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arXiv · 2202.13329

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

Jintao Wang·Xiaoqian Zhang·Chunqiu Li
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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.

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Jintao Wang, Xiaoqian Zhang, Chunqiu Li. 2022-02-27. Global martingale and pathwise solutions and infinite regularity of invariant measures for a stochastic modified Swift-Hohenberg equation. https://doi.org/10.1088/1361-6544/acc6b1

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