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

2024/04/23 by Wang, Jintao, Zhu, Dongdong, Li, Chunqiu
#37L30 #37L40 #37L60 #60H10 #Dynamical Systems (math.DS) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2404.14798

Abstract

We consider a stochastic lattice Cahn-Hilliard equation with nonautonomous nonlinear noise. First, we prove the existence of pullback random attractors in ℓ² 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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