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

Cheban David; Liu Zhenxin

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CHEBAN, David, LIU, Zhenxin. Periodic, quasi-periodic, almost periodic, almost automorphic, Birkhoff recurrent and Poisson stable solutions for stochastic differential equations. In: Journal of Difference Equations and Applications, 2020, nr. 4(269), pp. 3652-3685. ISSN 0022-6198. DOI: https://doi.org/10.1016/j.jde.2020.03.014

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Journal of Difference Equations and Applications

Numărul 4(269) / 2020 / ISSN 0022-6198 /ISSNe 1090-2732

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

DOI:https://doi.org/10.1016/j.jde.2020.03.014

Pag. 3652-3685

Cheban David¹², Liu Zhenxin¹

¹ School of Mathematical Sciences, Dalian University of Technology,
² Moldova State University

Disponibil în IBN: 25 septembrie 2020

Rezumat

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.

Cuvinte-cheie
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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<subject>asymptotic stability</subject>
<subject>Birkhoff recurrent solution</subject>
<subject>Bohr/Levitan almost periodic solution</subject>
<subject>Poisson stable solution</subject>
<subject>quasi-periodic solution</subject>
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