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Zbl 1033.60074
El-Borai, Mahmoud M.; Moustafa, Osama L.; Ahmed, Hamdy M.
Asymptotic stability of some stochastic evolution equations. (English)
[J] Appl. Math. Comput. 144, No. 2-3, 273-286 (2003). ISSN 0096-3003

The authors study the Hilbert space-valued stochastic differential equation $$dX+(A+Q)Xdt=BX dW,$$ where A is the generator of a strongly continuous semigroup $U(t)$, $W$ is a Hilbert space-valued Brownian motion, $B$ is a bounded linear operator and $Q$ is a general closed linear operator. Under the hypothesis of exponential stability of the semigroup $U(t)$ and a growth condition on $Q$ there exists a mild solution to this equation that has certain stability properties: for $t\to\infty$ the solution converges exponentially fast to zero in a mean square as well as in a pathwise sense.
[Markus Reiss (Berlin)]

MSC 2000:: *60H15 Stochastic partial differential equations

Keywords: stochastic evolution equations; exponential stability; mild solution

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