Logarithmic Harnack inequalities for transition semigroups in Hilbert spaces

TL;DR

This paper proves a logarithmic Harnack inequality for the transition semigroup of a stochastic differential equation in a Hilbert space, under less restrictive conditions than previous results, with applications discussed.

Contribution

It introduces a new logarithmic Harnack inequality for infinite-dimensional stochastic equations with weaker assumptions than prior work.

Findings

Established a logarithmic Harnack inequality for Hilbert space-valued SDEs.

Extended applicability of inequalities to broader classes of stochastic equations.

Provided applications demonstrating the utility of the inequalities.

Abstract

We consider the stochastic differential equation $$ \left\{ \begin{array}{lc} dX(t)=[AX(t)+F(X(t))]dt+C^{1/2}dW(t), & t>0;\\ X(0)=x \in \mathcal{X}; \end{array}\right. $$ where $\mathcal{X}$ is a Hilbert space, $\{W(t)\}_{t\geq 0}$ is a $\mathcal{X}$-valued cylindrical Wiener process, $A, C$ are suitable operators on $\mathcal{X}$ and $F:{\rm Dom}\,(F)\subseteq \mathcal{X}\to \mathcal{X}$ is a smooth enough function. We establish a logarithmic Harnack inequality for the transition semigroup $\{P(t)\}_{t\geq 0}$ associated with the stochastic problem above, under less restrictive conditions than those considered in the literature. Some applications to these inequalities are also shown.