Existence of Invariant Measures for Reflected SPDEs

TL;DR

This paper proves the existence of invariant measures for reflected stochastic partial differential equations with a single barrier, using tightness arguments and uniform Lp bounds, filling a gap in the literature.

Contribution

It extends previous results by establishing invariant measures for one-barrier reflected SPDEs, employing a novel approach to handle the lack of a priori bounds.

Findings

Proves existence of invariant measures for one-barrier reflected SPDEs.

Develops a method to obtain uniform Lp bounds without a priori solution bounds.

Uses Krylov-Bogolyubov theorem to establish tightness and existence.

Abstract

In this article, we close a gap in the literature by proving existence of invariant measures for reflected SPDEs with only one reflecting barrier. This is done by arguing that the sequence (u(t, .)) is tight in the space of probability measures on continuous functions and invoking the Krylov-Bogolyubov theorem. As we no longer have an a priori bound on our solution as in the two-barrier case, a key aspect of the proof is the derivation of a suitable Lp bound which is uniform in time.