Stochastic differential equations with coefficients in Sobolev spaces

Abstract

We consider the Itô stochastic differential equation dX t = m j =1 A j (X t ) dw j t + A 0 (X t ) dt on R d . The diffusion coefficients A 1 , . . . , A m are supposed to be in the Sobolev space W 1,p loc (R d ) with p > d, and to have linear growth. For the drift coefficient A 0 , we distinguish two cases: (i) A 0 is a continuous vector field whose distributional divergence δ(A 0 ) with respect to the Gaussian measure γ d exists, (ii) A 0 has Sobolev regularity W 1,p loc for some p > 1. Assume R d exp[λ 0 (|δ(A 0 )| + m j =1 (|δ(A j )| 2 + |∇A j | 2 ))] dγ d < +∞ for some λ 0 > 0. In case (i), if the pathwise uniqueness of solutions holds, then the push-forward (X t ) # γ d admits a density with respect to γ d . In particular, if the coefficients are bounded Lipschitz continuous, then X t leaves the Lebesgue measure Leb d quasi-invariant. In case (ii), we develop a method used by G. Crippa and C. De Lellis for ODE and implemented by X. Zhang for SDE, to establish existence and uniqueness of stochastic flow of maps.

Well-posedness of multidimensional diffusion processes with weakly differentiable coefficients

“…Moreover, we remark that, although the conditions above might read as perfect analogues of the notion of Regular Lagrangian flows [AC14, Definition 13], Stochastic Lagrangian Flows are not necessarily (neither expected to be) deterministic maps of the initial point only; this is evident when σ = 0 above and any probability concentrated on possibly non-unique solutions to the ODE give rise to a solution to the martingale problem. Despite this discrepancy, such a theory provides rather efficient tools to study stochastic differential equations under low regularity assumptions, in Euclidean spaces, and, together with [LBL08], which deals with analogous issues from a PDE point of view, has become the starting point for further developments, among which we quote [RZ10,Luo13,FLT10,Zha13]. Before we proceed with a more detailed description of our results and techniques, let us stress the fact that we are concerned uniquely with martingale problems, so we do not address nor compare our results with those obtained for strong solutions of equations under low regularity assumptions on the coefficients (see the seminal paper [Ver80] and [KR05,DFPR13] for more recent results).…”

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

“…It is clear thatlim n→∞ σ n − σ L 2q (µ) = 0 and lim n→∞ b n − b L 2q (µ) = 0.Combining these limits with Propositions 3.5 and 3.6, and making use of the uniform density estimate(3.20), we can finish the proof as in[18, Proposition 4.1].…”

“…Proof. We follow the line of arguments of [18,Theorem 5.2]. Denote by Z t = X t −X t and ξ t = |Z t | 2 where we omit the space variable x. Itô's formula leads to…”

Pathwise uniqueness of multi-dimensional stochastic differential equations with Hölder diffusion coefficients

“…[10,17]). There are also researches on the existence and uniqueness of generalized flows associated to SDEs with Sobolev coefficients (see [5,18]). The present work is motivated by [14], in which the author generalized the result of Yamada and Watanabe [15] to a particular class of multi-dimensional SDEs, that is, for each i ∈ {1, .…”