Exponential Ergodicity and Propagation of Chaos for Path-Distribution Dependent Stochastic Hamiltonian System

TL;DR

This paper establishes log-Harnack inequalities, exponential ergodicity, and propagation of chaos for path-distribution dependent stochastic Hamiltonian systems, advancing understanding of their long-term behavior and convergence properties.

Contribution

It derives the log-Harnack inequality for these systems and demonstrates exponential ergodicity and quantitative propagation of chaos, extending existing results to path-distribution dependent cases.

Findings

Log-Harnack inequality for path-distribution dependent systems

Exponential ergodicity in relative entropy

Quantitative propagation of chaos in Wasserstein distance

Abstract

By Girsanov's thoerem and using the existing log-Harnack inequality for distribution independent SDEs, the log-Harnack inequality is derived for path-distribution dependent stochastic Hamiltonian systems. As an application, the exponential ergodicity in relative entropy is obtained by combining with transportation cost inequality. In addition, the quantitative propagation of chaos in the sense of Wasserstein distance, which together with the coupling by change of measure implies the quantitative propagation of chaos in total variation norm as well as relative entropy are obtained.