A Modified log-Harnack inequality and asymptotically strong Feller property

TL;DR

This paper introduces a new modified log-Harnack inequality that implies the asymptotically strong Feller property, demonstrated through an example involving the 2D stochastic Navier-Stokes equation with degenerate noise.

Contribution

It presents a novel functional inequality that generalizes existing criteria for the asymptotically strong Feller property, with applications to stochastic PDEs.

Findings

The new inequality implies the ASF property.

Application to 2D stochastic Navier-Stokes equation with degenerate noise.

Demonstrates the inequality via asymptotic coupling.

Abstract

We introduce a new functional inequality, which is a modification of log-Harnack inequality established in [20] and [29], and prove that it implies the asymptotically strong Feller property (ASF). This inequality seems to generalize the criterion for ASF in [Proposition 3.12,14]. As a example, we show by an asymptotic coupling that 2D stochastic Navier-Stokes equation driven by highly degenerate but \emph{essentially elliptic} noises satisfies our modified log-Harnack inequality.