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On exponential stability criteria of stochastic partial differential equations. (English) Zbl 0997.60065
Some criteria for the mean square and almost sure exponential stability of the zero solution of the nonlinear stochastic partial differential equation $$X_t=X_0+\int _0^tA(s,X_s) ds + \int _0^tB(s,X_s) dW_s$$ are given, where $A(t,\cdot): V\to V'$, $B(t,\cdot):V\to \Cal L(K,H)$, $V$ is a Banach space and $H,K$ are real separable Banach spaces such that $V\hookrightarrow H=H'\hookrightarrow V'$, where the injections are continuous and dense. The coefficients $A,B$ are assumed to satisfy the usual coercivity, boundedness, monotonicity, hemicontinuity and measurability conditions. The main results obtained by {\it T. Caraballo} and {\it J. Real} [Stochastic Anal. Appl. 12, No. 5, 517-525 (1994; Zbl 0808.93069)] are improved, which in particular concerns the case of non-autonomous equation. Several illustrating examples are also given.

MSC:
 60H15 Stochastic partial differential equations
Full Text:
References:
 [1] Caraballo, T.; Real, J.: On the pathwise exponential stability of non-linear stochastic partial differential equations. Stochast. anal. Appl. 12, No. 5, 517-525 (1994) · Zbl 0808.93069 [2] Chow, P. L.: Stability of nonlinear stochastic evolution equations. J. math. Anal. appl. 89, 400-419 (1982) · Zbl 0496.60059 [3] Haussmann, U. G.: Asymptotic stability of the linear itô equation in infinite dimension. J. math. Anal. appl. 65, 219-235 (1978) · Zbl 0385.93051 [4] Mao, X.R., 1994. Exponential Stability of Stochastic Differential Equations. Marcel Dekker, Inc., New York. · Zbl 0806.60044 [5] Pardoux, E., 1975. Equations aux dérivées partielles stochastiques nonlinéaires monotones. Thesis, Université Paris Sud.