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Existence, uniqueness, and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces. (English) Zbl 1009.34074
The authors consider the following abstract stochastic evolution equation in Hilbert space with time delay $$ dX(t)=(-AX(t)+f(t,X_t))dt+g(t,X_t)dW(t),\quad t\ge t_0,$$ where $-A$ is the generator of an analytic semigroup; $X_t(u):=X(t+u)$ for $u\in [-r,0]$ is the segment of the process from $t-r$ to $t$; $f$ and $g$ are some suitably regular functions and $W$ is a Hilbert space-valued Wiener process with a nuclear covariance operator. Using semigroup theory, the authors give conditions for the existence and uniqueness of mild solutions to this equation and then find conditions for the $p$th moment and almost sure exponential stability of the solutions. Finally, the results are applied to a stochastic delay reaction-diffusion equation on a bounded interval with zero boundary conditions.

34K50Stochastic functional-differential equations
34D08Characteristic and Lyapunov exponents
60H15Stochastic partial differential equations
35K57Reaction-diffusion equations
Full Text: DOI
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