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Zbl 1009.34074
Taniguchi, Takeshi; Liu, Kai; Truman, Aubrey
Existence, uniqueness, and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces. (English)
[J] J. Differ. Equations 181, No.1, 72-91 (2002). ISSN 0022-0396

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.
[Markus Reiss (Berlin)]

MSC 2000:: *34K50 Stochastic delay equations
34D08 Lyapunov exponents
60H15 Stochastic partial differential equations
35K57 Reaction-diffusion equations

Keywords: mild solution; exponential stability; Lyapunov exponent; stochastic delay reaction-diffusion equation

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Zentralblatt MATH Berlin [Germany]