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

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:269

Article

Cheban, David^1,2  Liu, Zhenxin¹ 

[1] Dalian Univ Technol, Sch Math Sci, Dalian 116024, Peoples R China

[2] State Univ Moldova, Fac Math & Informat, Dept Math, A Mateevich St 60, MD-2009 Kishinev, Moldova

关键词: Stochastic differential equation;    Quasi-periodic solution;    Bohr/Levitan almost periodic solution;    Almost automorphic solution;    Birkhoff recurrent solution;    Poisson stable solution;    Asymptotic stability;   

DOI  :  10.1016/j.jde.2020.03.014

来源: Elsevier

【 摘 要 】

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 equation dx(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. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free