Existence and large-time asymptotics for solutions of semilinear parabolic systems with measure data

We study the Cauchy–Dirichlet problem for monotone semilinear uniformly elliptic second-order parabolic systems in divergence form with measure data. We show that under mild integrability conditions on the data, there exists a unique probabilistic solution of the system. We also show that if the operator and the data do not depend on time, then the solution of the parabolic system converges as t → ∞ to the solution of the Dirichlet problem for an associated elliptic system. In fact, we prove some results on the rate of the convergence.


Introduction
Let D ⊂ R d , d ≥ 2 be an open bounded domain. In the present paper, we study systems of the form where σ · σ T = a, X = {(X, P s,x ); (s, x) ∈ R + × D} is a time-inhomogeneous Markov process associated with the operator A t , ζ s is the first exit time of (X, P s,x ) from D, i.e., ζ s = inf{t ≥ s : X t / ∈ D}, (1.7) and A μ is the additive functional of X in the Revuz correspondence with μ. In the last integral in (1.6), B is a standard d-dimensional Brownian motion and ∇ Xū stands for the stochastic gradient ofū (see [14] or Sect. 5). Formula (1.6) can be regarded as a nonlinear Feynman-Kac formula, because taking t = s and integrating it with respect to the measure P s,x , we get whenever the above integrals exist. We would like to stress that our probabilistic solution u to (1.1) may be considered as some generalization of the notion of renormalized (or entropy) solution, because if f u ∈ L 1 (D T ), then u ∈ T 0,1 2 , u ∈ L q (0, T ; W 1,q 0 (D)) for q ∈ [1, d+2 d+1 ) and u is a renormalized (and entropy as well) solution to (1.1) (see Remark 5.14). Perhaps, also the following comment is appropriate at this point, although the probabilistic solution of (1.1) is in general weak and at first glance its definition seems complicated, it is actually very convenient to deal with.
Our results on the existence and uniqueness of solutions of (1.1) generalize known results in the sense that we consider semilinear parabolic systems with measure data (semilinear elliptic systems with measure data are considered in [15,23]). We should also stress that our results are proved for systems with f satisfying quite general condition (1.3) for which the usual monotonicity methods do not apply and we only require f to satisfy mild integrability condition (1.4) analogous to the integrability condition considered for elliptic equations or systems in [2,15,23]. We also allow f to depend on x.
In the second part of the paper, we investigate the asymptotic behavior as t → ∞ of probabilistic solutions of (1. we have for q.e. x ∈ D. As a matter of fact, we prove that there is c depending only on d, such that for every t > 0 and q ∈ (0, 1), 916 T. Klimsiak J. Evol. Equ.
for q.e. x ∈ D, where R is the potential operator of −A on D. From this, it follows in particular that there exists c = c( , d, |D|) such that for every t > 0, (1.11) because it is known that P x (ζ 0 > t) ≤ ae −bt for some a, b > 0 depending only on d, and |D| (the Lebesgue measure of D) and Rμ L 1 ≤ C(d, λ, |D|) μ T V ( μ T V stands for the total variation ofμ). For instance, if f (·, 0) = 0 and μ(dx) = 0, then the rate of convergence in (1.11) is the same as in the classical case of one linear equation (see [8]). We also show that in fact (1.10) holds for every x from the set (1. 12) In case N = 1, the large-time asymptotic behavior of solutions to parabolic equations with measure or L 1 data was investigated in [21,[27][28][29] (in [27], the case of general, possibly singular measures is considered). In all these papers in proofs, some comparison results are used. Therefore, the methods of [21,[27][28][29] cannot be applied to systems considered in the present paper. Let us also point out that these methods do not provide estimates of the difference between solutions to parabolic equations and the corresponding stationary solutions.
Although the main results of the paper concern systems of PDEs and are analytic in nature, the methods of proofs are those of stochastic analysis, Markov processes, and especially the theory of backward stochastic differential equations. Therefore, in Sects. 2-4, we give relevant background material concerning these topics. Then, in Sect. 5, we prove our results on the existence and uniqueness of solutions of (1.1), and in Sect. 6 results on their large-time behavior. Our idea of using the methods of backward stochastic differential equations to the study of large-time behavior of semilinear parabolic equations is new. It seems likely that it can be applied to wider that (1.1) class of equations.

Preliminary results
Let us fix a probability space ( , F, P) equipped with a filtration {F t } satisfying the usual conditions. By B, we denote a standard d-dimensional {F t }-Brownian motion. By A, we denote the set of all {F t } progressively measurable real-valued processes and by V (respectively, V c ) the subspace of A consisting of all increasing càdlàg (respectively, continuous) processes Y such that Y 0 = 0. M is the space of all processes Z ∈ A such that P( T 0 |Z t | 2 dt < ∞) = 1 for every T > 0. M p , p > 0, is the subspace of Vol. 14 (2014) Existence and large-time asymptotics 917 M consisting of all processes such that E( ∞ 0 |Z r | 2 dr ) p/2 < ∞. By D (respectively, S), we denote the space of all càdlàg (respectively, continuous) processes in A, and by D p (respectively, S p ), p > 0, the space of all processes Y ∈ D (respectively, Y ∈ S) such that E sup t≥0 |Y t | p < ∞. We say that a process Y is of class (D) if Y ∈ A and the family {Y τ , τ ∈ T }, where T is the set of all finite {F t }-stopping times, is uniformly integrable. For a càdlàg process Y , we write In the paper, we adopt the following convention. If S is a space of real functions and N , d ∈ N, then by [S] N (respectively, [S] N ×d ), we denote the space of all functions of the form The following multidimensional version of the Itô-Tanaka formula will be frequently used in the paper.
PROPOSITION 2.1. Let X be a progressively measurable process such that Then, there is L ∈ V such that for every p ≥ 1, Proof. The proof is a modification of the proof of [3, Lemma 2.2]. Obviously, it suffices to prove the formula for t ∈ [0, T ]. Set u ε (x) = (|x| 2 + ε 2 ) 1/2 , x ∈ R N , ε > 0. A straightforward computation shows that where I is the n-dimensional identity matrix. By the Itô-Meyer formula, Since 0<t<T | X s | ≤ |K T |, applying the Lebesgue dominated convergence theorem shows that for every t ∈ [0, T ], By what has already been proved, it follows that I ε 3,2 (t) is convergent. Put L t ( p) = lim ε→0 I ε 3,2 (t). Then, L is a càdlàg increasing process, and as in the proof [3, Lemma 2.2], one can show that if p > 1, then L t ( p) = 0 for t ∈ [0, T ], which completes the proof.
This implies that the process Under the assumptions of Proposition 2.1, for every 0 ≤ t ≤ T and p ≥ 1,

Backward stochastic differential equations
Let B denote a standard d-dimensional {F t }-Brownian motion. Let σ be a bounded {F t }-stopping time, ξ be an F σ -measurable random variable, A ∈ V and let f : Let us recall that a pair (Y, Z ) consisting of an R N -valued process Y and an R N ×dvalued processes Z is called a solution of BSDE(ξ, . By putting Y t = ξ, t ≥ σ, Z t = 0, t ≥ σ , we may and will assume in the sequel that the processes Y, Z are defined for t ≥ 0. We also adopt the convention that b a = 0 for a ≥ b. Then, the stochastic equation in (c) is satisfied for every t ≥ 0.
Let us consider the following assumptions.
In [3,Theorem 4.2], it is proved that under (A1)-(A4) with p > 1, there exists a unique solution (Y, Z ) ∈ S p ⊗ M p of BSDE(ξ, σ, f ). We will show how to modify the proof of [3,Theorem 4.2] to get the existence and uniqueness in the general case, i.e., for p ≥ 1 and nonzero process A.
The proof of [ (A) There is μ ∈ R and a nonnegative progressively measurable process For a process A ∈ V, we denote by |A| t its variation on the interval [0, t].
Proof. The proof goes through as for [3, Lemma 3.1], with obvious changes.
Proof. It suffices to repeat, with obvious changes, arguments from the proof of [ Vol. 14 (2014) Existence and large-time asymptotics 921 for some c ≥ 0 and Proof. By Corollary 2.3, Proof. Using standard arguments, one can prove the existence of a unique solution Furthermore, from [26], it follows that under the assumptions of the lemma, there We are now ready to formulate and prove the existence and uniqueness result in case p > 1.
Let us define h n as in the first step of the proof of [3, Theorem 4.2] (with r > a). Then, in much, the same way as in the proof of that theorem, but using Lemma 3.
Step 2. We define ξ n , f n as in the second step of the proof of [3, Theorem 4.2] and set A n We now turn to the case p = 1. We first prove the uniqueness result.
Proof Without the loss of generality, we may assume that By the Itô-Meyer formula and (A2), Taking the conditional expectation with respect to F t on both sides of the above inequality and then letting k → ∞ and using the fact that Y is of class (D), we conclude that |Y t | = 0, t ≥ 0.
For k > 0, let us put Proof. Without the loss of generality, we may assume that μ ≤ 0. Set By the Itô-Meyer formula, for t ≥ 0, we have Vol. 14 (2014) Existence and large-time asymptotics 923 the last inequality being a consequence of monotonicity of f n with respect to y.
Conditioning with respect to F t and using the fact that δY is of class (D), we conclude from the above inequality that To complete the proof, it suffices now to repeat step by step the arguments following Eq. (12) in the proof of [3, Proposition 6.4].

Markov processes and potential theory
To make our exposition in the next sections self-contained, in this section, we recall some useful facts about diffusions associated with the operator A t defined by (1.2), their additive functionals and the Revuz correspondence between these functionals and soft measures.

Time-inhomogeneous diffusions
, and then we define F s,t as the completion of F 0 s,t in F s,∞ with respect to P. Let p denote the fundamental solution for the operator A t defined by (1.2). It is known (see [31]) that there exists a unique time-inhomogeneous Markov process Namely, X is a unique Markov process for which p is the transition density function, i.e., It is known (see [32]) that X admits the so-called strict Fukushima decomposition, i.e., for every (s, where M is a two-parameter martingale additive functional (MAF) of X of finite energy and A is a two-parameter continuous additive functional (CAF) of X of zero energy. Moreover, which implies in particular that the process where σ · σ T = a is a Brownian motion under P s,x . It is also known (see [20]) that B s,· is an {F s,t } t≥s -Brownian motion.

Time-homogeneous diffusions
In what follows we will also make substantial use of time-homogeneous Markov process X associated with the operator ∂ ∂t + A t (see [24]). A brief sketch of a useful construction of X on an extension of is given below.
We set and consider the process X on defined as It is known (see [25]) that X admits the strict Fukushima decomposition, i.e., for every (s, x) ∈ R + × R d , where A is a CAF of X of zero energy and M is a MAF of X of finite energy. It is also known (see [19]) that and for t ≥ 0 define τ (t) : → R + by putting for ω = (s, ω). From now on, we adopt the convention that if ξ is a random variable on , then ξ(ω ) = ξ(ω) for ω = (s, ω) ∈ . With this convention Vol. 14 (2014) Existence and large-time asymptotics 925 We put By (4.1) and (4.6), for every (s, x) ∈ R + × R d . Therefore, B is a Brownian motion under P s,x for every (s, x) ∈ R + × R d . In fact, by [20], it is an {F t } t≥0 -Brownian motion.

Capacity, soft measures, and quasi-continuity
Let W be the space of all u ∈ L 2 (R + ; H 1 0 (D)) such that ∂u ∂t ∈ L 2 (R + ; H −1 (D)) endowed with the usual norm u W = u L 2 (R + ;H 1 0 (D)) + ∂u ∂t L 2 (R + ;H −1 (D)) . We define the parabolic capacity of an open set U ⊂ R + × D as From now on, we say that some property is satisfied for quasi-every (q.e. for short) x ∈ D (respectively, (s, x) ∈ R + × D) if it is satisfied except for some Borel subset of D (respectively, R + × D) of cap N (respectively, cap) capacity zero.
Let μ be a Radon measure on D (respectively, R + × D). Following Let B be a Borel subset of R + × D (respectively, D) and u : B → R be a Borel measurable function. We say that u is quasi-continuous if for every ε > 0, there exists a closed set F ε ⊂ B such that cap(B \ F ε ) < ε (respectively, cap N (B \ F ε ) < ε) such that u |F ε is continuous. In the paper, we shall mostly work with functions on B = D or B = D T . We adopt the convention that for u defined on B = D ( respectively, is continuous under P s,x (respectively, P 0,x ) for q.e. (s, x) ∈ D T (respectively, q.e. x ∈ D). We will also consider quasi-càdlàg functions on D T , i.e., Borel functions u on D T , such that the process [0, ζ τ ] t → u(X t ) is càdlàg for q.e. (s, x) ∈ D T .

Additive functionals and soft measures
Let E s,x (respectively, E s,x ) denote the expectation with respect to P s,x (respectively, P s,x ) and let m 1 denote the Lebesgue measure on R + × R d . Let ζ 0 be defined by (1.7) with s = 0, i.e., Let us recall that a positive AF A of X and a positive soft measure μ on R + × D are in the Revuz correspondence if (4.8) for every f ∈ B + (R + × D). If μ, 1 < ∞, then A is called integrable. It is known (see [22,30]) that under (4.8), the family of all integrable positive AFs of X and the family of all bounded positive soft measures on R + × D are in one-to-one correspondence. In what follows the additive functional corresponding to a positive bounded soft measure μ will be denoted by A μ .
Let p D denote the transition density of the process X killed on exiting R + × D. It is known that A μ corresponds to μ iff for q.e. (s, x) ∈ R + × D, for every f ∈ B + (R + × D) (see [22]). Suppose now that the coefficients of the operator (1.2) do not depend on time, i.e., a i j (t, x) = a i j (x) for (t, x) ∈ D T . We then set It is well known that the fundamental solution for A has the property that p(s, x, t, y) = p(t − s, x, y) for any t > s, x = y. Consequently, where P x = P 0,x is a time-homogeneous Markov process with the transition density p(t, x, y) = p(0, x, t, y). It is also known (see, e.g., [9]) that in the time-homogeneous Vol. 14 (2014) Existence and large-time asymptotics 927 case, a positive CAF A of X and a positive soft measure μ on D are in the Revuz correspondence if for every f ∈ B + (D), where E x denotes the expectation with respect to P x . If μ, 1 < ∞, then A is called integrable. In [9], it is proved that the family of all integrable positive CAFs of X and the family of all bounded positive soft measures on D are in one-to-one correspondence via formula (4.11). Let A μ denote the additive functional corresponding to a positive bounded soft measure μ, p D denote the transition density of the process X killed on exiting D, and let G D (x, y) = ∞ 0 p D (t, x, y) dt be the Green function for D. Then, for q.e. x ∈ D, (4.12) for every f ∈ B + (D). In the whole paper for a fixed Borel positive measure μ on D T (respectively, D), we denote Finally, let us recall that there is c depending only on d, such that p(t, x, y) ≤ Ct −d/2 for t > 0, x, y ∈ R d (see, e.g., [1]). Therefore, by [ where |D| denotes the Lebesgue measure of D, whereas from Corollary to Proposition 1.18 in [5], it follows that there exists constants a > 0, b > 0 depending only on d, and |D| such that for every t > 0, (4.14)

Markov-type BSDEs and PDEs
Let us fix T > 0 and set D T = [0, T ] × D. In this section, we show existence and uniqueness results for systems of PDEs of the form where A t is given by (1.2). Let f : D T × R N → R N . We consider the following hypotheses. (H1) f (·, ·, y) is measurable for every y ∈ R N and f (t, x, ·) is continuous for a.e. t, x, y ), y − y ≤ α|y − y | 2 for every y, y ∈ R N and a.e. (t, REMARK 5.1. It is known (see [17,Proposition 3.6]) that if f ∈ L 1 (D T ), then R 0,T | f | < ∞, m 1 -a.e. Therefore, (H4) is satisfied if sup |y|≤r | f (·, ·, y)| ∈ L 1 (D T ) for every y ∈ R N and r ≥ 0. However, the class of functions f ∈ B(D T ) such that R 0,T | f | < ∞, m 1 -a.e. is wider than L 1 (D T ). It includes in particular the space [17,Example 5.2]).
It is known that in case N = 1, one can find solutions of problems of the form (5.1) in the (nonlinear) space T 0,1 2 of all Borel measurable functions u on D T such that T k (u) ∈ L 2 (0, T ; H 1 0 (D)) for every k ≥ 1 (see [2,6]). In the case of systems, the problem is more difficult, because we do not know whether the solutions or its truncation have gradients in the usual sense (i.e., locally in some Sobolev space). This is related to the lack of integrability of f u . The same problem appears in the case of elliptic systems. In the scalar case, it is known that a solution of (5.1) belongs to W 1,q 0 (D) for every q ∈ [1, d d−1 ) (see [34]) but in case N > 1, the problem whether a solution belongs to W 1,q 0 (D) for some q ≥ 1 is open. To overcome the difficulty in [15], the existence and uniqueness of elliptic systems with measure data similar to (5.1) is proved in some wider than T 0,1 2 (see Corollary 5.6) linear space. The space introduced in [15] makes essential use of the Markov process X associated with the operator A defined by (4.10) and therefore may be called a stochastic Sobolev space. In what follows we extend the ideas from [15] to parabolic systems. We begin with the definition of the stochastic Sobolev space of functions depending on time.
Let W 0,1 (X D T ) denote the set of all u ∈ F M (definition below) for which there exists a sequence {u n } ⊂ C ∞ c (D T ) such that for q.e. (s, x) ∈ D T , In [14], it is proved that for every u ∈ Given u ∈ W 0,1 (X D T ), we denote by ∇ X u the unique function v satisfying (5.4). Notice that directly from the construction of ∇ X u, it follows that ∇ X u = ∇u a.e. if u ∈ L 2 (0, T ; H 1 0 (D)). By F M, we denote the space of Borel measurable functions u on D T such that for q. e. (s, x)

REMARK 5.2. (i) Let O(→) be the topology generated by the convergence in
. This topology is metrizable by the F-norm Indeed, by the Kantorovich-Kisyński theorem (see [11,12]), Assume that u n → u in O(→) and (u n ) does not converge to u in | · | 1 . Then, there exists ε > 0 and subsequence (n k ) ⊂ (n) such that On the other hand, by (5.5), there exists a subsequence (n k l ) ⊂ (n k ) such that u n k l → u in W 0,1 (X D T ). Hence, by the Lebesgue dominated convergence theorem, |u n k l − u| 1 → 0 as l → ∞, which contradicts (5.6). Now, assume that |u n − u| 1 → 0 as n → ∞ and let (n k ) ⊂ (n). By [16,Proposition 3.3], there exists a subsequence (n k l ) ⊂ (n k ) such that u n k l → u in W 0,1 (X D T ). Since (n k ) ⊂ (n) was arbitrary, (5.5) implies that u n → u in O(→). (ii) By [14,Proposition 4.6], the space (W 0,1 (X D T ), O(→)) is complete.
Proof. Since u ∈ W 0,1 (X D T ), there exists a sequence {η n } ⊂ C c (D T ) such that η n → u and ∇η n → ∇ X u in F M. Using the assumptions on θ , one can easily show that θ(η n ) → θ(u) and ∇θ(η n ) = θ (η n )∇η n → θ (u)∇ X u in F M, which proves the desired result. LEMMA 5.4. Let k ∈ R and u ∈ W 0,1 (X D T ). Then, u ∧ k, u ∨ k ∈ W 0,1 (X D T ) and Proof. We will prove the lemma for u ∧ k. The proof for u ∨ k is analogous. Set If we repeat the above arguments with σ n replaced byσ n defined aŝ Let F B denote the space of Borel measurable functions on D T such that for q.e. (s, x) ∈ D T , P s,x (ess sup r ∈[0,ζ τ ] |u(X r )| < ∞) = 1. Observe that every quasi-càdlàg function belongs to F B.
One of the reason why the space T 0,1 2 has been introduced is that the standard Sobolev space L 2 (0, T ; H 1 0 (D)) lacks the property that u ∈ L 2 (0, T ; H 1 0 (D)) if u is quasi-càdlàg (natural class for solutions of equations with measure data) and T k (u) ∈ L 2 (0, T ; H 1 0 (D)) for every k ≥ 0. The following lemma shows the stochastic Sobolev space has this remarkable feature. LEMMA 5.5. If u ∈ F B and T k (u) ∈ W 0,1 (X D T ) for every k ≥ 0, then u ∈ W 0,1 (X D T ).
Proof. First, observe that for q.e. (s, x) ∈ D T and ε > 0, which shows that T k (u) → u in F M. By Lemma 5.4, for k < l, we have Observe that |u(X r )| ≥ k). Vol. 14 (2014) Existence and large-time asymptotics 931 By the assumption that u ∈ F B, the right-hand side of the above inequality tends to zero as k → ∞, which shows that In [14], it is shown by examples that in general neither T 0,1 2 ⊂ W 0,1 (X D T ) nor T 0,1 2 ⊃ W 0,1 (X D T ). However, we have the following corollary to Lemma 5.5. COROLLARY 5.6. If u ∈ T 0,1 2 and u ∈ F B, then u ∈ W 0,1 (X D T ).
Besides being nonlinear, another drawback to the space T 0,1 2 is that it is sometimes too small in practice. For instance, in [14], it is proved that solutions of some types of the obstacle problem are quasi-continuous and belong to W 0,1 (X D T ) but do not belong to T 0,1 2 .
for every α ∈ R. If α ≥ 2, then v is not locally integrable, so there is no sense to speak about its distributional derivative. But let us observe that v ∈ T 0,1 2 for every α ∈ R.
(ii) Put u(t, x) = sin v(t, x), (t, x) ∈ D T . Then, by Lemma 5.3, u ∈ W 0,1 (X D T ) and for every α ∈ R. One can check that for every ε > 0 and α > 1, which shows that if α > 1, then u / ∈ T 0,1 2 since T k (u) = u for k ≥ 1. However, u ∈ L 1 (D T ) and one can check that its distributional derivative is given by the formula i.e., for every η ∈ C ∞ 0 (D), Accordingly, even if the distributional derivative of u ∈ W 0,1 (X D T ) exists, it is not a function in general.
Following [18], we adopt the following definition.
DEFINITION. We say that a measurable function f : D T → R is quasi-integrable if the function [0, ζ τ ] t → f (X t ) belongs to L 1 ([0, ζ τ ]) P s,x -a.s. for q.e. (s, x) ∈ D T . The set of all quasi-integrable functions on D T will be denoted by q L 1 (D T ).
We say that a measurable function u on D T is of class (FD) if for q.e. (s, x) ∈ D T , the process u(X) on [0, ζ τ ] is of class (D) under the measure P s,x . REMARK 5.8. It is known (see [19]) that if f is quasi-integrable in the analytic sense, i.e., if for every ε > 0, there exists an open set G ε ⊂ D T such that cap(G ε ) < ε and DEFINITION. We say that a measurable function u : In what follows for a given function u on D T , we set and given μ ∈ M 0,b (D T ), we denote byμ the unique measure in M 0,b (D T ) such that DEFINITION. We say that u is a solution of (1.1) on [0, T ] ifū is a solution of (5.1) with f u replaced byf u , μ replaced byμ and a replaced byā. Observe that from the above two definitions, it follows that if u is a solution of (5.1), then for q.e. (s, is a solution of BSDE s, x (ϕ, D, f +dμ). In the rest of this section, we are going to prove that under (H1)-(H4), this statement can be reversed in the following sense. For q.e. (s, x) ∈ D T , there exists a unique solution (Y s,x , Z s,x ) of BSDE s,x (ϕ, D, f +dμ) and if we set u(s, x) = E s,x Y s,x 0 for (s, x) ∈ D T , then u is a solution of (5.1). Moreover, Let us first state the following corollary to Theorem 3.7.
The crucial issue in the proof of representation (5.8) and existence of solution to system (5.1) is regularity of the function D T (s, x) → u(s, x) = E s,x Y s,x 0 . In most papers concerning probabilistic solutions for PDEs or stochastic representation for solutions of PDEs, the regularity is proved by using the results of probabilistic potential theory which may be applied when u(s, x) = E s,x ζ τ 0 d A r for some AF of X . Here, such approach cannot be applied because in general, u does not admit the last representation [in general, integrals on the right-hand side of (1.8) do not exists]. We cope with the problem of regularity of u in Propositions 5.10, 5.11. PROPOSITION 5.10. Let F be a Borel subset of D T such that cap(D T \ F) = 0. Assume that for every (s, x) ∈ F, the real process Y s,x is a continuous semimartingale under P s,x such that Y s,x t∨ζ = 0, t ≥ 0, and there exists a Borel function v on D T such that for every (s, x) ∈ F and every t ∈ [0, where L k (respectively, L −k ) is the local time of the process u(X) at k (respectively, −k). By the first step of the proof, T k (u) ∈ W 0,1 (X D T ) for every k ≥ 0. Since u is quasi-continuous, u ∈ F B and hence, by Lemma 5.5, u ∈ W 0,1 (X D T ). By the first step of the proof, Hence, by Lemma 5.4, for every k ≥ 0, which implies that The rest of the proof runs as in the first step. Proof. By Proposition 5.9, for every (s, x) ∈ F, there exists a solution (Y s,x , Z s,x ) of BSDE s,x (ϕ, D, f + dμ) such that (Y s,x , Z s,x ) ∈ D q ⊗ M q for q ∈ (0, 1) and Y s,x is of class (D). By [19,Proposition 3.5], u(X t ) = Y s,x t , P s,x -a.s. for every (s, x) ∈ F and every t ∈ [0, T τ ], where u(s, x) = E s,x Y s,x 0 . Our aim is to show that u is quasi-càdlàg, belongs to W 0,1 (X D T ) and representation (5.8) holds q.e. Note that we cannot apply directly Propositions 5.10, 5.11 because we do not know that u(X) is continuous. To overcome the difficulty, let us consider a solution (Y 1,s,x , Z 1,s,x ) ∈ D q ⊗ M q , q ∈ (0, 1), of the BSDE s,x (ϕ, D, μ) such that Y 1,s,x is of class (D). By [19,Proposition 3.7], u 1 (s, x) = E s,x Y 1,s,x 0 is quasi-càdlàg, u 1 ∈ T 0,1 2 and for every (s, x) ∈ F, u 1 (X t ) = Y 1,s,x t , t ∈ [0, ζ τ ], P s,x -a.s., σ ∇u 1 (X) = Z 1,s,x , dt ⊗ P s,x -a.e. It follows in particular that u is quasi-càdlàg. Since u 1 is quasi-càdlàg, it belongs to F B. Consequently, by Corollary 5.6, u 1 ∈ W 0,1 (X D T ). Therefore, from (5.18) and Proposition 5.11 applied to each coordinate of the function v, it follows that u ∈ W 0,1 (X D T ) and σ ∇ X u(X) = Z s,x , dt ⊗ P s,x -a.e.
for every (s, x) ∈ F. Thus, u is a solution of (5.1). Uniqueness follows from Theorem 3.6.
By [3, Lemma 6.1], for every q ∈ (0, 1), for every q ∈ (0, 1). Since q ∈ (0, 1), we have By the above, D T |∇ X u(r, y)| q p D (s, x, r, y) dr dy = E s,x ζ τ 0 |∇ X u(X r )| q dr < ∞ for q.e. (s, x) ∈ D T , where p D is the transition density of the process X killed on exiting D. Therefore, the desired result follows from the well-known fact that p D (s, x, ·, ·) is continuous and strictly positive on (s, T ] × D (see [1]).