Quasi-Invariant Flow Generated by Stratonovich SDE with BV Drift Coefficient

Abstract

We generalize the results of Ambrosio [1] on the existence, uniqueness, and stability of regular Lagrangian flows of ordinary differential equations to Stratonovich stochastic differential equations with BV drift coefficients. We then construct an explicit solution to the corresponding stochastic transport equation in terms of the stochastic flow. The approximate differentiability of the flow is also studied when the drift is a Sobolev vector field.

A unified treatment for ODEs under Osgood and Sobolev type conditions

“…Proof of Theorem 4.1. We follow the ideas of [6,Remark 2.4] (see also [14,Proposition 5.2]). ψ r (r) dy ≤ 1.…”

Lagrangian flows driven by $$BV$$ fields in Wiener spaces

“…The work of DiPerna and Lions did not cover the case of bounded variation (BV ) fields, which arise naturally in many contexts: it was settled by L. Ambrosio in [Amb04] and since then, BV fields were considered in other settings, e.g. SDEs, in [LL12], or Fokker-Planck equations, in [Luo13].…”

Degenerate irregular SDEs with jumps and application to integro-differential equations of Fokker-Planck type

“…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 [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].…”

“…Let µ i t , i = 1, 2 be two solutions of PIDE (5.2) with the same initial values. Then by (5.3), we have 16 In particular,…”