On the Strong Feller Property of Stochastic Delay Differential Equations with Singular Drift

TL;DR

This paper establishes the strong Feller property for stochastic delay differential equations with singular drift by extending existing methods and analyzing convergence in topological spaces.

Contribution

It introduces an extension of Maslowski and Seidler's approach to handle equations with singular, discontinuous drift coefficients.

Findings

Proves strong Feller property for a class of stochastic delay equations

Develops convergence techniques for discontinuous drifts

Extends existing methods to more general equations

Abstract

In this paper, we prove the strong Feller property for stochastic delay (or functional) differential equations with singular drift. We extend an approach of Maslowski and Seidler to derive the strong Feller property of those equations. The argumentation is based on the well-posedness and the strong Feller property of the equations' drift-free version. To this aim, we investigate a certain convergence of random variables in topological spaces in order to deal with discontinuous drift coefficients.