Regularities and Exponential Ergodicity in Entropy for SDEs Driven by Distribution Dependent Noise

TL;DR

This paper develops new techniques to establish regularity inequalities and ergodic properties for McKean-Vlasov SDEs driven by distribution-dependent noise, extending previous results to more general noise settings.

Contribution

It introduces a noise decomposition method to prove log-Harnack inequality and Bismut formula for distribution-dependent noise SDEs, including degenerate cases.

Findings

Established log-Harnack inequality for distribution-dependent noise SDEs

Derived Bismut formula in both non-degenerate and degenerate cases

Proved exponential ergodicity in entropy for these systems

Abstract

As two crucial tools characterizing regularity properties of stochastic systems, the log-Harnack inequality and Bismut formula have been intensively studied for distribution dependent (McKean-Vlasov) SDEs. However, due to technical difficulties, existing results mainly focus on the case with distribution free noise. In this paper, we introduce a noise decomposition argument to establish the log-Harnack inequality and Bismut formula for SDEs with distribution dependent noise, in both non-degenerate and degenerate situations. As application, the exponential ergodicity in entropy is investigated.