Periodic, quasi-periodic, almost periodic, almost automorphic, Birkhoff recurrent and Poisson stable solutions for stochastic differential equations

The paper is dedicated to studying the problem of Poisson stability (in particular stationarity, periodicity, quasi-periodicity, Bohr almost periodicity, Bohr almost automorphy, Birkhoff recurrence, almost recurrence in the sense of Bebutov, Levitan almost periodicity, pseudo-periodicity, pseudo-rec...

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Bibliographic Details
Published in:Journal of Differential Equations 2020-08, Vol.269 (4), p.3652-3685
Main Authors: Cheban, David, Liu, Zhenxin
Format: Article
Language:English
Subjects:
Almost automorphic solution
Asymptotic stability
Birkhoff recurrent solution
Bohr/Levitan almost periodic solution
Poisson stable solution
Quasi-periodic solution
Stochastic differential equation
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Summary:The paper is dedicated to studying the problem of Poisson stability (in particular stationarity, periodicity, quasi-periodicity, Bohr almost periodicity, Bohr almost automorphy, Birkhoff recurrence, almost recurrence in the sense of Bebutov, Levitan almost periodicity, pseudo-periodicity, pseudo-recurrence, Poisson stability) of solutions for semi-linear stochastic equationdx(t)=(Ax(t)+f(t,x(t)))dt+g(t,x(t))dW(t)(⁎) with exponentially stable linear operator A and Poisson stable in time coefficients f and g. We prove that if the functions f and g are appropriately “small”, then equation (⁎) admits at least one solution which has the same character of recurrence as the functions f and g. We also discuss the asymptotic stability of these Poisson stable solutions.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2020.03.014