Quasi-invariant flow generated by Stratonovich SDE with BV drift coefficients

TL;DR

This paper extends the theory of regular Lagrangian flows to Stratonovich SDEs with BV drift coefficients, providing explicit solutions and analyzing flow differentiability.

Contribution

It generalizes existing deterministic flow results to stochastic settings with BV drifts and constructs explicit solutions for stochastic transport equations.

Findings

Established existence and uniqueness of flows for Stratonovich SDEs with BV drifts

Constructed explicit solutions to stochastic transport equations

Analyzed approximate differentiability of stochastic flows

Abstract

We generalize the results of Ambrosio [Invent. Math. 158 (2004), 227--260] on the existence, uniqueness and stability of regular Lagrangian flows of ordinary differential equations to Stratonovich stochastic differential equations with BV drift coefficients. Then we 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.