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

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 $\phi$ 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 $\phi$ is asymptotically stable.

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• Levitan Almost Periodic Solutions of Linear Differential Equations