Almost automorphy and various extensions for stochastic processes

TL;DR

This paper explores different notions of pseudo almost automorphy for stochastic processes, demonstrating their limitations and establishing existence and uniqueness of solutions for certain stochastic differential equations with almost automorphic coefficients.

Contribution

It compares various modes of pseudo almost automorphy for stochastic processes and proves existence and uniqueness of solutions with these properties for specific SDEs.

Findings

Square-mean pseudo almost automorphy is too restrictive for SDEs.

Existence and uniqueness of solutions with almost automorphic properties are established.

Counterexample shows limitations of certain automorphy notions for stochastic equations.

Abstract

We compare different modes of pseudo almost automorphy and variants for stochastic processes: in probability, in quadratic mean, or in distribution in various senses. We show by a counterexample that square-mean (pseudo) almost automorphy is a property which is too strong for stochastic differential equations (SDEs). Finally, we consider two semilinear SDEs, one with almost automorphic coefficients and the second one with pseudo almost automorphic coefficients, and we prove the existence and uniqueness of a mild solution which is almost automorphic in distribution in the first case, and pseudo almost automorphic in distribution in the second case.