Invariant sample measures and sample statistical solutions for nonautonomous stochastic lattice Cahn-Hilliard equation with nonlinear noise

TL;DR

This paper studies a nonautonomous stochastic lattice Cahn-Hilliard equation with nonlinear noise, proving the existence of attractors, constructing invariant measures, and introducing a new statistical solution concept.

Contribution

It introduces a novel approach to invariant sample measures and statistical solutions for nonautonomous stochastic PDEs with nonlinear noise.

Findings

Existence of pullback random attractors in $\u2113^2$

Construction of time-dependent invariant sample measures

Development of a stochastic Liouville type equation

Abstract

We consider a stochastic lattice Cahn-Hilliard equation with nonautonomous nonlinear noise. First, we prove the existence of pullback random attractors in $\ell^2$ for the generated nonautonomous random dynamical system. Then, we construct the time-dependent invariant sample Borel probability measures based on the pullback random attractor. Moreover, we develop a general stochastic Liouville type equation for nonautonomous random dynamical systems and show that the invariant sample measures obtained satisfy the stochastic Liouville type equation. At last, we define a new kind of statistical solution -- sample statistical solution corresponding to the invariant sample measures and show that each family of invariant sample measures is a sample statistical solution.