Electron. J. Differential Equations, Vol. 2023 (2023), No. 35, pp. 1-21.

Asymptotic behavior of stochastic functional differential evolution equation

Jason Clark, Oleksandr Misiats, Viktoriia Mogylova, Oleksandr Stanzhytskyi

In this work we study the long time behavior of nonlinear stochastic functional-differential equations in Hilbert spaces. In particular, we start with establishing the existence and uniqueness of mild solutions. We proceed with deriving a priory uniform in time bounds for the solutions in the appropriate Hilbert spaces. These bounds enable us to establish the existence of invariant measure based on Krylov-Bogoliubov theorem on the tightness of the family of measures. Finally, under certain assumptions on nonlinearities, we establish the uniqueness of invariant measures.

Submitted June 22, 2022. Published April 12, 2023.
Math Subject Classifications: 35R60, 60H15, 92C35.
Key Words: Stochastic integral; mild solution; semigroup; white noise; delay differential equation; invariant measure.
DOI: https://doi.org/10.58997/ejde.2023.35

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Jason Clark
College of Engineering
Oregon State University
Corvallis, OR 97331, USA
email: jason.clark@oregonstate.edu
Oleksandr Misiats
Department of Mathematics
Virginia Commonwealth University
Richmond, VA 23284, USA
email: omisiats@vcu.edu
Viktoriia Mogylova
Department of Physics and Mathematics
Igor Sikorsky Kyiv Polytechnic Institute, Ukraine
email: mogylova.viktoria@gmail.com
Oleksandr Stanzhytskyi
Department of Mathematics
Taras Shevchenko National University of Kyiv, Ukraine
email: ostanzh@gmail.com

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