Malliavin and flow regularity of SDEs. Application to the study of densities and the stochastic transport equation

TL;DR

This paper establishes conditions for the regularity of strong solutions to SDEs driven by Brownian motion, improving density regularity and applying results to solve the stochastic transport equation with Lipschitz drift.

Contribution

It introduces a new regularity condition for SDE solutions, conjectures its optimality, and applies it to enhance density regularity and solve the stochastic transport equation.

Findings

Regularity condition for SDE solutions in space and Malliavin sense.

Improved regularity results for the densities of solutions.

Construction of classical solutions to the stochastic transport equation with Lipschitz drift.

Abstract

In this work we present a condition for the regularity, in both space and Malliavin sense, of strong solutions to SDEs driven by Brownian motion. We conjecture that this condition is optimal. As a consequence, we are able to improve the regularity of densities of such solutions. We also apply these results to construct a classical solution to the stochastic transport equation when the drift is Lipschitz.