tτ x ă su, the process pX x v q vě0 hits the barrier L strictly before time s, therefore E " hpu `τ x s`t , X x τ x s`t q ˇˇF τ x s ‰ 1 tτ x s ăsu " hpu `τ x s , Lq1 tτ x s ăsu .

On the other hand, on the set tτ x s ě su " tτ u,x ě su, and using Lemma 5.5, we have E " hpu `τ x s`t , X ´x from which we deduce using standard inequalities and the fact that the coefficients b and σ are bounded that τ x Ñ `8 in probability as x Ñ ´8 in the sense that lim xÑ´8 Ppτ x ě Kq " 1 for every K ą 0. Since h is bounded, we easily get (5.5) |Erhpu `τ x , Lq1 tτ x ďtu s| ď |h| 8 Ppτ x ď tq Ñ 0, x Ñ ´8.

Moreover, using the decomposition Erhpu `t, X x t q1 tτ x ětu s " Erhpu `t, X x t qs ´Erhpu `t, X x t q1 tτ x ďtu s with (5.5), we see that it remains to prove that Erhpu `t, X x t qs goes to zero as x Ñ ´8, u Ñ `8 or x Ñ ´8 and u ě 0. Let K ą 0. We decompose this term as follows Erhpu `t, X Letting K Ñ `8 allows to conclude lim xÑ´8,uÑ`8 Erhpu`t, X x t qs " 0 and the same argument gives lim xÑ´8 Erhput , X x t qs " 0 for u ě 0. This completes the proof.

and using a similar argument as in Step 1, the probability that the path of X or X N is in A is zero. This implies that T L pXq " r T L pXq and for all N ě 1, T L pX N q " r T L pX N q.

As explained on page 460 of [START_REF] Whitt | Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues[END_REF], X N ñ X in pCr0, 8q, M 2 q is equivalent to X N ñ X in pCr0, 8q, J 1 q. Therefore, in the following, all convergence in Cr0, 8q means convergence in the J 1 topology. At this point, it is not clear that the process X P D u . To overcome this issue, we assume without loss of generality that the probability space pΩ, Pq is rich enough to support an independent Brownian motion W . We consider the processes XN t " X N T ^t `Wt ´WT ^t, Xt " X T ^t `Wt ´WT ^t.

The processes XN and X induce a family of probability measures PN and P on Cr0, 8q.

Lemma 5.6. The family of measures P and PN satisfies the following properties (i) PN ñ P or equivalently XN ñ X. (ii) Under P, the set D u is of measure one. (iii) Under P, the set A is of measure zero, where A " tw : TL pwq ă T L pwqu.

Proof. Given a path in w P Cr0, 8q and T ą 0, we denote by w T the path stopped at the terminal time T , that is for all t ě 0, w T t " w t^T . (i) It is clear that the map px, yq Ñ px T , y ´yT q is continuous map from the space pCr0, 8q ˆCr0, 8q, d 1 _ d 2 q to itself, where d i for i " 1, 2 are the uniform metric on Cr0, 8q. It is know this metric also induces the product J 1 -topology on Cr0, 8q ˆCr0, 8q. By Corollary 12.7.1 in [START_REF] Whitt | Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues[END_REF], we have that the addition map px, yq Ñ x `y is continuous. This shows that the map px, yq Ñ x T `y ´yT is a continuous map (in the J 1 -topology) from Cr0, 8q ˆCr0, 8q to Cr0, 8q. Using the fact that W is independent of X N and X, we have pX N , W q ñ pX, W q (see p.26 in Billingsley [START_REF] Billingsley | Convergence of Probability Measures[END_REF]). Therefore by continuous mapping theorem, we have XN ñ X, or equivalently PN ñ P. (ii) We show that under P, the set of paths for which the supremum increases to infinity as time goes to infinity is of probability one. That is Pptw : S 8 pwq " 8uq " PpS 8 p Xq " 8q " PpS 8 p X ´X T q " 8q " Ppmax sąT pW s ´WT q " 8q " 1 where the last equality, follows from the law of iterated logarithm. (iii) The proof is similar to that of Step 1 or Bass [START_REF] Bass | Diffusions and Elliptic Operators[END_REF], p.66.

From Theorem 13.6.4 in [START_REF] Whitt | Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues[END_REF] and the continuous mapping theorem, one obtain for any f P C b pR `q, lim N Epf pT L pX N q ^T qq " lim N Epf pT L p XN q ^T qq " Epf pT L p Xq ^T qq " Epf pT L pXq ^T qq.

One can also replace T L by r T L since these two times coincide for X and X N . Hence, we conclude that (5.8) is valid. Now, from Theorem 4.2, the following representation holds

  (ii) b P C 1 b pRq and a P C 2 b pRq. The constants C and c may change from line to line. The constant C depends on the coefficients b, σ through their norms whereas c depends only on a, a. When the constant C depends on the time horizon T , we use the notation C T . 3.1. Expansion for the semigroup.

  (ii) b : R ÝÑ R is bounded measurable and a is η-Hölder continuous on R for some η P p0, 1s that is there exists a finite positive constant C such that sup xPR |bpxq| `sup px,yqPR 2 ,x‰y |apxq ´apyq| |x ´y| η ă C. The results on weak existence and uniqueness of a Markovian solution under (H2) can be found in Stroock an Varadhan [SV69]. 4.1. Expansion for the semigroup.

  ´t`rq hpu `s, Lqf y τ px, sqds " ż p0,T ´t`rq hpu `s, LqB x f y τ px, sqds " 2hpu `T ´t `r, LqgpapyqpT ´t `rq, L ´xq ´2 ż p0,T ´t`rq B 1 hpu `s, Lqgpapyqs, L ´xqds

Theorem 4. 1 .

 1 Let T ą 0. Assume that (H2) holds and that b is continuous on p´8, Ls. Then, for every h P C 8,0 b pR `ˆRq, one hasP T hpu, xq " hpu, xq1 txěLu `1txăLu ÿ ně0 I n T hpu, xqwhere the series converges absolutely and uniformly for pu, xq P R `ˆR.

  ż L ´8 dy |hpu `T, yq| ÿ ně0 | p D,n T px, yq| ă 8 By using the fact that h is bounded, one has ÿ ně0 ż T 0 dt hpu `t, Lq|p K,n px, tq| ď |h| 8 ÿ ně0 ż T 0 dt |p K,n px, tq|.

Theorem 4. 3 .

 3 Under assumption (H2), all statements of Theorem 4.1, Theorem 4.2 and Corollary 4.2 hold.

  (4.27) |B x p K px, tq| ď C t gpct, L ´xq and |B x p D T px, zq| ď C T 1{2 gpcT, z ´xq.

  Lemma 5.3. Given y, y ˚P R and ȳ P ry ^y˚, y _ y ˚s, suppose |y ˚´y| 2 ď v, then for any ε ą 0gpCv, x ´ȳq ď ˆ1 `1 ε ˙1{2 e ε C gpC ε v, x ´y˚q .

Γ

  On some Beta type integral. Lemma 5.4. Let b ą ´1 and a P r0, 1q. Then for any t 0 ą 0, n p1 ´aqΓp1 `bq Γp1 `b `np1 ´aqq Proof. Using the change of variables s " ut, one has ż t 0 s b pt ´sq ´ads " t b`1´a ż 1 0 u b p1 ´uq ´adu " t b`1´a Bp1 `b, 1 ´aq

  P N T hpu, xq " hpu, xq1 txěLu `1txăLu ż u`T u ż L ´8 hpt, zq p N T pu, x, dt, dzq. with p N T pu, x, dt, dzq :" p K,N px, t ´uqδ L pdzqdt `pD,N T px, zqδ u`T pdtqdz and p K,N px, tq :" ÿ ně0 p K,n,N px, tq and p D,N T px, zq :" ÿ ně0 p D,n,N T px, zq.

  Then the infinitesimal generator of pP t q tě0 is

	Lhpu, xq " 1 txăLu ˆbpxqB 2 hpu, xq	`1 2	apxqB 2 2 hpu, xq `B1 hpu, xq ˙.
	Similarly, the infinitesimal generator of p Pt q tě0 writes
	Ly hpu, xq " 1 txăLu ˆ1 2	apyqB 2 2 hpu, xq `B1 hpu, xq	˙.

Let h P C 1,2 b pR `ˆp´8, Lsq. Assume that the coefficients b, σ are continuous on p´8, Ls.

Proof. For x ě L, one has P t hpu, xq " Pt hpu, xq " hpu, xq so that Lhpu, xq " Ly hpu, xq " 0. Now, assume x ă L, the Itô formula yields hpu `τ x ^t, X x τ x ^tq ´hpu, xq " phpu `τ x ^t, X x τ x ^tq ´hpu, xqq 1 txăLu " 1 txăLu

  tτ x ăsu sds

	which in turn implies
	ˇˇˇP t hpu, xq ´hpu, xq t	´Lhpu, xq ˇˇˇď 1 txăLu	1 t	ż t 0	|ErbpX x s^τ x qB 2 hpu `s ^τ x , X x s^τ x q ´bpxqB 2 hpu, xqs|ds
		`1txăLu	1 t	ż t 0	|ErB 1 hpu `s ^τ x , X x s^τ x q ´B1 hpu, xqs|ds
		`1txăLu	1 t	ż t 0	|ErapX x s^τ x qB 2 2 hpu `s ^τ x , X x s^τ x q ´apxqB 2 2 hpu, xqs|ds
		`1txăLu CPpτ

x ď tq where C is a positive constant depending on |B 2 hp., Lq| 8 , |B 2 2 hp., Lq| 8 , |B 1 hp.

  Lemma 3.1. Assuming that a " σ 2 is strictly positive on p´8, Ls and let y P p´8, Ls. The kernel Pp Xx t P dz, max vPr0,ts Xx v ă Lq :" qy t px, zqdz is given by qy t px, zq " pgpapyqt, z ´xq ´gpapyqt, z `x ´2Lqq 1 txďLu 1 tzďLu , and f y τ px, sq " B s Ppτ x ď sq " 2B s gpapyqs, L ´xq and f y τ px, sq " B x gpapyqs, L ´xq for x ď L. Proof. In what follows, we may assume without loss of generality that σ is positive on p´8, Ls. We write

				L 2πapyqs 3{2 ´x a	exp	ˆ´pL ´xq 2 2apyqs ˙1txďLu .
	Furthermore, the following boundary conditions are satisfied: qy t pL, zq " qy t px, Lq " f y τ pL, sq " 0 together with B x f y τ px, sq " Pp Xx t ď z, max vPr0,ts Xx v ă Lq " P ˆWt ď z ´x σpyq , max vPr0,ts σpyq W v ă L ´x "
			Φ	ˆz σpyq ´x ?	t	˙`Φ	σpyq ? ˆ2pL ´xq ´pz ´xq t	˙´1
	where we used the joint distribution pW t , max vPr0,ts W v q coming from the reflection principle. Now differentiating w.r.t z yields the result. The probability density function s Þ Ñ f y τ px, sq is then easily deduced from the identity Ppτ x ą sq " Ppmax vPr0,ss Xx v ă Lq " 2Φ ´L´x σpyq ? 2apyqs s ¯´1 " 1 ´erfc ˆL´x ? ˙.
	We introduce the following operators defined for h P B b pR `ˆp´8, Lsq and y P p´8, Ls by
						ż t
		K y t hpu, xq " 1 txăLu	hpu `s, Lqf y τ px, sqds,
						0
						ż L
		S y t hpu, xq " 1 txăLu	´8 hpu `t, zqq y t px, zqdz
	which respectively correspond to the exit time operator and the killed diffusion operator. As a consequence of
	Lemma 3.1 and the very definition of the Markov semigroup p P y t q tě0 , we get the following results.
	Corollary 3.1. For all pu, xq P R `ˆR, one has
		P y t hpu, xq " 1 txěLu hpu, xq `Sy t hpu, xq `Ky t hpu, xq
			ż u`t	ż L
	(3.1)	" 1 txěLu hpu, xq `1txăLu	u		´8 hps, zq rδ L pdzqf y τ px, s ´uqds `δu`t pdsqq y t px, zqdzs .

  For all x ě L and u ě 0, one clearly has P T hpu, xq ´P T hpu, xq " 0 and the right hand side of (3.6) is also 0. Hence, we restrict to the case x ă L for the rest of the proof. We now compute B s PT ´sP s hpu, xq as the limit of

	Now, combining the previous computations, we write the following decomposition as ε Ó 0
	PT ´s´ε	P ε	´I ε	P s hpu, xq "	ż L ´8 qx T ´s´ε px, zq	P ε	´I ε	P s hpu `T ´s ´ε, zqdz
								"	ż L ´8 qx T ´s´ε px, zq	P ε	´I ε	p Pη `pI ´P η qqP s hpu `T ´s ´ε, zqdz
									ż L	ż
								Ñ	´8 dz	R	dy qx T ´spx, zqpB 1 P s hpu `T ´s, yqgpη, y ´zq ´Ps hpu `T ´s, yqbpzqB 2 gpη, y ´zq
								`1 2	P s hpu `T ´s, yqapxqB 2 2 gpη, y ´zqqdz
								`żR	pErLhpu `τ x s , X x τ x s qs ´ErLhpu `τ x s , z `Xx τ x s qsqgpη, zqdz
	Let h P C 1,2 " ż L ´8 ż R b pR `ˆp´8, Lsq. Assume that (H1) holds. Then, one has dzdy qx T ´spx, zqB 1 P s hpu `T ´s, yqgpη, y ´zq P T hpu, xq ´P T hpu, xq " ż T 0 ˘. `ż L ´8 ż R dzdy " B 2 apzqq x T ´spx, zq ˙´B z pbpzqq x T ´spx, zqq  P s hpu `T ´s, yqgpη, y ´zq z ˆ1 2 ds `K T ´sP s hpu, xq `S T ´sP s hpu, xq ´żR dyP s hpu `T ´s, yqB z p 1 apzqq x T ´spx, zqq |z"L gpη, y ´Lq 2 Proof. the following quantity (3.6) `żR pErLhpu `τ x s , X x τ x s qs ´ErLhpu `τ x s , z `Xx τ x s qsqgpη, zqdz
				1 ε	`P T ´s´ε P s`ε hpu, xq ´P T ´sP s hpu, xq	˘" PT ´s´ε ´P T ´s ε	P s hpu, xq `P T ´s´ε	P ε	´I ε	P s hpu, xq
	as ε Ó 0. We will start with the second term appearing in the right-hand side of the above equality. Let us note
	that Lemma 2.1 does not guarantee neither that P t h P C 1,2 b pR `ˆp´8, Lsq nor that LP t h can be written in a
	differential form. We use a regularization technique that we now explain. We introduce the smoothing operator
	Pη hpu, xq " particular, note that B 1 P s hpu, xq " pP s B 1 hqpu, xq due to time homogeneity and the fact that h P C 1,2 ş R hpu, zqgpη, z ´xqdz so that pu, xq Þ Ñ Pη P s hpu, xq " ş R P s hpu, zqgpη, z ´xqdz P C 1,2 b pR `ˆRq. In b pR `ˆp´8, Lsq.
	We also remark that one has
									ż
									P ε Pη P s hpu, xq " Er	P s hpu `τ x ε , X x τ x ε `zqgpη, zqdzs
									R
									ż
									"	P ε P s hp., z `.qpu, xqgpη, zqdz
									R
	so that, by dominated convergence, one gets
								pP ε ´Iq ε	Pη P s hpu, xq "	ż R	P ε	´I ε	P s hp., z `.qpu, xqgpη, zqdz
									ż
									Ñ	P s Lhp., z `.qpu, xqgpη, zqdz, ε Ó 0
									R
	which clearly implies
			pP ε ´Iq ε	pI ´P η qP s hpu, xq Ñ	ż	R	pP s Lhpu, xq ´Ps Lhp., z `.qpu, xqq gpη, zqdz
									ż
									"	pErLhpu `τ x s , X x τ x s qs ´ErLhpu `τ x s , z `Xx τ x s qsqgpη, zqdz, ε Ó 0.
									R
	However, putting the spatial derivatives on the smoothing kernel, one also gets
	pP ε ´Iq ε	Pη P s hpu, xq Ñ	ż R	" B 1 P s hpu, zqgpη, z ´xq ´Ps hpu, zqbpxqB 2 gpη, z ´xq	`1 2	P s hpu, zqapxqB 2 2 gpη, z ´xq		dz, ε Ó 0.

  , zqB 1 P s hpu `T ´s, yqgpη, y ´zq Ñ P s hpu, xq is continuously differentiable for h P C 1,2 b pR `ˆp´8, Lsq, the differentiability of t Þ Ñ qx t px, zq and the continuity of v Þ Ñ P s hpu `v, Lqf x τ px, vq. Now, combining the two limits with the relation B t qx t px, zq " 1 2 apxqB 2

	From (3.1), one has
	PT ´s´ε ´P T ´s ε	P s hpu, xq "	ż L ´8 P s hpu `T ´s ´ε, zq ´Ps hpu `T ´s, zq ε	qx T ´s´ε px, zqdz
								`ż L ´8 P s hpu `T ´s, zq	qx T ´s´ε px, zq ´q x T ´spx, zq ε	dz
								ż T ´s
								´1 ε	T ´s´ε	P s hpu `v, Lqf x τ px, vqdv
								Ñ	´ż L ´8 B 1 P s hpu `T ´s, zqq x T ´spx, zqdz	´ż L ´8 P s hpu `T ´s, zqB t	qx T ´spx, zqdz
								´Ps hpu `T ´s, Lqf x τ px, T ´sq, as ε Ó 0,
	where we used that u Þ Ñ
								ż L
								´8 dz qx T ´spx, zqB 1 P s hpu `T ´s, zq,
	ż R	dyP s hpu `T ´s, yqB z p	1 2	apzqq x T ´spx, zqq |z"L gpη, y ´Lq Ñ P s hpu `T ´s, LqB z p	1 2	apzqq x T ´spx, zqq |z"L
	and that	ş L ´8 ş	R dzdy	"	B 2 z `1 2 apzqq x T ´spx, zq ˘´B z pbpzqq x T ´spx, zqq	‰	P s hpu `T ´s, yqgpη, y ´zq converges to
						ż L ´8 dz	"	B 2 z ˆ1 2	apzqq x T ´spx, zq ˙´B z pbpzqq x T ´spx, zqq		P s hpu `T ´s, zq
	as η Ñ 0. Hence, one concludes that
	lim εÑ0 PT ´s´ε	P ε	´I ε	P s hpu, xq " PT ´sLP s hpu, xq
								"	ż L ´8 dz	"	B 2 z ˆ1 2	apzqq x T ´spx, zq ˙´B z pbpzqq x T ´spx, zqq 	P s hpu `T ´s, zq
								`ż L ´8 dz qx T ´spx, zqB 1 P s hpu `T ´s, zq
								´Ps hpu `T ´s, LqB z p	1 2	apzqq x T ´spx, zqq |z"L

z qx t px, zq and Lemma 3.1 finally yield B s `P T ´sP s hpu, xq ˘" lim εÑ0 1 ε `P T ´s´ε P s`ε hpu, xq ´P T ´sP s hpu, xq " KT ´sP s hpu, xq `S T ´sP s hpu, xq. From the continuity of t Þ Ñ Pt hpu, xq, P t hpu, xq, (3.4) and (3.5) (which implies that s Þ Ñ KT ´sP s hpu, xq, ST ´sP s hpu, xq P L 1 pr0, T sq and Fubini's theorem, one gets P T hpu, xq ´P T hpu, xq " ż T 0 B s `P T ´sP s hpu, xqdu ˘ds " ż T 0 KT ´sP s hpu, xq `S T ´sP s hpu, xq ( ds " ż T 0 KT ´sP s hpu, xq `S T ´sP s hpu, xq ( ds This concludes the proof. Given that P s hpu, Lq " E " hpu `τ L s , X L τ L s q ‰ " hpu, Lq, we observe the following important property KT ´sP s hpu, xq " 1 txăLu papLq ´apxqq apxq f x τ px, T ´sqP s hpu `T ´s, Lq " KT ´shpu, xq and KT ´s Ps hpu, xq " KT ´shpu, xq. Hence, (3.6) may be simplified as follows

  txďLu 1 tzďLu and introduce the quantity θt px, zq " θ t px, zqΛ t px, zq. With these notations, from (3.23), for every bounded measurable function h P B b pR `ˆp´8, Lsq, one obtains ST hpu, xq " Erhpu `T, Xx T qθ T px, Xx T q1 tτ x ěT u s " Erhpu `T, Xx

	St px, zq " 1 txăLu	" B 2 z	"	2 1	papzq ´apxqqq x t px, zq 	´Bz rbpzqq x t px, zqs *
		" 1 txăLu	" p	2 1	a 2 pzq ´b1 pzqqq x t px, zq `pa 1 pzq ´bpzqqB z	qx t px, zq	`1 2	papzq ´apxqqB 2 z	qx t px, zq	*
	(3.23)	" θ t px, zqq x t px, zq.
	Here, using (5.1) and (5.2), we explicitly have
	θ t px, zq " 1 txăLu	"ˆ1 2	a 2 pzq ´b1 pzq ˙`pa 1 pzq ´bpzqqµ 1 t px, zq	`1 2	* t px, zq papzq ´apxqqµ 2	,
	µ 1 t px, zq " H 1 papxqt, z ´xq	´1 apxqt	2pL ´xq apxqt pexpp´2 pz´LqpL´xq	q ´1q	,
	µ 2
		0ďvďt	W v ď	L σpxq ´x	ˇˇW t "	z σpxq ´x	¯" " 1 ´exp ˆ´2	pL ´xqpL ´x ´pz ´xqq tapxq	˙* 1

t px, zq " H 2 papxqt, z ´xq `1 a 2 pxqt 2 4pz ´LqpL ´xq pexpp´2 pz´LqpL´xq apxqt q ´1q

. We also use the distribution function of the supremum of the Brownian bridge Λ t px, zq :" P ´max

  1 ¸P sn hpu, xq " e λT Erhpu `pζ n `τ ζn, Xπ ζn q ^T, Xπ Ssi´si`1 ¸K sn´1´sn hpu, xq " e λT Erhpu `ζn´1 `τ

								n´1
						pζn`τ	ζn, Xπ ζn q^T	q1 tN "nu	j"0 ź	λ ´1 θζj`1´ζj p Xπ ζj , Xπ ζj`1 qs
	and for n ě 1
			˜n´2
	ż	∆npT q	ds n	ź i"0	ζn´1, Xπ ζ n´1 , Lq1 tN "n´1u 1	tτ	ζ n´1 , Xπ ζ n´1 ďT ´ζn´1u
				ˆpapLq ´ap Xπ ζn´1 qq ap Xπ ζn´1 q	n´2 j"0 ź	λ ´1 θζj`1´ζj p Xπ ζj , Xπ ζj`1 qs

  is sufficient to show that the last two terms satisfy the statements of the lemma. By using integration by parts and the relationship between the Lévy distribution and the complementary error function, we have for x ď L,

	Proof. We recall from (3.1) that the function P y T ´t`r hpu, xq can be decomposed into
					#		+
					ż
		P y T ´t`r hpu, xq " hpu, xq1 txěLu `1txăLu	hpu `T ´t `r, zqq y T ´t`r px, zqdz
					p´8,Lq
				#			+
				ż
	(4.1)		`1txăLu	hpu `s, Lqf y τ px, sqds
				p0,T ´t`rq
	so that it ż
		hpu `s, Lqf y τ px, sqds
		p0,T ´t`rq
			˜L	¸´ż	˜L	¸ds.
	(4.2)	" hpu `T ´t `r, Lqerfc	´x 2apyqpT ´t `rq a	p0,T ´t`rq	B u hpu `s, Lqerfc	´x 2apyqs a
		ż
		B u	hpu `s, Lqf y τ px, sqds
		p0,T ´t`rq
				˜L		¸´ż	˜L	¸ds.
	(4.3)	" B u hpu `T ´t `r, Lqerfc	´x 2apyqpT ´t `rq a	p0,T ´t`rq	B 2 u hpu `s, Lqerfc	´x 2apyqs a

2,0 b pR `ˆRq holds. Then the function pt, u, xq Þ Ñ P y T ´t`r hpu, xq belongs to C 1,1,2 b pr0, T s ˆR`ˆp ´8, Lsq.

  t. x, we get

	ż
	B 2 x
	p0,T ´t`rq
				˜L	2a
					´x
				a 2pT ´t `rqapyq
					˜L	¸ds
		´1pyq	ż p0,T ´t`rq	B 2 1 hpu `s, Lqerfc	´x 2pT ´sqapyq a
	which allows to conclude that the second derivative with respect to x is continuous and uniformly bounded on
	r0, T s ˆR`ˆp ´8, Ls. For x ă L, from the fundamental theorem of calculus, one has
	ż
	B t	hpu `s, Lqf y τ px, sqds " ´hpu `T ´t `r, Lqf y τ px, T ´t `rq
	p0,T ´t`rq
	which is clearly jointly continuous and uniformly bounded in pt, u, xq P r0, T s ˆR`ˆp ´8, Ls.

hpu `s, Lqf y τ px, sqds " 2hpu `T ´t `r, LqH 1 papyqpT ´t `rq, L ´xqgpapyqpT ´t `rq, L ´xq ´2a ´1pyqB 1 hpu `T ´t `r, Lqerfc

  We also want to make an important remark concerning the linear maps defined above. For t ą 0, it is clear from the estimates of B 2 x qy t px, zq and B x qy t px, zq given in Lemma 5.2 give together with the hypothesis (H2) that by bounded convergence theorem, x Þ Ñ Ŝy t hpu, xq is continuous and equal to zero at x " L. The continuity of Ky t h is slightly more involved. In fact, from the indicator function in (4.5), we see that the function x Þ Ñ Ky t hpu, xq is zero for x ě L and non-zero for x ă L. We make use of integration by parts formula twice and Lemma 3.1 in order to write From (4.6), we also have the following estimate Proof. From Lemma 2.1 we have for y P p´8, Ls, the explicit form of the generator of pP t q tě0 and p P yThe integrability in time of the above expression follows from the estimates in Lemma 5.2. To obtain (4.8), we rewrite the above expression using (4.1) for x ă L which gives In order to write the backward parametrix expansion of the Markov semigroup pP t q tě0 , we need to define the following integral operators for any bounded measurable function h : R `ˆR Ñ R

	Ŝy t px, zq :" Ky t px, sq :" and define for all h P C 2,0 (4.4) (4.5) b pR `ˆRq ˆ1 2 ˆ1 2 Ŝy t hpu, xq :" ż R dz hpu `t, zq Ŝy rapxq ´apyqs B 2 x rapxq ´apyqs B 2 x f y qy t px, zq `bpxqB x τ px, sq `bpxqB x f y qy t px, zq ˙1txăLu 1 tzăLu , τ px, sq ˙1txăLu 1 tsďtu t px, zq and Ky t hpu, xq :" ż R`d s hpu `s, Lq Ky t px, sq. Ky t hpu, xq " rapxq ´apyqs apyq hpu `t, Lqf y τ px, tq ´rapxq ´apyqs apyq ż t 0 ds B 1 hpu `s, Lqf y τ px, sq `2bpxqhpu `t, Lqgpapyqt, L ´xq ´2bpxq ż t 0 ds B 1 hpu `s, Lqgpapyqs, L ´xq (4.6) " rapxq ´apyqs apyq hpu `t, Lqf y τ px, tq ´rapxq ´apyqs apyq B 1 hpu `t, Lqerfc ˜L ´x a 2apyqt ŗapxq ´apyqs apyq ż t 0 ds B 2 1 hpu `s, Lqerfc ˜L ´x a 2apyqs ¸`2bpxqhpu `t, Lqgpapyqt, L ´xq ´2bpxq ż t | Ky t hpu, xq| (4.7) ď C ! |h| 8 f y τ px, tq `|B 1 h| 8 erfc ˜L ´x a 2apyqt ¸`|b| 8 |hpu `t, Lq| ? t `|b| 8 |B 1 h| 8 ż t 0 ds gpapyqs, L ´xq ) . This bound will be used in future calculations. C 2,0 b pR `ˆRq, @pu, xq P R `ˆp´8, Lq, P T P y r hpu, xq ´P y T `r hpu, xq " ż T 0 ds P s p Ŝy T ´s`r h `K y T ´s`r hqpu, xq. (4.8) t q tě0 for functions h P C 1,2 b pR `ˆp´8, Lsq. For h P C 2,0 b pR `ˆRq, we have from Lemma 4.1 that for r ą 0, the function pu, xq Þ Ñ P y r hpu, xq belongs to C 1,2 b pR `ˆp´8, Lsq. Therefore we have by differentiating the composition pP s T ´s`r hpu, xq " 1 txăLu # ż p´8,Lq hpu `T ´s `r, zq ˆ1 2 rapxq ´apyqs B 2 x qy T ´s`r px, zq `bpxqB x qy T ´s`r px, zq ˙+ `1txăLu # ż T ´s`r 0 ds hpu `s, Lq ˆ1 2 rapxq ´apyqs B 2 x f y τ px, sq `bpxqB x f y τ px, sq ˙+ " p Ŝy T ´s`r h `K y T ´s`r hqpu, xq. S t hpu, xq :" ż L ´8 dz hpu `t, zqq z t px, zq, K t hpu, xq :" ż t 0 ds hpu `s, Lqf L τ px, sq, (4.10) Ŝt hpu, xq :" ż L ´8 dz hpu `t, zq Ŝz t px, zq, Kt hpu, xq :" ż t 0 ds hpu `s, Lq KL t px, sq. (4.11) P pL ´L y q P y where Ŝz

0

ds B 1 hpu `s, Lqgpapyqs, L ´xq and from the second equality above, it is clear that Ky t hpu, xq has finite non-zero left limit at x " L as h P C 2,0 b pR `p´8, Lsq. Therefore, in general, Ky t hpu, L´q ‰ Ky t hpu, Lq " 0 and for fixed u, t ą 0, the map x Þ Ñ Ky t hpu, xq is right continuous with left limit at x " L. Lemma 4.2. Assume that (H2) holds and b is continuous on p´8, Ls. For any r ą 0, y P p´8, Ls and h P y T ´s`r hpu, xqq 0ďsďT with respect to the time variable s, P T P y r hpu, xq ´P y T `r hpu, xq " ż T 0 ds P s pL ´L y q P y T ´s`r hpu, xq. (4.9) pL ´L y qhpu, xq " 1 txăLu ˆbpxqB x `1 2 rapxq ´apyqsB 2 x ˙hpu, xq. t px, zq and KL t px, sq are given in (4.4) and (4.5

  While, if h P C 2,0 b pR `ˆp´8, Lsq and suppphq Ď R `ˆp´8, Lq, then

	(4.17)	P T hpu, xq "	#	S T hpu, xq 0	`şT 0 ds P s ŜT ´shpu, xq, x ă L, x ě L.

16) P T hpu, xq " # pS T `KT qhpu, xq `şT 0 ds P s p ŜT ´s `K T ´sqhpu, xq, x ă L, hpu, xq x ě L.

  y ´zqhpu 1 `r, zqq y

					r px 1 , zq	`ż r	τ px 1 , sq ds gpε, L ´yqhpu 1 `s, Lqf y )
						0
	It is clear from Lemma 5.2 and the following	ş r 0 ds integrable bound
	(4.18)	f y τ px, sq ď C	L	´x ās	gpās, L ´xq " CB x gpās, L ´xq

  To take the limit as ε Ó 0, we again use the fact that ş R dy gpε, L ´yqhpu `s, Lqf y τ px, sq ď C|h| 8 B x gpās, L ´xq, It is clear that |S T `r hpu 1 , x 1 q| ď |h| 8 and |K T `r hpu 1 , x 1 q| ď |h| 8 . Therefore by using fact that lim rÓ0 K t`r hpu, xq " K t hpu, xq and lim rÓ0 S t`r hpu, xq " S t hpu, xq, we conclude that

	which is	ş L ´8 dz integrable. Therefore by dominated convergence theorem
			ż L				ż	ż L
			´8 dz lim εÓ0	R	dy gpε, z ´yqhpu `T `r, zqq y T `r px, zq (	"	´8 dz hpu `T `r, zqq z T `r px, zq (
									" S T `r hpu, xq.
									ż
									lim rÓ0	lim εÓ0	dy P y T `r pgpε, y ´¨qhqpu, xq " pK
									1 and the continuity of the integral
	we conclude that
									ż
									lim rÓ0	lim εÓ0	R	dy P T	P y
	We now consider the term lim rÓ0 lim εÓ0 fact that ş T `r 0 ds |hpu `s, Lq|f y τ px, sq ď |h| 8 to obtain ş T `r pgpε, y ´¨qhqpu, xq. We first apply Fubini's theorem by using the R dy P y
		ż	dy	ż T `r	τ px, squ " ds tgpε, L ´yqhpu `s, Lqf y	ż T `r	ds	ż
		R			0				0
	which is	ş T `r 0	ds integrable. Therefore by dominated convergence theorem, we obtain
	ż T `r 0	ds lim εÓ0	R ż	dy tgpε, L ´yqhpu `s, Lqf y τ px, squ "	0 ż T `r	ds hpu `s, Lqf L τ px, sq	(	" K T `r hpu, xq.
	Similarly, we note that from Lemma 5.2, Fubini's theorem can be applied to obtain
	ż								ż L	ż
									(	"	´8 dz

r pgpε, y ´¨qhqpu, xq " P T hpu, xq. R dy tgpε, L ´yqhpu `s, Lqf y τ px, squ . R dy ż L ´8 dz gpε, z ´yqhpu `T `r, zqq y T `r px, zq R dy gpε, z ´yqhpu `T `r, zqq y T `r px, zq ( To take the limit as r Ó 0, we notice that again from Lemma 5.2 ż R dy ˇˇgpε, z ´yqhpu `T `r, zqq y T `r px, zq ˇˇď |h| 8 gpcpT `rq, x ´zq. T `ST qhpu, xq.

  vq ˇˇ1 tx 1 ăLu pu, x, du 1 , dx 1 q integrable. Therefore we can take the limit as ε Ó 0 using dominated convergence and consider, where we use again (4.6) and integration by parts Finally, to take the limit as r Ó 0, it is sufficient to use (4.15), to obtain | KT ´s`r hpu 1 , x 1 q| ď C T p|h| 8 , |B 1 h| 8 q

	ď Cgpε, L ´yq ! |h| 8 B x 1 gpāpT ´s `rq, x 1 ´Lq `|B 1 h| 8 ď Cgpε, L ´yq 1 T ´s `r ş R dy ş T 0 ds ş ş T 0 ds ş P s lim which is εÓ0 ż R dy gpε, L ´yq Ky T ´s`r pgpε, L ´¨qhqpu 1 , x 1 q " lim εÓ0 ż R dy gpε, L ´yq `|b| 8 |h| 8 ? T ´s `r `|b| 8 |B 1 h| 8 rapx 1 q ´apyqs apyq hpu 1 `T ´s `r, Lqf y ? T ´s `r) τ px 1 , T ´s `rq ´lim εÓ0 ż R dy gpε, L ´yq rapx 1 q ´apyqs apyq ż T ´s`r 0 ds B s hpu 1 `s, Lqf y τ px 1 , sq `lim εÓ0 ż R dy gpε, L ´yq2bpx 1 qhpu 1 `T ´s `r, Lqgpapyqt, L ´xq ´lim εÓ0 ż R dy gpε, L ´yq2bpx 1 q ż T ´s`r 0 ds B s hpu 1 `s, Lqgpapyqs, x 1 ´Lq " KT ´s`r hpu 1 , x 1 q 1 pT ´sq 1´η 2 which is independent of r and ş T 0 ds ş P s pu, x, du 1 , dx 1 q integrable. Therefore one can conclude that ż T P which is lim rÓ0 lim εÓ0 0 dsP s tpS y T ´s`r `Ky

s pu, x, du 1 , dx 1 q integrable. It is clear from above that ż R dy | Ky T ´s`r pgpε, y ´¨qhqpu, xq| ď 1 T ´s `r T ´s`r qpgpε, y ´¨qhqupu, xq " ż T 0 ds P s p ŜT ´sh `K T ´shqpu, xq.

  hold for any Borel function h defined on R `ˆp´8, Ls as soon as the above integrals are finite.

			Lq|	1 ? s	gpcs, x ´Lq	`ż L ´8 dz |hpT, zq|gpcT, x ´zq,
	|B x Erhpτ x T , X x τ x T qs| ď	ż T 0	ds|hps, Lq|	1 s	gpcs, x ´Lq	`ż L ´8 dz |hpt, zq|	1 ? T	gpcT, x ´zq,

  , tq :" H1 papyqt, L ´xqf y τ px, tq, and write Kt px, sq " θL px, sqf L τ px, sq1 tsătu with θL px, sq :" 1 2 rapxq ´apLqs H2 papLqs, L´xq`bpxq H1 papLqs, L´xq. With the above notations, performing the change of variables t ´ti " s n´i`1 and z i " y n´i`1 for i " 0, . . . , n, we can write I K,n px, tq " dy n θL py n , t ´sn qf L τ py n , t ´sn qq y1 s1 px, y 1 q

		B 2 x f y τ px, tq "	"	pL ´xq 2 apyq 2 t 2 ´3 apyqt		f y τ px, tq :" H2 papyqt, L ´xqf y τ px, tq,
	ż		ż			n´1
	∆ n ptq	ds n	p´8,Ls n

  Cε η{2 |h| 8 which converges to zero as ε Ó 0.In order to prove the convergence of the parametrix series, we need to study the two proxy kernels: the proxy killed diffusion kernel and the proxy exit time kernel. The density qy t and its derivatives are given by qy " L or z " L, it is understood that we are always taking left-hand derivatives.To derive the bound with the |L ´z| β term, we consider first the case where 4|L ´z| 2 ď t, to estimate qy t px, zq " gpapyqt, x ´zq ´gpapyqt, x ´p2L ´zqq one apply the mean value theorem to gpapyqy, x ´zq with respect to the points z and 2L ´z to obtain for some θ P r0, 1s, where in the second line we have used the space-time inequality and fact that 2 1´β |L ´z| 1´β ă t

	and
	(5.1)
	B z	qy t px, zq "	ˆ´z apyqt ´x	gpapyqt, z ´xq	`z `x ´2L apyqt	gpapyqt, L ´x `L ´zq ˙1txďLu 1 tzďLu ,
	(5.2)
	B 2 z	qy t px, zq "	ˆˆpz ´xq 2 apyq 2 t 2 ´1 apyqt	˙gpapxqt, z ´xq	´ˆpL ´z `L ´xq 2 apyq 2 t 2	´1 apyqt	˙gpapxqt, 2L ´x ´zq ˙1txďLu 1 tzďLu
	and at x Lemma 5.2. Assume that (H1) (i) or (H2) (i) holds. For any β P r0, 1s, there exists C, c ą 1, such that for any
	px, zq P p´8, Ls 2 and r " 0, 1, 2, the following estimates hold:
	(5.3)		|B r x	qy t px, zq| ď C	ˆ|L ´z| β t r`β 2	^1 t r 2	˙gpct, x ´zq, and |B r x f y τ px, tq| ď	t	C 2 r`1	gpct, L ´xq.
	Proof. From the expression of qy t px, zq, the following estimates for qy t px, zq and its derivatives hold
								|q t px, zq| ď Cgp2āt, z ´xq `Cgp2āt, z `x ´2Lq
								|B x	qy t px, zq| ď C	1 t 1 2	gp2āt, z ´xq	`C 1 2 t 1	gp2āt, L ´x `L ´zq
								|B 2 x	qy t px, zq| ď C	1 t	gp2āt, z ´xq	`C 1 t	gp2āt, L ´x `L ´zq.
	Proof. It is sufficient to write
							ż L
							´8 dz hpu `ε, zqq z ε py, zq
							"	ż L ´8 dz hpu `ε, zqq y ε py, zq |q y t px, zq| " |tz ´p2L ´zquB x gpapyqt, x ´θz ´p1 ´θqp2L ´zqq| `ż L ´8 dz hpu `ε, zq rq z ď C |z ´L| β 2 t β gp2āt, x ´θz ´p1 ´θqp2L ´zqq
										ď C	|z ´L| β 2 t β	gp4āt, x ´zq
										1´β
										2
	Hence, it is sufficient to show that the second term converges to zero. Using (5.3) and the Hölder regularity of a,
	we can bound the second term by
										ż L
		|h| 8 ε py, zq| ď B x ´8 dz |q z ε py, zq ´q y qy t px, zq " ˆz ´x apyqt gpapyqt, z ´xq ´L ´x `L apyqt gpapyqt, L ´x `L ´zq ´z	Ḃ2
		x	qy t px, zq "	ˆˆpz ´xq 2 apyq 2 t 2 ´1 apyqt	˙gpapyqt, z ´xq	´ˆpL ´z `L ´xq 2 apyq 2 t 2	´1 apyqt	˙gpapyqt, L ´x `L ´zq ˙

Lemma 5.1. For all pu, yq P R `ˆp´8, Lq and h P C b pR `ˆp´8, Lsq, one has lim εÑ0 ż L ´8 dz hpu `ε, zqq z ε py, zq " hpu, yq. ε py, zq ´q y ε py, zqs " Erhpu `ε, Xy ε q1 tτ y ěεu s `ż L ´8 dz hpu `ε, zq rq z ε py, zq ´q y ε py, zqs . By the dominated convergence theorem and the fact that τ y ą 0 a.s. since y ă L, one gets lim εÓ0 Erhpu `ε, Xy ε q1 tτ y ěεu s " hpu, yq. t px, zq " gpapyqt, z ´xq ´gpapyqt, z `x ´2Lq, Furthermore, for px, zq P p´8, Ls 2 , one has gpct, z ´x ´2pL ´xqq ď gpct, z ´xq, since in the exponent pz ´xq 2 ´4pL ´xqpz ´xq `4pL ´xq 2 " pz ´xq 2 ´4pL ´xqpz ´L `L ´xq `4pL ´xq 2 " pz ´xq 2 `4pL ´xqpL ´zq ě pz ´xq 2 .

  by using the Markov property and time homogeneity, the above is equal to Finally, on the set tτ x s ă su, the process pX x v q vě0 hits the barrier L before time s, therefore P t hpu `τ x s , X x τ x s q1 tτ x s ăsu " P t hpu `τ x s , Lq1 tτ x s ăsu " hpu `τ x s , Lq1 tτ x s ăsu . This completes the proof of (5.4) and therefore of the Chapman-Kolmogorov relation for pP t q tě0 follows. Moreover, by dominated convergence, one has P t hpu, xq Ñ hpu, xq as t Ó 0. It now remains to prove that P t C 0 pR `, Rq Ă C 0 pR `, Rq. Second step: Continuity of pu, xq Þ Ñ P t hpu, xq Let pu n , x n q Ñ pu, xq. From the Lipschitz continuity of the coefficients b, a, we deduce that max 0ďtďT X xn t Ñ max 0ďtďT X x t in L 2 pPq as n Ñ `8 which in turn implies the convergence τ xn Ñ τ x in distribution. Moreover, since τ x ‰ t a.s., by the continuous mapping theorem we obtain Erhpu n `τ xn , Lq1 tτ xn ďtu s Ñ Erhpu `τ x , Lq1 tτ x ďtu s, n Ñ `8. By similar arguments, one gets Erhpu n `t, X xn t q1 tτ xn ětu s Ñ Erhpu `t, X x t q1 tτ x ětu s, n Ñ `8. Hence we conclude that pu, xq Þ Ñ P t hpu, xq is continuous. Third step: lim |pu,xq|Ñ`8 P t hpu, xq " 0 For u ě 0, x ě L, one has P t hpu, xq " hpu, xq so that lim uÑ`8,xÑ`8 P t hpu, xq " lim uÑ`8 P t hpu, xq " lim xÑ`8 P t hpu, xq " 0 for u ě 0 and x ě L. By dominated convergence theorem, one gets lim uÑ`8 P t hpu, xq " 0, for all x P R. Hence it remains to prove that lim uÑ`8,xÑ´8 P t hpu, xq " lim xÑ´8 P t hpu, xq " 0 for u ě 0. Standard estimates on (1.1) shows that sup xPR Ermax 0ďtďK |X x t ´x|s ă 8, for every K ą 0. Now, for every K ą 0 and x ă 0, one gets Ppτ x ď Kq " Pp max

	x τ x s`t q ˇˇF τ x s	‰ 1 tτ x s ěsu " E " hpu `τ x s`t , X x τ x s`t q ˇˇF s	‰	1 tτ x s "su
						" E "	hpu `s	`τ s,X x s t	, X s`τ s,X x s s,X x s t	q ˇˇF s	‰	1 tτ x s "su
	E " hpu `s `τ s,y t , X s,y s`τ s,y t	q	‰ˇˇy	"X x s	1 tτ x s "su " P t hpu `s, X x s q1 tτ x s "su .
	Putting the above computations together, we obtain
	E " hpu `τ x s`t , X x τ x s`t q ˇˇF τ x s	‰	" hpu `τ x s , Lq1 tτ x s ăsu `Pt hpu `s, X x s q1 tτ x s "su
	0ďtďK	X x t ´x ě L ´xq ď	sup xPR Ermax 0ďtďK |X x t ´x|s L

and

  x t qs " Erhpu `t, x `Xx t ´xq1 t|X x t ´x|ďKu s `Erhpu t, X x t q1 t|X x t ´x|ąKu s. By dominated convergence, for u ě 0, one has lim

	xÑ´8,uÑ`8	Erhpu `t, X x t q1 t|X x t ´x|ďKu s " lim xÑ´8	Erhpu `t, X x t q1 t|X x t ´x|ďKu s " 0
	which combined with
	Erhpu `t, X x t q1 t|X x t ´x|ěKu s ď |h| 8 Pp|X x t ´x| ě Kq ď |h| 8	sup xPR Er|X x t ´x|s K
	yield
		lim sup xÑ´8,uÑ`8	|Erhpu `t, X x t qs| ď |h| 8	sup xPR Er|X x t ´x|s K	.

For exact definitions of these spaces, see Section

2.1.

This inequality will be used at several places throughout the article and we will omit to refer to it.

5.4. Proof of Theorem 4.3. In this section, we will adopt the notation which appears in [START_REF] Whitt | Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues[END_REF]. We will prove that Theorem 4.2 is true under (H2). Similar arguments also gives Theorem 4.1. By Theorem 174 of Kestelman [START_REF] Kestelman | Modern theories of integration[END_REF] p.111, there exists a sequence of continuous functions pb N q N ě1 such that (5.7) Let X N " pX N t q tě0 be the unique weak solution to the following one-dimensional SDE

σpX N s qdW s , t ě 0 and τ N be its first hitting time of the barrier L. Let T ą 0. We first prove that for any h P C 8,0 b pR `ˆRq

We remark that Erhpu `T, X N τ N ^T qs " Erhpu `T, X N T q1 tτ N ąT u s `hpu `T, LqPpτ N ď T q and Erhpu `τ N T, Lqs " Erhpu `τ N , Lq1 tτ N ďT u s `hpu `T, LqPpτ N ą T q so that adding the two decompositions we obtain Erhpu `T, X N τ N ^T qs `Erhpu `τ N ^T, Lqs " Erhpu `τ N ^T, X N τ N ^T qs `hpu `T, Lq. Consequently, we can make use of the following decomposition: Erhpu`τ N ^T, X N τ N ^T qs " Erhpu`T, X N τ N ^T qs`Erhpu`τ N ^T, Lqs´hpu`T, Lq and prove the convergence of both terms: Erhpu `T, X N τ N ^T qs and Erhpu `τ N ^T, Lqs.

'

Step 1: Convergence of pErhpu `T, X N τ N ^T qsq N ě1 . Let Ω " Cpr0, 8q, Rq equipped with the topology of uniform convergence on bounded intervals and X t pwq " wptq. If pP N q N ě1 denotes the sequence of probability measures on Ω induced by the sequence pX N q N ě1 , we know from Theorem 11.3.3 of Stroock and Varadhan [SV79] that pP N q N ě1 converges weakly to the measure P (unique) solution of the martingale problem, induced by X the (unique) weak solution to the SDE with drift coefficient b. Define the mapping g : Ω Ñ Ω by gpwqptq " X τ ^tpwq " X t 1 ttăτ u `L1 ttěτ u . Then, g is discontinuous at w if and only if w leaves rL, 8q after τ without visiting pL, 8q, that is if T L ˝θτ pωq ą 0, where T L is the first hitting time associated to X of the set pL, 8q and θ denotes the shift operator (see Bass [START_REF] Bass | Diffusions and Elliptic Operators[END_REF], p.66 for a similar argument). By the strong Markov property, one has P x pT L ˝θτ ą 0q " E x rP L pT L ą 0qs " P L pT L ą 0q. Since b, σ are bounded on R and a " σ 2 is uniformly elliptic, one has P L pT L ą 0q " 0. Hence, if C g is the set of discontinuities of g one has PpC g q " 0 so that by the continuous mapping theorem: pP N ˝g´1 q N ě1 converges weakly to P ˝g´1 . As a consequence, lim N Ñ`8 Erhpu `T, X N τ N ^T qs " Erhpu `T, X τ ^T qs. This completes the first step of the proof.

'

Step 2: Convergence of pErhpu `τ N ^T, Lqsq N ě1 . Without loss of generality, we assume that the initial condition x 0 and the barrier L satisfy 0 ď x 0 ă L. For the case that L is negative, we consider the hitting time of |L| for the process ´X. Let D u be the subspace of Dr0, 8q (the set of all R-valued functions on r0, 8q that are càdlàg for all t P r0, 8q) that are unbounded above and have non-negative initial value. (Definitions given on p.532, section 13.6 in Whitt [START_REF] Whitt | Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues[END_REF]).

We consider the maps r T , T : Cr0, 8q Ñ Cr0, 8q, where T t pwq :" min ts ą 0 : w s ą tu r T t pwq :" min ts ą 0 : w s ě tu and the map S : w Ñ S ¨pwq " max 0ďsď¨ws . The map S is continuous on Dr0, 8q in the J 1 topology (we refer to section 3.3 in [START_REF] Whitt | Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues[END_REF] for the definition of the J 1 metric). For the definition of the M 2 topology, we refer to p.504 of [START_REF] Whitt | Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues[END_REF]. Our aim is to apply Theorem 13.6.4 [START_REF] Whitt | Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues[END_REF] that we now recall.

Theorem 5.1. (continuity of first-passage-time-functions) Let w P D u that is not equal to z ą 0 throughout the interval pT z pwq ´ε, T z pwqq for any ε ą 0. If w n Ñ w in pD, M 2 q then as n Ñ 8, T z pw n q Ñ T z pwq.

Remark 5.2. Note that the set of paths which does not take the value L through the interval pT L pwq ´ε, T L pwqq for any ε ą 0 is the complement of the set A :" tw : r T L pwq ă T L pwqu " tw : w leaves rL, 8q immediately upon hitting Lu

Here, p D,1,N T px, zq " qz T px, zq, p K,1,N px, tq " f L τ px, tq, and

ŜN,zi si´si`1 pz i`1 , z i q, p K,n,N px, tq " I K,n,N px, tq :"

ŜN,zi si´si`1 pz i`1 , z i q and KL,N and Ŝz,N are the kernels defined by (4.5) and (4.4) with drift coefficient b N instead of b. Moreover, for a fixed N the series defining p K,N px, tq, p D,N T px, zq converge absolutely and uniformly for px, t, zq P R ˆKT ˆR, where K T is any compact subset of p0, T s. Importantly, note that from (5.7), the positive constants C, c ą 1 appearing in (4.12) and (4.13) (with b N instead of b) do not depend on N . Consequently, for all pt, zq P p0, T s ˆp´8, Ls, the following Gaussian upper-bounds hold p K,N px, tq ď Ct ´1{2 gpct, L ´xq and p