Stochastic Homeomorphism Flows of SDEs with Singular Drifts and Sobolev Diffusion Coefficients

In this paper we prove the stochastic homeomorphism flow property and the strong Feller property for stochastic differential equations with sigular time dependent drifts and Sobolev diffusion coefficients. Moreover, the local well posedness under local assumptions are also obtained. In particular, we extend Krylov and R\"ockner's results in \cite{Kr-Ro} to the case of non-constant diffusion coefficients.


Introduction and Main Result
Consider the following stochastic differential equation (SDE) in R d : where b : R + × R d → R d and σ : R + × R d → R d × R d are two Borel measurable functions, and {W t } t 0 is a d-dimensional standard Brownian motion defined on some complete filtered probability space (Ω, F , P; (F t ) t 0 ). When σ is Lipschitz continuous in x uniformly with respect to t and b is bounded measurable, Veretennikov [14] first proved the existence of a unique strong solution for SDE (1.1). Recently, Krylov and Röckner [10] proved the existence and uniqueness of strong solutions for SDE (1.1) with σ ≡ I d×d and T 0 R d |b t (x)| p dx q p dt < +∞, ∀T > 0, (1.2) provided that d p More recently, following [10], Fedrizzi and Flandoli [4] proved the α-Hölder continuity of x → X t (x) for any α ∈ (0, 1) basing on Girsanov's theorem and Khasminskii's estimate. In the case of non-constant and non-degenerate diffusion coefficient, the present author [15] proved the pathwise uniqueness for SDE (1.1) under stronger integrability assumptions on b and σ (see also [6] for Lipschitz σ and unbounded b). Moreover, there are many works recently devoted to the study of stochastic homeomorphism (or diffeomorphism) flow property of SDE (1.1) under various non-Lipschitz assumptions on coefficients (see [3,16,5] and references therein). We first introduce the class of local strong solutions for SDE (1.1). Let τ be any (F t )-stopping time and ξ any F 0 -measurable R d -valued random variable. Let S τ b,σ (ξ) be the class of all R dvalued (F t )-adapted continuous stochastic process X t on [0, τ) satisfying P ω : . 1 and such that We now state our main result as follows: Theorem 1.1. In addition to (1.2) with p, q ∈ (1, ∞) satisfying (1.3), we also assume that (H σ 1 ) σ t (x) is uniformly continuous in x ∈ R d locally uniformly with respect to t ∈ R + , and there exist positive constants K and δ such that for all (t, x) ∈ R + × R d , with the same p, q as required on b, where ∇ denotes the generalized gradient with respect to x. Then for any (F t )-stopping time τ (possibly being infinity) and x ∈ R d , there exists a unique strong solution Moreover, for almost all ω and all t 0, and for any t > 0 and bounded measurable function φ, x, y ∈ R d , where C t > 0 satisfies lim t→0 C t = +∞. Remark 1.2. The uniqueness proven in this theorem means local uniqueness. We want to emphasize that global uniqueness can not imply local uniqueness since local solution can not in general be extended to a global solution.
By localization technique (cf. [15]), as a corollary of Theorem 1.1, we have the following existence and uniqueness of local strong solutions. Theorem 1.3. Assume that for any n ∈ N and some p n , q n ∈ (1, ∞) satisfying (1.3), (i) |b t |, |∇σ t | ∈ L q n loc (R + ; L p n (B n )), where B n := {x ∈ R d : |x| n}; (ii) σ ik t (x) is uniformly continuous in x ∈ B n uniformly with respect to t ∈ [0, n], and there exist positive constants δ n such that for all (t, Then for any x ∈ R d , there exist an (F t )-stopping time ζ(x) (called explosion time) and a unique strong Proof. For each n ∈ N, let χ n (t, x) ∈ [0, 1] be a nonnegative smooth function in R + × R d with χ n (t, x) = 1 for all (t, x) ∈ [0, n] × B n and χ n (t, x) = 0 for all (t, x) [0, n + 1] × B n+1 . Let b n t (x) := χ n (t, x)b t (x) and By Theorem 1.1, for each x ∈ R d , there exists a unique strong solution X n t (x) ∈ S ∞ b n ,σ n (x) to SDE (1.1) with coefficients b n and σ n . For n k, define τ n,k (x, ω) := inf{t 0 : |X n t (ω, x)| k} ∧ n. It is easy to see that . By the local uniqueness proven in Theorem 1.1, we have P{ω : X n t (ω, x) = X k t (ω, x), ∀t ∈ [0, τ n,k (x, ω))} = 1, which implies that for n k, Hence, if we let ζ k (x) := τ k,k (x), then ζ k (x) is an increasing sequence of (F t )-stopping times and for n k, P{ω : b,σ (x) and (1.4) holds. The aim of this paper is now to prove Theorem 1.1. We organize it as follows: In Section 2, we prove two new estimates of Krylov's type, which is the key point for our proof and has some independent interest. In Section 3, we prove Theorem 1.1 in the case of b = 0. For the stochastic homeomorphism flow, we adopt Kunita's simple argument (cf. [11]). For the strong Feller property, we use Bismut-Elworthy-Li's formula (cf. [2]). In Section 4, we use Zvonkin's transformation to fully prove Theorem 1.1. In Appendix, we recall some well known facts used in the present paper.

Two estimates of Krylov's type
We first introduce some spaces and notations. For p, q ∈ [1, ∞) and 0 S < T < ∞, we denote by L q p (S , T ) the space of all real Borel measurable functions on [S , T ]×R d with the norm For m ∈ N and p 1, let H m p be the usual Sobolev space over R d with the norm where ∇ denotes the gradient operator, and · L p is the usual L p -norm. We also introduce for Here and below, we use the convention that the repeated indices in a product will be summed automatically. Moreover, the letter C will denote an unimportant constant, whose dependence on the functions or parameters can be traced from the context. We first prove the following estimate of Krylov's type (cf. [8, p.54, Theorem 4]).

Theorem 2.1. Suppose that σ satisfies (H σ 1 ) and b is bounded measurable. Fix an (F t )-stopping time τ and an
Let ρ be a nonnegative smooth function in R d+1 with support in {x ∈ R d+1 : |x| 1} and R d+1 ρ(t, x)dtdx = 1. Set ρ n (t, x) := n d+1 ρ(nt, nx) and extend u(s) to R by setting u(s, ·) = 0 for s T and u(s, ·) = u(0, ·) for s 0. Define . Then by (2.4) and the property of convolutions, we have Now using Itô's formula for u n (t, x), we have In view of sup The proof is thus completed by (2.7) and letting n → ∞.
Next, we want to relax the boundedness assumption on b. The price to pay is that a stronger integrability assumption is required.
Fix an (F t )-stopping time τ and an F 0 -measurable R d -valued random variable ξ and let X t ∈ S τ b,σ (ξ). Given T 0 > 0, there exists a positive constant C = C(K, δ, d, p, q, T 0 , b L q p (T 0 ) ) such that for all f ∈ L q p (T 0 ) and 0 S < T T 0 , Proof. Following the proof of Theorem 2.1, we let r = d + 1 and assume that f ∈ L r r (T 0 ) ∩ L q p (T 0 ). Below, for N > 0, we write x), u(T, x) = 0. Moreover, for some constant C 1 = C 1 (K, δ, d, p, q, T 0 , N), and for some constant For R > 0, define Let u n be defined by (2.6). As in the proof of Theorem 2.1 (see (2.8)), by (2.12), we have Hence, by (2.12) and (2.13), where C = C(K, δ, d, p, q, T 0 , b L q p (T 0 ) ) is independent of n and R, N. Observe that for fixed N > 0, by (2.11), lim n→∞ f N n − f L r r (T ) = 0, and for fixed R > 0, by the dominated convergence theorem, Taking limits for both sides of (2.14) in order: n → ∞, N → ∞ and R → ∞, we obtain (2.10).

SDE with Sobolev diffusion coefficient and zero drift
In this section we consider the following SDE without drift: We first prove that:

In particular, there exists a unique strong solution for SDE (3.1).
Proof. Set Z t := X t − Y t . By Itô's formula, we have If we set Here and below, we use the convention that 0 0 ≡ 0. Thus, if we can show that t → M t∧τ + A t∧τ is a continuous semimartingale, then the uniqueness follows. For this, it suffices to prove that for any t 0, E|M t∧τ | 2 < +∞, EA t∧τ < +∞. Set σ n s (x) := σ s * ρ n (x), where ρ n is a mollifier in R d as used in Theorem 2.1. By Fatou's lemma, we have By estimate (2.3), we have and also, I 3 (t) = 0. For I 1 (t), we have Combining the above calculations, we obtain that for all t 0, Similarly, we can prove that where the star denotes the transpose of a matrix. The existence of a unique strong solution now follows from the classical Yamada-Watanabe theorem (cf. [7]).
Below, we prove better regularities of solutions with respect to the initial values. 1 ) and (H σ 2 ), let X t (x) be the unique strong solution of SDE (3.1). For any T > 0, γ ∈ R and all x y ∈ R d , we have
In particular, by Novikov's criterion, is a continuous exponential martingale. Hence, by Hölder's inequality, we have where C is independent of ε and x, y.
Since σ is bounded, the following lemma is standard, and we omit the details.
where C 1 = C 1 (K, γ, T ), and for any γ 1 and t, s 0, Basing on Lemmas 3.2 and 3.3, it is by now standard to prove the following theorem (cf. [11,Theorem 4.5.1]). For the reader's convenience, we sketch the proof here.
For any x, y, x ′ , y ′ ∈ R d with x y, x ′ y ′ and s t, it is easy to see that By Lemmas 3.2 and 3.3, for any γ 1 and s, t ∈ [0, T ], we have Choosing γ > 4(d + 1), by Kolmogorov's continuity criterion, there exists a continuous version to the mapping (t, x, y) → R t (x, y) on {(t, x, y) ∈ R + × R d × R d : x y}. In particular, this proves that for almost all ω, the mapping x → X t (ω, x) is one-to-one for all t 0.
As for the onto property, let us define As above, using Lemmas 3.2 and 3.3, one can show that (t, x) → J t (x) admits a continuous version. Thus, (t, x) → X t (ω, x) can be extended to a continuous map from R + ×R d toR d , wherê Hence, X t (ω, ·) :R d →R d is homotopic to the identity mapping X 0 (·) so that it is an onto map by the well known fact in homotopic theory. In particular, for almost all ω, x → X t (ω, x) is a homeomorphism onR d for all t 0. Clearly, the restriction of X t (ω, ·) to R d is still a homeomorphism since X t (ω, ∞) = ∞.
Now we turn to the proof of the strong Feller property.

Theorem 3.5.
Under (H σ 1 ) and (H σ 2 ), let X t (x) ∈ S ∞ 0,σ (x) be the unique strong solution of SDE (3.1), then for any bounded measurable function φ, T > 0 and x, y ∈ R d , Proof. Define σ n t (x) := σ t * ρ n (x), where ρ n is a mollifier in R d . By (H σ 1 ), it is easy to see that for all (t, x) ∈ R + × R d , Let X n t (x) ∈ S ∞ 0,σ n (x) be the unique strong solution of SDE (3.1) corresponding to σ n . By the monotone class theorem, it suffices to prove (3.3) for any bounded Lipschitz continuous function φ. First of all, by Bismut-Elworthy-Li's formula (cf. [2]), for any h ∈ R d , we have where for a smooth function f , we denote  4) and (3.5), which implies that for all t ∈ (0, T ] and x, y ∈ R d , |E(φ(X n t (x))) − E(φ(X n t (y)))| C T φ ∞ √ t |x − y|, (3.6) where C T is independent of n. Now for completing the proof, it only needs to take limits for (3.6) by proving that for any Set Z n t (x) := X n t (x) − X t (x) and η n (s) := M|∇σ n s |(X n s (x)) + M|∇σ n s |(X s (x)) 2 .
By Theorem 5.1, we have