Harnack Inequality for Distribution Dependent Stochastic Hamiltonian System

TL;DR

This paper establishes a dimension-free Harnack inequality for distribution-dependent stochastic Hamiltonian systems with specific regularity conditions on the drift, extending previous results to more general measure-dependent drifts.

Contribution

It extends the Harnack inequality to systems with drifts that are Hölder-Dini continuous in the measure variable, broadening the class of systems where such inequalities hold.

Findings

Proved dimension-free Harnack inequality for the system.

Extended existing results to drifts with Hölder-Dini continuity.

Applicable to degenerate and non-degenerate components with different regularity conditions.

Abstract

The dimension free Harnack inequality is established for the distribution dependent stochastic Hamiltonian system, where the drift is Lipschitz continuous in the measure variable under the distance induced by the H\"{o}lder-Dini continuous functions, which are $\beta (\beta>\frac{2}{3})$-H\"{o}lder continuous on the degenerate component and square root of Dini continuous on the non-degenerate one. The results extend the existing ones in which the drift is Lipschitz continuous in the measure variable under $L^2$-Wasserstein distance.