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Invariant measures for semilinear stochastic equations. (English) Zbl 0758.60057
The stochastic evolution $$dX=(AX+F(X))dt+B(X)dW(t)$$ in a Hilbert space is considered, where $$A$$ is the infinitesimal generator of a $$C_ 0$$- semigroup $$S(t)$$, $$t\geq 0$$, $$F$$ and $$B$$ are linear or nonlinear mappings, and $$W(\cdot)$$ is a Wiener process. The existence and uniqueness of invariant measures are proved for dissipative equations and linear stable equations, while the existence is proved for equations with compact $$S(\cdot)$$.

##### MSC:
 60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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##### References:
 [1] DOI: 10.1016/0022-247X(82)90110-X · Zbl 0496.60059 · doi:10.1016/0022-247X(82)90110-X [2] Da Prato, G. and Ichikawa, A. 1985.Stability and Quadratic control for linear stochastic equations with unbounded coefficients, Vol. 4-B, 987–1001. Bollettino U.M.I. · Zbl 0598.93062 [3] Da Prato G., Stochastics 23 pp 1– (1987) · Zbl 0634.60053 · doi:10.1080/17442508708833480 [4] Da Prato G., Stochastics 15 pp 271– (1985) · Zbl 0581.60048 · doi:10.1080/17442508508833360 [5] Da Prato G., Differential and Integral Equations · Zbl 0594.34062 [6] Da Prato G., A note on stochastic convolution · Zbl 0758.60049 [7] G3tarek D., Probability Theory and Related Fields [8] Ethier S.N., Markov Processes. Characterization and Convergence (1986) · doi:10.1002/9780470316658 [9] Freidlin M.I., TAMS 305 pp 665– (1988) [10] Funaki T., Nagoya Math. J 89 pp 129– (1983) · Zbl 0531.60095 · doi:10.1017/S0027763000020298 [11] Hasminskii R.Z., Stochastic Stability of Differential Equations (1980) · doi:10.1007/978-94-009-9121-7 [12] DOI: 10.1016/0022-247X(78)90211-1 · Zbl 0385.93051 · doi:10.1016/0022-247X(78)90211-1 [13] Ichikawa A., Stochastics 11 pp 143– (1983) · Zbl 0531.93065 · doi:10.1080/17442508308833282 [14] DOI: 10.1016/0022-247X(82)90041-5 · Zbl 0497.93055 · doi:10.1016/0022-247X(82)90041-5 [15] Ichikawa A., Stochastics 12 pp 1– (1984) · Zbl 0538.60068 · doi:10.1080/17442508408833293 [16] Kotelenez P., Stochastics 21 pp 345– (1987) · Zbl 0622.60065 · doi:10.1080/17442508708833463 [17] Kozlov S.M., Trudy Seminara Petrouskogo 4 pp 147– (1978) [18] DOI: 10.1080/17442508908833585 · Zbl 0683.60037 · doi:10.1080/17442508908833585 [19] Maslowski, B. 1990. On ergodic behaviour of solutions to systems of stochastic reaction-diffusion equations with correlated noise. Proceedings of the 8th Winterschool in Stochastic Processes and Control Theory. 1990, Georgenthal. · Zbl 0719.60059 [20] Maslowski B., LNiCIS 136 pp 210– (1989) [21] Manthey R., Qualitative behaviour of solutions of stochastic reaction-diffusion equations · Zbl 0761.60055 · doi:10.1016/0304-4149(92)90062-U [22] Marcus R., TAMS 198 pp 177– (1974) [23] Sowers R. B., Large deviations for the invariant measure of a reaction-diffusion equation with non-Gaussian perturbation · Zbl 0767.60025 · doi:10.1007/BF01300562 [24] Walsh J.B., Ecole d’Ete de Probabilite de Saint Flour XIV - 1984, LNiM 1180 pp 265– (1984) · doi:10.1007/BFb0074920 [25] Zabczyk J.Linear stochastic systems in Hilbert spaces; structural properties and limit behaviourReport 236 of the Institute of Mathematics 1981 Polish Academy of Sciences. Also in Banach Center Publications vol. 41 (1985) 591 609 [26] Zabczyk J., On the stability of infinite dimensional linear stochastic systems 5 (1979) · Zbl 0435.93047
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