Harnack Inequalities for Functional SDEs with Multiplicative Noise and Applications

TL;DR

This paper establishes log-Harnack inequalities for functional SDEs with multiplicative noise using a novel coupling method, leading to applications such as strong Feller properties and heat kernel estimates for the associated semigroup.

Contribution

It introduces a new coupling approach to prove Harnack inequalities for delay SDEs with multiplicative noise, expanding the theoretical understanding of their regularity properties.

Findings

Established log-Harnack inequality for functional SDEs with multiplicative noise

Derived strong Feller property and heat kernel estimates for the transition semigroup

Investigated dimension-free Harnack inequality in this context

Abstract

By constructing a new coupling, the log-Harnack inequality is established for the functional solution of a delay stochastic differential equation with multiplicative noise. As applications, the strong Feller property and heat kernel estimates w.r.t. quasi-invariant probability measures are derived for the associated transition semigroup of the solution. The dimension-free Harnack inequality in the sense of \cite{W97} is also investigated.