Generalized stochastic flow associated to the Itô SDE with partially Sobolev coefficients and its application

Abstract

We consider the Itô SDEs on R n with partially Sobolev coefficients. Assuming the exponential integrability of the negative part of the divergence of the drift coefficient and the partial gradient of the diffusion coefficient with respect to the Cauchy measure, we show the existence, uniqueness and stability of generalized stochastic flows associated to such equations. As an application, we prove the weak differentiability in the sense of measure of the stochastic flow generated by the Itô SDE with Sobolev coefficients.

The Itô SDEs and Fokker–Planck equations with Osgood and Sobolev coefficients

“…To simplify notations, we write b(x) = b(x) 1+|x| and σ(x) = σ(x) 1+|x| for x ∈ R d . Our first main result extends [34,Theorem 2.3].…”

“…First we prove an a-priori moment estimate on the solution flow X t to (1.1). This improves the result presented in [34,Lemma 2.4]. Lemma 3.1 (Moment estimate).…”

“…Therefore, the mixed Osgood and Sobolev regularity can only be imposed on the drift coefficient b in (1.1). The first main result of this paper also extends [34,Theorem 2.3], in which the drift coefficient b is required to be in the first order Sobolev space. We remark that when the equation (1.1) has non-degenerate diffusion coefficient σ, the existence of a unique strong solution can be proved under quite weak conditions on the drift b, see [28,21] for integrability conditions and [23,40] for weak continuity conditions.…”

“…This property shows that ψ δ is a concave function for any δ > 0. If ρ(s) ≡ s for all s ≥ 0, then ψ δ (ξ) = log 1 + ξ δ which is the auxiliary function used in the previous works [11,42,18,34]. Let q ≥ 1 be fixed.…”

“…This method was developed in [9] to show the well posedness of (1.2) when the gradient of b is given by some singular integral. Following this direct method, there are also studies on the SDE (1.1) with coefficients having Sobolev regularity [42,18,44,34]. Regarding the corresponding PDE, Le Bris and Lions studied in [30] the Fokker-Planck type equations with Sobolev coefficients; using Ambrosio's commutator estimate for BV vector fields, their results was slightly extended in [32] to the case where the drift coefficient has only BV regularity.…”