Stochastic equations with time-dependent singular drift

TL;DR

This paper establishes the unique solvability and Feller property for stochastic differential equations with a broad class of time-dependent singular drifts, including critical and weak integrability conditions.

Contribution

It proves well-posedness and regularity results for SDEs with highly singular, time-dependent drifts, extending previous classes of admissible vector fields.

Findings

Unique weak solutions are established for broad classes of singular drifts.

The Feller property is proven for these stochastic equations.

The results include drifts in the critical Ladyzhenskaya-Prodi-Serrin class and beyond.

Abstract

We prove unique weak solvability and Feller property for stochastic differential equations with drift in a large class of time-dependent vector fields. This class contains, in particular, the critical Ladyzhenskaya-Prodi-Serrin class, the weak $L^d$ class as well as some vector fields that are not even in $L^{2+\varepsilon}_{\rm loc}$, $\varepsilon>0$.