Release Time:2020-06-09  Hits:

Indexed by: Journal Papers

Date of Publication: 2020-08-05

Journal: JOURNAL OF DIFFERENTIAL EQUATIONS

Included Journals: SCIE

Volume: 269

Issue: 4

Page Number: 3652-3685

ISSN: 0022-0396

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

Abstract: 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.