Bohr/Levitan Almost Periodic and Almost Automorphic Solutions of Linear Stochastic Differential Equations without Favard's Separation Condition

TL;DR

This paper establishes conditions under which linear stochastic differential equations with almost periodic or automorphic coefficients have unique solutions that are also almost periodic or automorphic in distribution, without relying on Favard's separation condition.

Contribution

It proves the existence and uniqueness of almost periodic and automorphic solutions for linear stochastic equations under less restrictive conditions than previously required.

Findings

Unique Bohr/Levitan almost periodic solutions in distribution exist under stability and precompactness.

Solutions are characterized without Favard's separation condition.

The results extend the theory of stochastic differential equations with almost periodic coefficients.

Abstract

We prove that the linear stochastic equation $dx(t)=(A(t)x(t)+f(t))dt+g(t)dW(t)$ with linear operator $A(t)$ generating a continuous linear cocycle $\varphi$ and Bohr/Levitan almost periodic or almost automorphic coefficients $(A(t),f(t),g(t))$ admits a unique Bohr/Levitan almost periodic (respectively, almost automorphic) solution in distribution sense if it has at least one precompact solution on $\mathbb R_{+}$ and the linear cocycle $\varphi$ is asymptotically stable.