Degenerate Irregular SDEs with Jumps and Application to Integro-Differential Equations of Fokker-Planck type

We investigate stochastic differential equations with jumps and irregular coefficients, and obtain the existence and uniqueness of generalized stochastic flows. Moreover, we also prove the existence and uniqueness of $L^p$-solutions or measure-valued solutions for second order integro-differential equation of Fokker-Planck type.


Introduction
Recently, there are increasing interests to extend the classical DiPerna-Lions theory [7] about ordinary differential equations (ODE) with Sobolev coefficients to the case of stochastic differential equations (SDE) (cf. [14,15,10,26,27,28,9,16]). In [10], Figalli first extended the DiPerna-Lions theory to SDE in the sense of martingale solutions by using analytic tools and solving deterministic Fokker-Planck equations. In [14], Le Bris and Lions studied the almost everywhere stochastic flow of SDEs with constant diffusion coefficients, and in [15], they also gave an outline for proving the pathwise uniqueness for SDEs with irregular coefficients by studying the corresponding Fokker-Planck equations with irregular coefficients. In [26] and [28], we extended DiPerna-Lions' result to the case of SDEs by using Crippa and De Lellis' argument [6], and obtained the existence and uniqueness of generalized stochastic flows for SDEs with irregular coefficients (see also [9] for some related works). Later on, Li and Luo [16] extended Ambrosio's result [1] to the case of SDEs with BV drifts and smooth diffusion coefficients by transforming the SDE to an ODE. Moreover, a limit theorem for SDEs with discontinuous coefficients approximated by ODEs was also obtained in [20].
In this paper we are concerned with the following SDEs in [0, 1] × R d with jumps: 1] is a d-dimensional Brownian motion and N(dy, dt) is a Poisson random measure in R d \ {0} with intensity measure ν t (dy)dt,Ñ(dy, dt) := N(dy, dt) − ν t (dy)dt is the compensated Poisson random measure. The aim of the present paper is to extend the results in [26] to the above jump SDEs with Sobolev drift b and Lipschitz σ, f . Let us now describe the motivation. Suppose that f t (x, y) = y. Let L be the generator of SDE (1.1) (a second order integro-differential operator) given as follows: for ϕ ∈ C ∞ b (R d ), smooth function with bounded derivatives of all orders,  Let X t be a solution of SDE (1.1). The law of X t in R d is denoted by µ t . Then by Itô's formula (cf. [12] or [2]), one sees that µ t solves the following second order partial integro-differential equation (PIDE) of Fokker-Planck type in the distributional sense: subject to the initial condition: lim t↓0 µ t = Law of X 0 in the sense of weak convergence, (1.4) where L * t is the adjoint operator of L t formally given by where for a probability measure µ in R d \ {0} and y ∈ R d , τ y µ := µ(· − y). More precisely, for . If b and σ are not continuous, in order to make sense for (1.5), one needs to at least assume that The following two questions are our main motivations of this paper: (2 o ) If the initial distribution µ 0 has a density with respect to the Lebesgue measure, does µ t have a density with respect to the Lebesgue measure for any t ∈ (0, 1]? When there is no jump part and the diffusion coefficient is non-degenerate, in [3] the authors have already given rather weak conditions for the uniqueness of measure-valued solutions based upon the Dirichlet form theory. In [10], Figalli also gave some other conditions for the uniqueness of L 1 ∩ L ∞ -solutions by proving a maximal principle. In [22], using a representation formula for the solutions of PDE (1.3) proved in [10], which is originally proved by Ambrosio [1] for continuity equation, we gave different conditions for the uniqueness of measure-valued solutions and L p -solutions to second order degenerated Fokker-Planck equations. However, to the best of the author's knowledge, there are few results on the integro-differential equation of Fokker-Planck type. The non-local character of the operator L causes some new difficulties to analyze by the classical tools.
For answering the above two questions to equation (1.3), we shall use a purely probabilistic approach. The first step is to extend the almost everywhere stochastic flow in [14,26,28] to SDE (1.1) so that we can solve the above question (2 o ). In this extension, we need to carefully treat the jump size. Since even in the linear case, if one does not make any restriction on the jump, the law of the solution would not be absolutely continuous with respect to the Lebesgue measure (cf. [18, p.328, Example]). The next step is to prove a representation formula for the solution of (1.3) as in [10,Theorem 2.6]. This will lead to the uniqueness of PIDE (1.3) by proving the pathwise uniqueness of SDE (1.1). This paper is organized as follows: In Section 2, we collect some well known facts for later use. In Section 3, we study the smooth SDEs with jumps, and prove an a priori estimate about the Jacobi determinant of x → X t (x). In Section 4, we prove the existence and uniqueness of almost everywhere or generalized stochastic flows for SDEs with jumps and rough drifts. In Section 5, the application to second order integro-differential equations of Fokker-Planck type is presented.

Preliminaries
Throughout this paper we assume that d 2. Let M d×d be the set of all d × d-matrices. We need the following simple lemma about the differentials of determinant function. and (1 iσ(i) + tb iσ(i) ), (2.4) where S d is the set of all permutations of {1, 2, · · · , d} and sgn(σ) is the sign of σ.
If for some T > 0, where M c and M d are respectively continuous and purely discontinuous martingale parts of M, In Sections 3 and 4, we shall deal with the general Poisson point process. Below we introduce some necessary spaces and processes. Let (Ω, F , P; (F t ) t 0 ) be a complete filtered probability space and (U, U ) a measurable space. Let (W(t)) t 0 be a d-dimensional standard (F t )-adapted Brownian motion and (p t ) t 0 an (F t )-adapted Poisson point process with values in U and with intensity measure ν t (du)dt, a σ-finite measure on [0, 1] × U (cf. [12]). Let N(du, (0, t]) be the counting measure of p t , i.e., for any Γ ∈ U , The compensated Poisson random measure of N is given bỹ We remark that for Γ ∈ U with t 0 ν s (Γ)ds < +∞, the random variable N((0, t], Γ) obeys the Poisson distribution with parameter t 0 ν s (Γ)ds. Below, the letter C with or without subscripts will denote a positive constant whose value is not important and may change in different occasions. Moreover, all the derivatives, gradients and divergences are taken in the distributional sense.
The following lemma is a generalization of [ We also need the following technical lemma (cf. [28,Lemma 3.4]).

Lemma 2.4.
Let µ be a locally finite measure on R d and (X n ) n∈N be a family of random fields on Ω × R d . Suppose that X n converges to X for P ⊗ µ-almost all (ω, x), and for some p 1, there is a constant K p > 0 such that for any nonnegative measurable function ϕ ∈ L p µ (R d ), Then we have: Let ϕ be a locally integrable function on R d . For every R > 0, the local maximal function is defined by Then there exist C q,d > 0 and a negligible set A such that for all x, y ∈ A c with |x − y| R, Then there exist C d > 0 and a negligible set A such that for all x, y ∈ A c with |x − y| R, Then for some C d,p > 0 and any N,

SDEs with jumps and smooth coefficients
In this section, we consider the following SDE with jump: are measurable functions and smooth in the spatial variable x, and satisfy that Moreover, we assume that there exist two functions L 1 , where α ∈ (0, 1) is small and p ∈ (1, ∞) is arbitrary, and such that for all (s, Under conditions (3.2)-(3.4) with small α (saying less than 1 8d ), it is well known that SDE (3.1) defines a flow of C ∞ -diffeomorphisms (cf. [11,18], [17,Theorem 1.3]). Let Then J t satisfies the following SDE (cf. [11,18]): The following lemma will be our starting point in the sequel development.
Lemma 3.1. The Jacobi determinant det(J t ) has the following explicit formula: Proof. By (3.5), Itô's formula and Lemma 2.1, we have t is a continuous increasing process given by and and M t := M c t + M d t is a martingale given by and By Doléans-Dade's exponential formula (cf. [18]), we obtain the desired formula.
Below, we shall give an estimate for the p-order moment of the Jacobi determinant. For this aim, we introduce the following function of jump size control α: (3.10) Note that lim α↓0 β α = +∞.
In order to give an estimate for det(∇X −1 t (x)) in terms of det(J t (x)) = det(∇X t (x)), we shall use a trick due to Cruzeiro [5] (see also [4,28,9]). Below, let which means that for any nonnegative measurable function ϕ on R d , It is easy to see that for almost all ω and all (t, x) ∈ [0, 1] × R d , and (3.18) We need the following estimate: where the constant C is an increasing function with respect to its arguments.
Then for any p ∈ (0, β α ), where the constant C is inherited from Lemmas 3.2 and 3.3.
Proof. The estimate follows from , Hölder's inequality and Lemmas 3.2 and 3.3.

SDEs with jumps and rough drifts
We first introduce the following notion of generalized stochastic flows (cf. [15,26,28]
Let χ ∈ C ∞ (R d ) be a nonnegative cutoff function with The following lemma is direct from the definitions and the property of convolutions.

Lemma 4.4.
For some C > 0 independent of n, we have and The other estimates are similar.
Let X n t (x) be the stochastic flow of C ∞ -diffeomorphisms to SDE (3.1) associated with coefficients (b n , σ n , f n ). Lemma 4.5. let β α be defined by (3.10), where α is from (3.3) small enough so that β α > 1. Then for any p > 1 + 1 β α , there exists a constant C p > 0 such that for all non-negative function ϕ ∈ L p µ (R d ), Proof. The estimate follows from and Theorem 3.4 and Lemma 4.4.

Lemma 4.7.
For any R > 1, there exist constants C 1 , C 2 > 0 such that for all δ ∈ (0, 1) and n, m > 4/δ, . If there are no confusions, we shall drop the variable "x" below. Note that By Itô's formula, we have Noting that . By Burkholder's inequality and Fubini's theorem, we have As the treatment of I n,m 1 (t), by Lemma 4.5, we can prove that Combining (4.7)-(4.12), we obtain (4.6).
We are now in a position to give 13 Proof We have where G n,m R is defined as in Lemma 4.7. By Lemmas 3.3 and 4.4, the first term is less than where C is independent of n, m and R, and d 2.
For the second term, we make the following decomposition:  Cδ √ e η − 1.
Combining the above calculations, we obtain that Taking limits in order: n, m → ∞, δ → 0, η → ∞ and R → ∞ yields that lim n,m→∞ Thus, there exists an adapted cádlág process X t (x) such that By Lemma 2.4, it is standard to check that X t (x) solves SDE (3.1) in the sense of Definition 4.1.
For the uniqueness, let X i t (x), i = 1, 2 be two almost everywhere stochastic flows of SDE (3.1). As in the proof of Lemma 4.7, we have Below we consider the more general Lévy generator: where a i j t (x) := k σ ik t (x)σ jk t (x) and ν satisfies that We recall the following notion of Stroock and Varadhan's martingale solutions (cf. [24,25]).
is a P-martingale with respect to (F t ), which is equivalent that for all θ ∈ R d , and the law of X ε 0 is µ ε 0 . Here,b ε s (x) is defined by (5.1) with δ = 1 and replacing b by b ε . Let P ε be the law of t → X ε t in Ω. Since P ε (|w 0 | R) = µ ε 0 (B c R ) → 0 uniformly in ε as R → ∞, by [24, p.237, Theorem A.1], (P ε ) ε∈(0,1) is tight in P(Ω). Let P be any accumulation of (P ε ) ε∈(0,1) . Without loss of generality, we assume that P ε weakly converges to P as ε → 0. By taking weak limits for both sides of (5.6), it is clear that (5.5) holds.
For completing the proof, it remains to show that P is a martingale solution corresponding to (L , µ 0 ). That is, we need to prove that for any 0 s < t 1 and bounded continuous and F s -measurable function Φ s on Ω, ϕ ∈ C ∞ 0 (R d ), (L r ϕ)(w r )dr Φ s (w) = 0. 17 This will follow by taking weak limits for The more details can be found in [10, p.118, Step 3].

Definition 5.5. (Weak solution)
If there exists a filtered probability space (Ω, F , P; (F t ) t∈ [0,1] ) and an (F t )-adapted Brownian motion W t , an (F t )-adapted Poisson random measure N(dy, dt) with intensity measure ν t (dy)dt and an (F t )-adapted process X t on (Ω, F , P; (F t ) t∈ [0,1] ) such that for some δ > 0 and all t ∈ [0, 1], whereb δ s (x) is defined by (5.1), then we say (Ω, F , P; (F t ) t∈ [0,1] ) together with (W, N, X) a weak solution. By weak uniqueness, we means that any two weak solutions with the same initial law have the same law in Ω.
The following result gives the equivalence between weak solutions and martingale solutions.