Convergence of bi-spatial pullback random attractors and stochastic Liouville type equations for nonautonomous stochastic p-Laplacian lattice system

TL;DR

This paper investigates the long-term behavior and convergence of stochastic attractors and invariant measures for a nonautonomous stochastic p-Laplacian lattice system with multiplicative noise, establishing key continuity and convergence results.

Contribution

It proves the upper semi-continuity of pullback random attractors and demonstrates the convergence of invariant measures and stochastic Liouville equations for the system.

Findings

Proved upper semi-continuity of pullback random attractors.

Established convergence of invariant measures in .

Showed invariant measures satisfy stochastic Liouville equations.

Abstract

We consider convergence properties of the long-term behaviors with respect to the coefficient of the stochastic term for a nonautonomous stochastic $p$-Laplacian lattice equation with multiplicative noise. First, the upper semi-continuity of pullback random $(\ell^2,\ell^q)$-attractor is proved for each $q\in[1,+\infty)$. Then, a convergence result of the time-dependent invariant sample Borel probability measures is obtained in $\ell^2$. Next, we show that the invariant sample measures satisfy a stochastic Liouville type equation and a termwise convergence of the stochastic Liouville type equations is verified. Furthermore, each family of the invariant sample measures is turned out to be a sample statistical solution, which hence also fulfills a convergence consequence.