Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 35, pp. 1-21.

Title: Asymptotic behavior of stochastic functional differential evolution equation
       
Authors: Jason Clark (Oregon State Univ., Corvallis, OR, USA)
         Oleksandr Misiats (Virginia Commonwealth Univ., Richmond, USA)
         Viktoriia Mogylova (Igor Sikorsky Kyiv Polytechnic Institute, Ukraine)
         Oleksandr Stanzhytsky (Taras Shevchenko National Univ. of Kyiv, Ukraine)

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.