Log-Harnack Inequality for Stochastic Differential Equations in Hilbert Spaces and its Consequences

TL;DR

This paper proves a log-Harnack inequality for stochastic differential equations in Hilbert spaces with non-additive noise, leading to results like the strong Feller property and entropy-cost inequalities for the solution semigroup.

Contribution

It introduces a novel log-Harnack inequality for SDEs in Hilbert spaces with non-additive noise, expanding the understanding of their regularity properties.

Findings

Established a log-Harnack inequality for the semigroup of solutions.

Derived the strong Feller property for the solution semigroup.

Proved entropy-cost inequalities related to the semigroup.

Abstract

A logarithmic type Harnack inequality is established for the semigroup of solutions to a stochastic differential equation in Hilbert spaces with non-additive noise. As applications, the strong Feller property as well as the entropy-cost inequality for the semigroup are derived with respect to the corresponding distance (cost function).