Stochastic partial integral-differential equations with divergence terms*

We study a class of stochastic partial integral-differential equations with an asymmetrical non-local operator 1 2 ∆+a∆ α 2 +b ·∇ and a distribution expressed as divergence of a measurable field. For 0 < α < 2, the existence and uniqueness of solution is proved by analytical method, and a probabilistic interpretation, similar to the Feynman-Kac formula, is presented for 0 < α < 1. The method of backward doubly stochastic differential equations is also extended in this work.


Introduction
We consider the following stochastic partial integral-differential equation in this article For constants a > 0 and 0 < α < 2, the non-local operator A = 1 2 ∆ + a α ∆ α 2 + b · ∇ is a non-symmetric infinitesimal generator of a Markov process with jumps. Coefficients f , g = (g 1 , · · · , g d ) and h = h 1 , · · · , h d 1 are non-linear random functions. The differential term with ← − dB t refers to a backward stochastic integral with respect to a d 1 -dimensional Brownian motion on probability space Ω , F B , P , (B t ) t≥0 , so that the doubly stochastic framework introduced by Pardoux and Peng [15] could be applied.
This article is devoted to extending the current methods of stochastic partial integraldifferential equations (PIDEs for short) and backward doubly stochastic differential equations (BDSDEs for short) driven by some Lévy processes, while a singular term divg is involved in the equation which is understood in distributional sense. Throughout this paper a classic Sobolev weak solution is considered. The contribution of this study is threefold: types of forward-backward martingale decomposition and Fukushima decomposition, which correspond to an asymmetric Markov process with jumps, are presented respectively; a stochastic representation, similar to Feymann-Kac formula, for the solution of stochastic PIDE (1.1) is obtained; a connection between stochastic PIDEs with singular terms and a class of BDSDEs is built. In a special case that h ≡ 0, equation (1.1) turns out to be a parabolic partial differential equation (PDE for short). Then actually, the connection between this PDE and some backward stochastic differential equation (BSDE for short) is also obtained, which generalizes the classic connection between PDEs and BSDEs (see [13,14,16]). In particular, we prove the existence and uniqueness of solution to the stochastic PIDEs by analytical approach in Section 4.1. This is a generalization of the result in [5], where only symmetric operator was considered. From this point of view, Section 4.1 has its own independent interest.
Inspired by a serial of works (see, for example, [5,8,10,18]) on dealing with the singular term divg, we consider a stochastic PIDE with non-local operator that is associated with a perturbed Lévy process (X t ) t≥0 . As a generalization of the results in [8,18], we consider the non-local operator and define a stochastic *-integral for u, when u is in the domain of the bilinear form E associated with A. Precisely, for u ∈ D(A), where the forward and backward martingales are included. The last term on the right hand side comes from the asymmetry of A, and it may vanish when a symmetric operator is considered, since p m ≡ 1 if the perturbation b = 0 (see Section 3.1).
Moreover, a further decomposition of the zero-energy part of Fukushima decomposition of u(X t ) is given in Proposition 3.8. It holds that u(X t ) − u(X s ) = M u| t s + t s a α ∆ α 2 u(X r ) + b, ∇u (X r ) dr − 1 2 t s ∇u * dX, for s < t.
Due to this decomposition, we consider the solution u to stochastic PIDE (1.1), of which the existence and uniqueness is proved in Theorem 4.2, and find that the process u t (X t ) gives a solution to a class of BDSDEs, leave the problem when α ∈ (1, 2) to our future work. We also want to mention that the existence and uniqueness of solution to stochastic PIDE hold for 0 < α < 2, since analytical methods are applied there. The paper is organized as follows. In Section 2, we introduce the basic function spaces, Dirichlet forms and stochastic processes. Section 3 is the main part of the paper in which we define the stochastic *-integral and prove the forward-backward martingale decomposition corresponding to the non-symmetric non-local generator. Then, in Section 4, we prove the existence and uniqueness of solution to the stochastic PIDEs and give a probabilistic interpretation for this solution. The last section is devoted to building the connection between stochastic PIDEs and BDSDEs.

Preliminaries
Throughout this paper, we assume d ≥ 1 as an integer. Let L 2 (R d ) be the space of square integrable functions with respect to Lebesgue measure on R d , which is a Hilbert space equipped with the inner product and norm Since an evolution problem over a fixed time interval [0, T ] will be considered, we define the norm for a function in Hilbert space which is a Banach space equipped with the norm as follows, is the space of real functions which can be extended as infinite differential functions in For α ∈ (0, 2), the fractional Laplacian ∆ α 2 , which can be seen as the most basic integral-differential operator, can be defined in several equivalent ways (see [7]). For . Let F and F −1 denote Fourier and inverse Fourier transform on L 2 (R d ) respectively. It is known that Fourier transformation of ∆ be the Sobolev space of order s, which is equipped with the norm By interpolation and Sobolev embedding theorem (see [20]), for 0 < s < s , it holds that . For any γ > 0, we define the Kato class where |µ(dx)| is the total variation of the signed measure µ(dx) and B(x, ) is the ball in R d centred at x with radius . We say a function f ∈ K γ if the signed measure µ(dx) = f (x)dx ∈ K γ . It is easy to check that, if d > 2, for any function |f | ∈ L p (R d ), p > d, then |f | ∈ K d−1 and |f | 2 ∈ K d−2 .
By [3] and [19], we know that the bilinear form (E, H 1 (R d )) is a lower-bounded regular Dirichlet form and is related to a Hunt process (X t ) t≥0 whose infinitesimal generator A is of the following form In fact, by Theorem 3.25 in [3], for any v ∈ H 1 (R d ) and ∈ (0, 1), there exists C( ) > 0 such that Then, for any u, v ∈ H 1 (R d ), by Hölder's inequality and Kato-type inequality (2.1), one has where ∈ (0, 1) and C( ) > 0. By the first inequality in (2.2) and Sobolev embedding theorem, there exists a constant κ 0 > 0 and δ > 0 such that for any κ > κ 0 , Therefore, (E, H 1 (R d )) is a well-defined closed form on L 2 (R d ), and by [11], there are unique strong continuous semigroups (P t ) t≥0 and (P * t ) t≥0 with of which the corresponding generators A and A * satisfying It is known that the transition function is absolutely continuous with respect to the Lebesgue measure, i.e.
where p(t, x, y) is the density. By [2], when |b| ∈ K d−1 , the heat kernel p(t, x, y) is continuous on R + × R d × R d and for any t > 0 and x ∈ R d , R d p(t, x, y)dy = 1. Moreover, the following two-sided estimate holds: there exist constants C i , i = 1, 2, 3, 4 such that Let D(R + ; R d ) be the space of right continuous R d −valued functions on R + having left limits equipped with Skorokhod topology. There is a Lévy process (Ω, X t , θ t , F, F t , P x , x ∈ R d ) associated to the Dirichlet form (E, H 1 (R d )) on canonical paths space Ω := D(R + ; R d ).
For the process X, we have decomposition zÑ (dz, ds), is the martingale additive functional and N u| t s is the zero-energy additive functional. For any function u in the domain of generator A, i.e., u ∈ D(A), it is known that N u| t s = t s Au(X r )dr.

Decomposition with forward and backward martingales
In this section, we give a forward-backward martingale decomposition corresponding to the asymmetric non-local operator A, for which a stochastic *-integral is defined.
with the non-homogenous transition function We denote the density of Q 0,t by p Q (t, x, y) : .
For u, f, g ∈ D(∆) ∩ L ∞ (R d ), one can easily prove the following identity, Actually, we can have an identity corresponding to the fractional Laplacian operator similar as identity (3.1).
Then by [6], it follows that .
where the second equality is derived from Corollary 3.2, and the last equality is obtained . Dividing both sides of the above equality by p T (x), then by the definition of semigroup Q 0,t , the lemma is proved.
we define a process as follows, properties of which are given in the next proposition.
Proof. This proposition follows Proposition 3.1 in [8]. Hence, we just give the sketch of the proof. The first assertion can be obtained by the Recalling the Fukushima decomposition and adding the two equations above, we have Hence, the desired result is obtained.
The following theorem can be obtained by approximating u ∈ H 1 (R d ) with a sequence of u n ∈ D(∆) in H 1 (R d ).
Moreover, the following forward-backward martingale decomposition holds, Remark 3.6. Let µ be a probability measure on R d , then P µ (·) := R d P x (·) µ(dx) is a probability measure on (Ω, F) and the density of semigroup P t under P µ is p µ r (y) = R d p(r, x, y) µ(dx). The same proof as above implies that Theorem 3.5 also holds under P µ .
For any u ∈ H 1 (R d ) and α ∈ (0, 1), since H 1 (R d ) ⊂ H α (R d ), u lies in the domain of operator K. We define t s ∇u * dX := M u| t s +M u| t s + t s 2Ku(X r ) + G(u, p µ r ) p µ r (X r ) dr, (3.5) which actually does not depend on the initial measure µ. This stochastic *-integral has an important and helpful property shown in the following lemma.
Proof. By (3.4) and Fukushima's decomposition (2.5), we find that Hence, is a zero-energy functional. Hence, t s ∇u * dX is also a zero-energy functional.
For the second assertion, we only need to show that the zero-energy process is a martingale. Actually, we will prove that, for fixed t ∈ (0, T ), "backward" process is a backward martingale with respect to the backward filtration. Due to the Markovian property of the reversed process {X T −t }, we only need to prove, for any t ∈ [0, T ], With the help of Itô's formula, we have where the last but one equality is obtained by Lemma 3.1. The last equality is deduced by the condition ∆u = h in weak sense and f (r, Hence, we have proved that the process (3.6) is a backward martingale under probability P µ . Since {M u| t s , s ∈ [0, t]} is another backward martingale, we know that t s ∇u * dX + t s h(X r )dr is a backward martingale with zero energy. Hence, it is a null process.
Combing (2.5), (3.4) and (3.5), we obtain the following proposition, which gives a further description of the zero-energy functional in Fukushima decomposition. Ku(X r ) + b, ∇u (X r ) dr − 1 2 t s ∇u * dX, for s < t. Remark 3.9. When a function u ∈ H 1 loc (R d ) (i.e., for any ϕ ∈ C 1 c (R d ), uϕ ∈ H 1 (R d )) is considered, we can still define M u| t 0 ,M u| T t , for t ∈ [0, T ], which turn out to be (local) martingale and (local) backward martingale respectively (see [6,18]). In this case, we are used to denoting by M i andM i the local martingale and the local backward martingale which are associated with the coordinate function u i (x) = x i , for i = 1, · · · , d.
Since we consider the time evolution problem in later discussion, we define the stochastic Here, the backward stochastic integral is defined as where the limit is taken over the partition s = t 0 < t 1 < · · · < t n = t and δ = max Similarly as the proof of Lemma 3.7 and Lemma 3.1 in [18], we have the following result.
then it holds that t s ∇g * dX = − t s F r (X r ) dr, for s < t.
Denote Ω 2 := D([0, T ]; R d ) as the Skorohod space. The canonical processs (V t ) t and the shift operator θ 2 t can be defined similarly as (W t ) t and θ 1 t given above respectively.
is a Hunt process corresponding to Dirichlet form E 2 . We consider the sample space Ω := Ω 1 × Ω 2 and the process (X t ) t≥0 defined by X t (ω 1 , ω 2 ) = W t (ω 1 ) + V t (ω 2 ) for t ≥ 0. The shift operator Θ t : Ω −→ Ω is defined by Θ t (w 1 , w 2 )(s) = (w 1 (t + s), w 2 (t + s)), for any s, t ≥ 0. The σ-field F and filtration F t are given by F := F 1 × F 2 and F t := F 1 t × F 2 t . The family of probability measures {P x } x is defined by P x := P x 1 × P x 2 . We see that (Ω, X t , Θ t , F, F t , P x ) is a homogeneous Markov process related to the symmetric Dirichlet form (E, H 1 (R d )). For the process X, we have the following decomposition, for α ∈ (0, 1), zÑ (dz, ds) Since X is symmetric and Lebesgue measure m is invariant, then which induces that G(p m r , φ) = 0 for any φ ∈ H 1 (R d ). In this symmetric case, for any u ∈ H 1 (R d ), the forward-backward martingale decomposition in Theorem 3.5 is the decomposition proved in [6], Thus, by Proposition 3.8, we have the following relation, This equation coincides with the one obtained by applying Itô's formula to u(X t ).

A probabilistic interpretation of stochastic PIDE with divergence term
In this section, applying the forward-backward martingale decomposition obtained in Theorem 3.5, Fukushima decomposition obtained in Proposition 3.8 and letting u be the solution of (1.1), we will give a decomposition of the Dirichlet process u t (X t ) and then the probabilistic interpretation is presented.
T -predictable processes (u t ) 0≤t≤T such that the trajectories t → u t are in F a.s. and E u 2 T < ∞, where E denotes expectation under probability P .

Existence and uniqueness
If |b| is bounded, we definef (t, x, u, ∇u) := f (t, x, u, ∇u) + b, ∇u , then by Theorem where the constant C only depends on the structure coefficients of the stochastic PIDE (1.1).
In order to prove Theorem 4.2, we deal with the mild solution to stochastic PIDE firstly, the definition of which is given as follows. Please note that the term involving divergence will be defined later (see Lemma 4.5).

Definition 4.3.
We say that u ∈ H T is a mild solution to stochastic PIDE (1.1) with terminal condition u T = Φ, if the following equality is verified almost surely, for each t ∈ [0, T ],    Proof. Firstly, we assume that Φ ∈ D(A) and f ∈ D T . Then, it is clear that the maps t → P T −t Φ, t → T t P s−t f s ds belong to C([0, T ], D(A)) and are L 2 (R d )−differentiable with continuous derivatives. Then it follows that t → u t belongs to C([0, T ], D(A)), and it is also L 2 (R d )−differentiable with continuous derivative. Hence, we have Applying formula for integration by parts, we obtain that, for all ϕ ∈ D T , Since we also have that, for all t ∈ [0, T ], which proves the equality (4.4). Combing with the form of E, we get By applying Hölder's inequality, Kato-type inequality (2.1) and Cauchy-Schwartz's inequality, we have, for any 0 < < 1, > 0, Taking and small enough such that (1− − ) > 0, and applying Gronwall's inequality, we obtain that Taking the supremum of the above inequality, we get sup which implies that u ∈F . Finally, one can obtain the result in the general case by approximation.
To treat the divergence term, we need to give a precise definition of the integral T t P s−t divg s ds, which is just a formal writing with g ∈ L 2 ([0, T ], R d ). Obviously, this integral is well defined if g ∈ D T . We therefore define a operator U : D T →F by The next lemma proves that we can extend it by continuity. In the following discussion, we will use the formal expression T t P s−t divg s ds rather than U g.  Moreover, the following relation is satisfied (4.6) Proof. Assume that g ∈ D T firstly, then it is obvious that divg t ∈ D(A). By Lemma 4.4, we deduce that u = U g is a weak solution of (4.5). By (4.4) we get which yields equation (4.6) in this smooth case. Using Young's and Kato-type inequality, by the similar method in proof of Lemma 4.4, we have We choose , and small enough such that (1 − − − ) > 0. Thanks to Gronwall's lemma, we obtain Then taking the supremum of the above inequality, we get U g T ≤ C T 0 g s 2 ds, which implies U g ∈F and the continuity of operator U . The result with general g ∈ L 2 ([0, T ], R d ) follows by approximation.
We apply Itô's formula to β 2 and obtain Applying Kato-type inequality, Burkholder-Davies-Gundy's inequality and Gronwall's inequality, we prove β ∈ H T . By the density argument, we get assertions in general case.
The following proposition can be obtained by a similar argument as Proposition 4.9 in [4] or Proposition 7 in [5]. We therefore omit the proof here. Proof of Theorem 4.2: Let θ and δ be two positive numbers, which will be determined later. We introduce a norm of space L 2 (Ω × [0, T ]; H 1 (R d )), Choosing u, v ∈ H T , by Itô's formula, we have almost surely,  Therefore, it follows that Since 2γ + β 2 < 1, we can choose and small enough such that and then choose θ satisfying that .
We therefor conclude the result by fixed point theorem.

Probabilistic interpretation
By Theorem 4.2, we know that there is a unique function u satisfying stochastic PIDE (1.1). We then give a probabilistic interpretation for the Dirichlet process u t (X t ) in this section. Please note that we always assume that α ∈ (0, 1) in the following discussion.
In the case that g is independent of u and ∇u, by assumption (HD2), for any ϕ ∈ D T , we have T 0 (g s , ∇ϕ s ) ds ≤ g 2,2 · ∇ϕ 2,2 ≤ g 2,2 ·  Therefore, the weak solution u of stochastic PIDE (1.1) with linear g = g t (x) also satisfies the following equation It has been proved that the triplet of processes (Y, Z, U ), defined in (5.1) with u ∈ H T being the solution of equation (1.1), is the solution of non-linear BDSDE (5.2). Actually, the converse statement can also be proved. We therefore build a connection between the solutions for stochastic PIDEs and BDSDEs.
Theorem 5.1. Suppose Y t = w(t, X t ), Z t = φ(t, X t ), U (t, z) = ψ(t, X t− , z) is a solution of the non-linear BDSDE (5.2) defined as Definition 2 (2), then w ∈ H T represents a weak solution of stochastic PIDE (1.1) as defined in Definition 4.1.
Proof. We only give the proof in the linear case, i.e. f, g, h only depend on (t, x), and the nonlinear case follows easily.
Applying Itô's formula and taking expectation under P ⊗ P, it follows that |Ū s (z)| 2 ν(dz)ds = 0, P ⊗ P − a.s., which implies that Y t = Y t , P ⊗ P − a.s., then w(t, ·) = w (t, ·), dP ⊗ dx − a.e.; Z t = Z t , P ⊗ P − a.s., so that φ(t, ·) = ∇w (t, ·), dP ⊗ dx − a.e.. Hence w = w ∈ H T , dP ⊗ dt ⊗ dx − a.e.. coincides with the *-integral defined in [18] and the integrand can be generalized from the gradient of functions in H 1 (R d ) to all of the functions in L 2 (R d ). From this point of view, we actually generalized the results in [18] for non-local operators.