Asymptotic Log-Harnack Inequality and Applications for Stochastic Systems of Infinite Memory

TL;DR

This paper establishes the asymptotic log-Harnack inequality for various stochastic systems with infinite memory, leading to important properties like heat kernel estimates, uniqueness of invariant measures, and asymptotic regularity.

Contribution

It introduces the asymptotic log-Harnack inequality for complex stochastic systems with infinite memory, a novel extension in the field.

Findings

Derived asymptotic heat kernel estimates

Proved uniqueness of invariant probability measures

Established asymptotic strong Feller property

Abstract

The asymptotic log-Harnack inequality is established for several different models of stochastic differential systems with infinite memory: non-degenerate SDEs, Neutral SDEs, semi-linear SPDEs, and stochastic Hamiltonian systems. As applications, the following properties are derived for the associated segment Markov semigroups: asymptotic heat kernel estimate; uniqueness of the invariant probability measure; asymptotic gradient estimate and hence, asymptotically strong Feller property; and asymptotic irreducibilty.