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

zbMATH Open1499.34320arXiv1707.08723MaRDI QIDQ5063358

David Cheban

Publication date: 17 March 2022

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.

Full work available at URL: https://arxiv.org/abs/1707.08723

zbMATH Keywords

almost automorphic solutionsFavard theorylinear stochastic differential equationsBohr/Levitan almost periodic solutions

Mathematics Subject Classification ID

Stochastic ordinary differential equations (aspects of stochastic analysis) (60H10) Ordinary differential equations and systems with randomness (34F05) Linear differential equations in abstract spaces (34G10) Almost and pseudo-almost periodic solutions to ordinary differential equations (34C27) Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions (43A60)

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