Invariant sample measures and random Liouville type theorem for a nonautonomous stochastic <inline-formula><tex-math id="M1">$ p $</tex-math></inline-formula>-Laplacian equation

Abstract

&lt;p style='text-indent:20px;'&gt;We introduce invariant sample measures to nonautonomous random dynamical systems, and consider the dynamical behaviors of a nonautonomous stochastic &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ p $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-Laplacian equation with multiplicative noise on a bounded domain. We first use the asymptotic a priori estimate method to prove the existence of &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ (L^2, L^q) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-pullback random attractors for the generated nonautonomous random dynamical system. Then, we establish the existence of invariant sample measures and random Liouville type theorem in &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ L^2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; for this equation. Moreover, the invariant sample measures are carried by &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ W_0^{1, p}\cap L^q $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;.&lt;/p&gt;

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

“…References [3,4,[26][27][28][29][30][31] studied the limit property or stochastic stability of some invariant measures for stochastic processes as the noise vanishes or perturbs. There have been more references concerning the existence of invariant measures, such as [2,8,13,14,19,37,45,51,52,54] and their references. For the existence of invariant measures for stochastic partial differential equation, it often needs to assume that the drift terms to be dissipative.…”

Invariant measures for first order nonlocal lattice dynamical systems with delays

“…This is because measurements of many important aspects of the turbulent flows are actually the measurements of some time-average quantities. Nowadays, there have been a series of works on invariant measures of evolution equations; see [4,12,5,13,14,24,32,33,34,45,44,50,56,53] for continuous systems. By using the generalized Banach limit, Lukaszewicz, Real and Robinson [33] constructed invariant measures for general continuous dynamical systems on metric spaces.…”