Backward Stochastic Differential Equations and Dirichlet Problems of Semilinear Elliptic Operators with Singular Coefficients

In this paper, we prove that there exists a unique solution to the Dirichlet boundary value problem for a general class of semilinear second order elliptic differential operators which do not necessarily have the maximum principle and are non-symmetric in general. Our method is probabilistic. It turns out that we need to solve a class of backward stochastic differential equations with singular coefficients, which is of independent interest itself. The theory of Dirichlet forms also plays an important role.


Introduction
In this paper, we will use probabilistic methods to solve the Dirichlet boundary value problem for the semilinear second order elliptic partial differential equations (PDEs) of the following form: where D is a bounded domain in R d , f (·, ·) is a nonlinear function and ϕ ∈ C(∂D). The operator A is given by where a(x) = (a ij (x)) 1≤i,j ≤d is a Borel measurable, (not necessarily symmetric) matrixvalued function on R d satisfying and There is a big literature regarding the probabilistic approaches of solving boundary value problems. The pioneering work is traced back to Kukutani [12] who used Brownian motion to represent the solution of the classical Dirichlet problem for Laplacian operators. In the linear case, i.e. f (·, ·) = 0, ifb = 0, q ≤ 0, the problem (1.1) is solved by the Feynman-Kac formula 0 q(X(s))ds ϕ(X τ D )], where X = ( , F, F t , X t , P x , x ∈ R d ) is the diffusion process associated with the generator L 1 given by (1.4) and τ D is the first exit time of X from D. We refer the readers to Chen and Zhao [8] for details. The first result on probabilistic interpretation of viscosity solutions of semilinear parabolic PDEs was obtained by Peng in [17] and [16], through the backward stochastic differential equations, which is different from the weak solution considered in this paper.
Sinceb is merely measurable, some of the coefficients of the operator A are very singular (just distributions). Whenb = 0, f (·, ·) = 0 and the matrix a(x) is symmetric, the boundary value problem (1.1) was considered by Zhang [18]. The time reversal of symmetric Markov processes and the backward stochastic differential equations (BSDEs) with random terminal time play an important role in [18]. When a is non-symmetric and f (·, ·) = 0 (i.e. the linear case), Chen, Sun and Zhang [5] obtained a probabilistic representation of the solution to problem (1.1) under the condition that q − divb ≤ g (1.5) in the sense of distribution for a sufficient small non-negative function g ∈ L p (D).
In this paper, we consider the weak/Soblev solution (see the definition in the next section) of the semilinear problem (1.1). We will show that there exists a unique, continuous weak solution to the Dirichlet value problem (1.1) under appropriate conditions. We will use the h-transform method to tackle the singular term "div(b·)". To solve the semilinear boundary value problem (1.1), we first produce a candidate to the solution by appealing to the theory of BSDEs. More precisely, we will solve a class of BSDEs with very singular coefficients and random terminal time. The BSDEs are driven by the martingale part of a diffusion process. The study of this class of BSDEs is of independent interest. It turns out that the classical L 2 setting of BSDEs is not suitable here. We have to work in the framework of L 1 and deal with class D stochastic processes. We refer the readers to the nice article [4] for related results.
The rest of the paper is organized as follows. In Section 2, we set up the precise framework. Section 3 is devoted to establishing the existence and uniqueness of solutions of a backward stochastic differential equation with singular coefficients. The boundary value problem is solved in Section 4.

Framework
We assume that d ≥ 3. Hereafter we fix p > d 2 . We will use ·, · and (·, ·) to denote respectively the inner product of the Euclidean space R d and the inner product of the L 2 (D)-space. || · || ∞ denotes the L ∞ normal. Recall that W 1,2 (D) is the Sobolev space on D with the norm ||u|| W 1,2 (D) = ( D ∇u, ∇u dx + (u, u)) 1 2 and that W 1,2 0 (D) denotes the completion of C ∞ 0 (D) under the same norm || · || W 1,2 (D) . The operator A introduced in Section 1 is rigorously determined by the quadratic form (Q, D(Q)) with D(Q) = W 1,2 (D) and for u, v ∈ W 1,2 (D) Now we need to recall the notion of VMO functions in order to apply the W 1,p estimates for the divergence operators. A locally integrable function g is said to be in the where B r denotes a ball of radius r and .
be the diffusion process generated by the infinitesimal operator L 0 . It is well-known that (X t , P 0 x ) is a conservative Feller process on R d that has continuous transition density function which admits a twosided Aronsons heat kernel estimate (see Aronson [1,2]). Moreover, by Theorem IV.2.5 in [14], for quasi-everywhere (q.e.) x ∈ R d we have the Fukushima's decomposition where M 0 t is a martingale additive functional (MAF) and N 0 t is a continuous additive functional (CAF) of locally zero energy. However the decomposition (2.1) can be strengthened to hold for every x ∈ R d . This follows from the existence of the heat kernel, Theorem 3.5.4 in [15] and Theorem 2 in [11]. Let v be the function stated in Remark 2.1. By Sobolev where M v t is a MAF and N v t is a CAF of zero energy.
Assume that f(x,y) is a measurable function, which is continuous w.r.t. y and satisfies for some constant C and J, J 1 ∈ L p (D). Finally set

BSDEs with Singular Coefficients
In this section, we will obtain the existence and uniqueness of solutions of a class of BSDEs with singular coefficients and random terminal time. The results is of independent interest on its own right and will be used in the subsequent sections. Recall that X = ( , F, F t , X t , P x , x ∈ R d ) denotes the diffusion process associated with the generator L 1 given by where F t is the completed minimal admissible filtration generated by X t . Similar to (2.1) we have the following Fukushima's decomposition: for any x ∈ R d is a MAF of finite energy and N t is CAF of locally zero quadratic variation. Moreover, ij (X(s))ds.
Given a h ∈ L p (D). Let F be a Borel measurable function on D satisfying for some constant C > 0. Then similar to Theorem 3.18 and Theorem 4.6 in [9], we have the following result.
Proof Let G D (x, y) denote the Green function on the domain D associated with the operator L 1 . It is known from (2.14) in [5] that So h belongs to the class of functions K 1 defined in [6]. It thus follows from Theorem 2.2 in [6] and (3.4) that where C 1 is a positive constant. Moreover by (3.6) inf On the other hand, it was shown in [1,2] that the transition density function p(t, x, y) of (X t , P x , x ∈ R d ) has a Gaussian upper bound estimate: for some constants σ 1 , σ 2 > 0. This bound together with h ∈ L p (D) implies that sup x∈D E x [ δ 0 |h(X(s))|ds] < 1 for some δ > 0. Hence similar to the proof of Theorem 4.6 in [9] we can show that The remainder of the proof of (3.5) is the same as that of Theorem 3.18 in [9]. we omit the details.
Fix a probability measure μ on R d and let P = P μ denote the probability law of the diffusion process X starting with the initial distribution μ, namely, P (·) = R d P x (·)μ(dx). We now consider BSDEs driven by the martingale part M in (3.2) with random terminal time on the probability space ( , F, F t , P ). Denote by E the expectation under P .
Let g(t, y, ω) : [0, ∞] × R × → R be a progressively measurable function. For notational convenience, we omit the random parameter ω. Given a finite stopping time τ and a random variable ξ ∈ F τ . Let us recall the definition of a solution of the BSDEs with random terminal time.
if Y is a R-valued progressively measurable process and Z is a R d -valued predictable process such that: To prove the existence and uniqueness of a solution to the BSDE (3.10), we need the following lemma which can be found in [4].
Suppose that g is continuous with respect to the variable y and satisfies where K 1 (t), K 2 (t) ≥ 0, K 3 (t) ≥ 0 are progressively measurable stochastic processes. For any 0 < β < 1, denote by ϕ β the set of the real-valued, adapted, continuous processes Let denote the set of the stopping times. We say that a progressively measurable pro- Then it is known that the space of adapted continuous processes which belong to class (D) is complete under || · || 1 norm, see Chapter VI 21 in [10]. Here is the main result of this section.

Theorem 3.2 Assume
and Then it is easy to see that (Y, Z) is a solution to the BSDE (3.10) if and only if (Ỹ ,Z) satisfies the following BSDE: Therefore, we only need to establish the existence and uniqueness of the BSDE (3.16). Note thatg(t, y) is also continuous with respect to y and Moreover for any n ∈ N, where the first inequality is due to the fact K 1 (s) = K 2 (s) = 0 when s ≥ τ . By (3.11) and (3.15), for any r > 0, we have (3.18) Note that the martingale representation theorem with respect to the martingale part M is valid according to Theorem 2.1 in [18]. Thus, by (3.17)-(3.18) and Proposition 6.4 in [4] for any n ∈ N, there exists a solution pair (Ỹ N n ,Z N n ) to the following BSDẼ (3.19) and moreoverỸ N n ∈ Class (D), and δg n (s, y) = I s≤τ [g(s,Ỹ N n (s) + y) −g(s,Ỹ N n (s)) +g(s,Ỹ N n (n))I s≥n ]. Then (y 1 − y 2 )(δg n (t, y 1 ) − δg(t, y 2 )) ≤ 0. By (3.19) we have for any t ≥ 0 Take the conditional expectation with respect to F t to obtain here we have used the fact that Letting k → ∞ in (3.21) and noticing δg n (s, 0) = 0 for s > τ, it follows from (3.22) and (3.24) that Hence, as n → ∞. By (3.11), we have and δg(t, y) =g(t, y + Y N (t)) −g(t,Ỹ N (t)). Then by (3.17) y · δg(t, y) = y(g(t, y +Ỹ N (t)) −g(t,Ỹ N (t))) ≤ 0.
is also in Class (D), using Lemma 2.2 and the similar arguments as in the proof of (3.25), we obtain This together with (3.14) imply as N → ∞. Hence there exists a continuous processỸ ∈ Class (D) such that lim N→∞ ||Ỹ N −Ỹ || 1 = 0. For any 0 < β < 1, using Lemma 6.1 in [4] and (3.34) we get (3.38) (3.38), (3.14) and (3.35)imply that Letting N → ∞ in (3.31), it is easy to see that (Ỹ ,Z) is a solution to the BSDE (3.16). Now we proceed to prove the uniqueness of the BSDE (3.16). Suppose (Ỹ 1 ,Z 1 ) is another solution to (3.16) such thatỸ 1 ∈ Class (D). Let δỸ (t) = Y (t) −Ỹ 1 (t), δZ(t) =Z(t) −Z 1 (t). Choose a sequence of stopping times {T k } k≥1 that increases to ∞ and such that Taking the conditional expectation with respect to F t on both sides, we obtain Let f (x, y) : D × R → R be a given Borel measurable function. Assume that f is continuous w.r.t. the variable y and satisfies where J 1 (x), J 2 (x) ∈ L p (D) and C is a constant. We have the following important corollary:

Corollary 3.3 Assume there exists a x 0 ∈ D such that
Then for any F τ D -measurable bounded r.v. ξ and x ∈ D, the following two statements hold: (i) There exists a unique pair (Y x , Z x ) satisfying the following BSDE: (ii) There exists at most one solution (Y x , Z x ) satisfying the BSDE (3.41) such that Y x is a bounded process. Remark 3.1 Please be ware of the subtle difference of (i) and (ii). We do not assume The statement (ii) is needed in the next section. X(s))| + J 2 (X(s)))ds] < ∞. By Lemma 3.1 and (3.40) we see that all the conditions in Theorem 3.2 are satisfied. Hence for any x ∈ D, there exists a unique solution (Y x , Z x ) to (3.41) such that e − t 0 J 1 (X(s))ds Y x (t) ∈ Class (D). Next we prove the last statement of the Corollary (i). SetỸ (X(t))y+ e − t 0 J 1 (X(s))ds f (X(t), e t 0 J 1 (X(s))ds y),ξ = e − τ 0 J 1 (X(s))ds ξ as in Theorem 2.1. It follows from (3.41) thatỸ Taking the conditional expectation with respect to F t on both sides of the above inequality, we have (3.44) In particular, we have (ii) Suppose that (Y x , Z x ) and (Y x , Z x ) are solutions to BSDE (3.41) such that Y x and Y x are both bounded processes. Since J 1 ∈ L p (D), there exists constant α > 0, γ > 1 such that ||γ (−J 1 − α) + || L p (D) < 1 σ where σ is defined in (3.6). Then by (3.6) and Kahamiskii's inequality, Moreover by (3.7) and (3.8) (3.45) for some positive constants γ 1 , and Y x (s) = e s∧τ D 0 Choose a sequence of stopping times {τ n } n≥1 that increases to ∞ and such that Note that  Taking the expectation on the both sides of (3.47) we get Notice that lim n→∞ |δY x (τ n ∧ τ D )| = e τ D 0 (−J 1 −α)(X(s))ds |ξ − ξ | = 0, P x -a.e.. Using the uniformly integrability of {δY x (t ∧ τ n ∧ τ D )} n≥1 and {δY x (τ n ∧ τ D )} n≥1 , letting n → ∞ in (3.48) we find

By Lemma 3.2 and (3.46), it holds that
Furthermore we deduce that t 0 δZ x (s), dM s = 0 and hence Z x = Z x . The uniqueness is proved.

Linear Case
Consider the following second order elliptic operator The quadratic form generated by L 2 is Since |b| 2 , q ∈ L p for p > d 2 , it is known that (E q , D(E q )) is a coercive closed form and there exist a α 0 > 0 such that ∂u ∂x j dx and E q α 0 (u, u) = E q (u, u)+α 0 (u, u). By Theorem 3.2 in [13] or Lemma 4.1 in [7], for sufficient large α > 0, the α-resolvent operator of (E q , D(E q )) is given by where X = (X t , P x , x ∈ D) is the diffusion associated with the operator L 1 in (1.4). Let F be a measurable function satisfying for some constant C. Take ϕ ∈ C(∂D) and consider the boundary value problem We have the following result: q(X(s))ds)] < ∞.

q(X(s))ds F (X(t))dt]. (4.5)
Proof It is known from Lemma 2.1 in [5] that u 1 (x) = E x [ϕ(X(τ D ))] is the unique continuous weak solution of the problem (X(t))dt]. By Lemma 3.1 and (4.3) u 2 is well defined. Let u(x) be defined as (4.5). Then u = u 1 + u 2 . Hence, to prove that u is a continuous weak solution to (4.4) we only need show that u 2 is a continuous weak solution of the following problem: Since u 1 is bounded, by (4.3) we have |F 1 (x)| ≤ c 1 (1 + |q(x)|) for some c 1 > 0. By Lemma 5.7 in [8] and Lemma 2.1 in [5], we know that the semigroup generated by L 2 is strong Feller. Hence, using the similar proof as that of Theorem 3.18 in [9] we obtain that u 2 ∈ C(D).
Next we show that u 2 ∈ W 1,2 0 (D). Since q ∈ L p for p > d 2 and since the transition density function of (X t , P x , x ∈ R d ) has the Gaussian upper bound estimate by (3.9), we can easily deduce that αG q α is strongly continuous on L p .

Semilinear Case
Recall that M 0 is the martingale part of the diffusion process (X t , P 0 x ) generated by L 0 . Define the exponential martingale

b(X(s)) ds).
Then we have dP x dP 0 x Let g(x, y) : D × R → R be a measurable function such that g is continuous w.r.t. y and satisfies for some constant C and K, K 1 ∈ L p (D).
Consider the following semilinear boundary value problem: Proof Set g 1 (x, y) = g(x, y) + q(x)y. By (4.10) and Corollary 3.3 for any x ∈ D, there exists a unique solution (Y x , Z x ) to the following BSDE: . By the strong Markov property of X and the uniqueness of the BSDE (4.11), it is easy to see that P x -a.e. (4.12) Consider the following linear problem: Since u 0 is bounded by Corollary 3.3, by Theorem 4.1 the above equation admits a unique continuous weak solution u ∈ W 1,2 (D). On the other hand, by the Fukushima's decomposition we have for q.e. x ∈ D, P x -a.e. u(X(s)), dM s are P x -uniformly integrable martingales. It follows from (4.11) and (4.13) that Taking the conditional expectations respect to F t on both sides of the above equations, we get (4.14) Since u ∈ C(D) and since Y x (t) is bounded (see (4.12)), let T n → ∞ in (4.14) to obtain Hence u is a weak solution to (4.9). Now we proceed to prove the uniqueness of the boundary value problem (4.9). Suppose u 1 is another continuous weak solution of (4.9). Then by Fukushima's decomposition (u 1 (X(t)), u 1 (X(t))) is also a solution of BSDE (4.11) for q.e. x ∈ D. Since u(X(t)) and u 1 (X(t)) are bounded, by Corollary 3.3(ii) we have u 1 (X(t)) = u(X(t)). In particular, 0))|] = 0 for q.e. x ∈ D. By the continuity of u 1 and u, it holds for every x ∈ D. The uniqueness is proved.
Let f (x, y) : D × R → R be a measurable function that is continuous w.r.t. y and satisfies for some constant C and J, J 1 ∈ L p (D). Consider the following semilinear elliptic PDEs: Now we can state the main result in this section.
where E 0 x 0 stands for the expectation under P 0 x 0 , N v t is the zero-energy part of the Fukushima's decomposition defined in (2.2). Then there exists a unique continuous weak solution to problem (4.16). Moreover, if f (x, y) = F (x), the solution is given by Proof Introduce the following operator on L 2 (D): Set h = e v and consider the following semi-linear elliptic PDE: Note that: (4.20) Thus condition (4.17) is equivalent to (4.21) Since h(x) > 0, (4.15) implies

Example 4.3
If q − J − divb ≤ g for some g ∈ L 1 (D) in the sense of distribution and there exists a x 0 ∈ D such that Then condition (4.17) in Theorem 4.2 is satisfied.
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