Jason Clark, Oleksandr Misiats, Viktoriia Mogylova, Oleksandr Stanzhytskyi
Abstract:
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 |
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Oleksandr Misiats Department of Mathematics Virginia Commonwealth University Richmond, VA 23284, USA email: omisiats@vcu.edu |
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Viktoriia Mogylova Department of Physics and Mathematics Igor Sikorsky Kyiv Polytechnic Institute, Ukraine email: mogylova.viktoria@gmail.com |
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Oleksandr Stanzhytskyi Department of Mathematics Taras Shevchenko National University of Kyiv, Ukraine email: ostanzh@gmail.com |
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