## 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.

### MSC:

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

 [1] Curtain, R.F.; Pritchard, A.J., Functional analysis in modern applied mathematics, (1977), Academic Press New York · Zbl 0448.46002 [2] Curtain, R.F., Stability of stochastic partial differential equations, J. math. anal. appl., (1981) [3] M. El Borai, A note on some stochastic initial value problems, in: Proceedings of the First Arabic Conference in Physics and Mathematics, Baghdad, 1978 [4] Flandoli, F.; Schaumlffel, K.U., Stochastic parabolic equations in bounded domains: random evolution operators and Lyapunov exponents, Stoch. stoch. rep., 29, (1990) [5] Gihman, I.I.; Skorhod, A.V., Stochastic differential equations, (1972), Springer Verlag Berlin · Zbl 0242.60003 [6] Haussmann, U.G., Asymptotic stability of the linear ito equation in infinite dimensions, J. math. anal. appl., (1978) · Zbl 0385.93051 [7] Ikeda, N.; Watanabe, S., Stochastic differential equations and diffusion processes, (1981), North Holland Publishing Company · Zbl 0495.60005 [8] Klimov, G., Probability theory and mathematical statistics, (1986), Mir Publishers Moscow · Zbl 0658.60002 [9] Krylov, N.V.; Rozovskii, B.L., Ito equations in Banach space and strongly parabolic stochastic partial differential equations, Soviet mathematics doki., (1979) · Zbl 0435.60061 [10] Krylov, N.V.; Rozovskii, B.L., Stochastic evolution equations, J. soviet math., 16, (1981) · Zbl 0474.60049 [11] Mohammed, S.E.A.; Scheutozow, M.K.R., Lyapunov exponents of linear stochastic functional differential equations, Annals prob., 25, 2, (1997)
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