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arXiv · 2404.14798

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

Jintao Wang·Dongdong Zhu·Chunqiu Li
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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.

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Jintao Wang, Dongdong Zhu, Chunqiu Li. 2024-04-23. Invariant sample measures and sample statistical solutions for nonautonomous stochastic lattice Cahn-Hilliard equation with nonlinear noise. https://arxiv.org/abs/2404.14798

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