Pseudo almost automorphic solutions and invariant measures for non-autonomous stochastic evolution variational inequalities driven by Lévy noise

Source abstract

This paper introduces a novel notion of pseudo almost automorphic invariant measures for non-autonomous stochastic systems. Aided by this theoretical framework, we analyze the pullback asymptotic statistical behavior of a stochastic evolution variational inequality, with a focus on a system comprising a Lévy-noise-driven stochastic evolution equation coupled with a stochastic variational inequality. We first establish the existence and uniqueness of a mild solution on R by applying the Banach fixed point theorem to the stochastic evolution variational inequality. Subsequently, we prove that this solution exhibits square-mean pseudo almost automorphy and derive its general stability properties, encompassing exponential, polynomial and logarithmic decay. Owing to the fundamental functional-analytic differences between evolution equations and variational inequalities, we are led to introduce the notion of a square-mean pseudo-weakly almost automorphic process and to develop original analytical methods. Building on this foundation, we further prove that the distribution of the square-mean pseudo almost automorphic solution is the unique pseudo almost automorphic invariant measure, with the general stability established in terms of the Wasserstein metric. Notably, the pullback statistical behavior characterized by pseudo almost automorphic invariant measures has not yet been studied. This paper thus fills this gap, thereby offering an analytical framework applicable to a broader class of inhomogeneous Markov processes.