Global martingale and pathwise solutions and infinite regularity of invariant measures for a stochastic modified Swift-Hohenberg equation

TL;DR

This paper proves the existence of solutions and invariant measures with increasing regularity for a stochastic modified Swift-Hohenberg equation, confirming a conjecture about infinite regularity under certain conditions.

Contribution

It establishes the existence of martingale and pathwise solutions, invariant measures, and their infinite regularity for a stochastic Swift-Hohenberg equation, advancing understanding of its long-term behavior.

Findings

Existence of local and global solutions in Sobolev spaces

Existence of invariant and ergodic measures

Invariant measures possess infinite regularity under certain conditions

Abstract

We consider a 2D stochastic modified Swift-Hohenberg equations with multiplicative noise and periodic boundary. First, we establish the existence of local and global martingale and pathwise solutions in the regular Sobolev space $H^{2m}$ for each $m\geqslant1$. Associated with the unique global pathwise solution, we obtain a Markovian transition semigroup. Then, we show the existence of invariant measures and ergodic invariant measures for this Markovian semigroup on $H^{2m}$. At last, we improve the regularity of the obtained invariant measures to $H^{2(m+1)}$. With appropriate conditions on the diffusion coefficient, we can deduce the infinite regularity of the invariant measures, which was conjectured by Glatt-Holtz \textit{et al.} in their situation.