Asymptotic stability of some stochastic evolution equations. (English) Zbl 1033.60074

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.


60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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