Non-local parabolic equations with singular (Morrey) time-inhomogeneous drift

TL;DR

This paper establishes Sobolev regularity for solutions to non-local parabolic equations with singular, time-inhomogeneous drifts, providing new bounds and regularity estimates relevant for stochastic processes and McKean-Vlasov equations.

Contribution

It introduces novel Sobolev regularity estimates for non-local parabolic equations with minimal assumptions on unbounded drifts, and derives Krylov bounds for associated stochastic processes.

Findings

Sobolev regularity estimates for solutions with unbounded drifts

Krylov bounds for Feller stable processes

A priori regularity estimates for McKean-Vlasov equations

Abstract

We obtain Sobolev regularity estimates for solutions of non-local parabolic equations with locally unbounded drift satisfying some minimal assumptions. These results yield Krylov bound for the corresponding Feller stable process as well as some a priori regularity estimates on solutions of McKean-Vlasov equations. A key element of our arguments is a parabolic operator norm inequality that we prove using some ideas of Adams and Krylov.