Parabolic equations and SDEs with time-inhomogeneous Morrey drift

TL;DR

This paper establishes the unique weak solvability of certain stochastic differential equations with time-inhomogeneous Morrey class drifts, using Sobolev regularity theory for associated parabolic equations, expanding the class of solvable SDEs.

Contribution

It proves the existence and uniqueness of weak solutions for SDEs with drifts in the largest Morrey class, a significant extension in stochastic analysis.

Findings

Weak solutions form a Feller evolution family.

Solutions exist for drifts with minimal integrability close to the critical threshold.

The approach relies on advanced Sobolev regularity theory for parabolic PDEs.

Abstract

We prove the unique weak solvability of stochastic differential equations with time-inhomogeneous drift in essentially the largest (scaling-invariant) Morrey class, i.e.\,with integrability parameter $q>1$ close to $1$. The constructed weak solutions constitute a Feller evolution family. The proofs are based on a detailed Sobolev regularity theory of the corresponding parabolic equation.