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Keywords:
dissipative system; compact semigroup; exponential ergodicity; spectral gap
dissipative system; compact semigroup; exponential ergodicity; spectral gap
Summary:
We study ergodic properties of stochastic dissipative systems with additive noise. We show that the system is uniformly exponentially ergodic provided the growth of nonlinearity at infinity is faster than linear. The abstract result is applied to the stochastic reaction diffusion equation in $\mathbb{R}^d$ with $d\le 3$.
References:
[1] S. Aida: Uniform positivity improving property, Sobolev inequalities and spectral gaps. J. Funct. Anal. 158 (1998), 152–185. MR 1641566 | Zbl 0914.47041
[2] R. Arima: On general boundary value problem for parabolic equations. J. Math. Kyoto Univ. 4 (1964), 207–243. DOI 10.1215/kjm/1250524714 | MR 0197997 | Zbl 0143.13902
[3] Mu-Fa Chen: Equivalence of exponential ergodicity and $L^2$-exponential convergence for Markov chains. Stochastic Process. Appl. 87 (2000), 281–297. MR 1757116
[4] A. Chojnowska-Michalik and B. Goldys: Existence, uniqueness and invariant measures for stochastic semilinear equations in Hilbert spaces. Probab. Theory Related Fields 102 (1995), 331–356. DOI 10.1007/BF01192465 | MR 1339737
[5] A. Chojnowska-Michalik and B. Goldys: Nonsymmetric Ornstein-Uhlenbeck semigroup as second quantized operator. J. Math. Kyoto Univ. 36 (1996), 481–498. DOI 10.1215/kjm/1250518505 | MR 1417822
[6] G. Da Prato: Large asymptotic behaviour of Kolmogorov equations in Hilbert spaces. Partial Differential Equations (Praha, 1998), Chapman & Hall/CRC, Boca Raton, 2000, pp. 111–120. MR 1713879 | Zbl 0946.47027
[7] G. Da Prato: Poincaré inequality for some measures in Hilbert spaces and application to spectral gap for transition semigroups. Ann. Scuola Norm. Sup. Pisa Cl. Sci. $(4)$ 25 (1997), 419–431. MR 1655525 | Zbl 1039.60053
[8] G. Da Prato and J. Zabczyk: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge, 1992. MR 1207136
[9] G. Da Prato and J. Zabczyk: Ergodicity for Infinite Dimensional Systems. Cambridge University Press, Cambridge, 1996. MR 1417491
[10] G. Da Prato, A. Debussche and B. Goldys: Invariant measures of non symmetric dissipative stochastic systems. (to appear).
[11] G. Da Prato, D. Gątarek and J. Zabczyk: Invariant measures for semilinear stochastic equations. Stochastic Anal. Appl. 10 (1992), 387–408. DOI 10.1080/07362999208809278 | MR 1178482
[12] D. Gątarek and B. Goldys: On invariant measures for diffusions on Banach spaces. Potential Anal. 7 (1997), 539–553. MR 1467205
[13] B. Goldys and B. Maslowski: Ergodic control of semilinear stochastic equations and the Hamilton-Jacobi equation. J. Math. Anal. Appl. 234 (1999), 592–631. DOI 10.1006/jmaa.1999.6387 | MR 1689410
[14] N. Jain and B. Jamison: Contributions to Doeblin’s theory of Markov processes. Z. Wahrscheinlichkeitstheorie und Verw. Geb. 8 (1967), 19–40. MR 0221591
[15] S. Jacquot and G. Royer: Ergodicité d’une classe d’équations aux dérivées partielles stochastiques. C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), 231–236. MR 1320362
[16] A. Lasota and M. C. Mackey: Chaos, Fractals and Noise. Springer-Verlag, New York, 1994. MR 1244104
[17] B. Maslowski: Strong Feller property for semilinear stochastic evolution equations and applications. Stochastic Systems and Optimization (Warsaw, 1988). Lecture Notes in Control Inform. Sci. Vol. 136, Springer, Berlin, 1989, pp. 210–224. MR 1180781 | Zbl 0686.60053
[18] B. Maslowski: On ergodic behaviour of solutions to systems of stochastic reaction-diffusion equations with correlated noise. Stochastic Processes and Related Topics (Georgenthal, 1990), Akademie-Verlag, Berlin, 1991, pp. 93–102. MR 1127885 | Zbl 0719.60059
[19] B. Maslowski: On probability distributions of solutions of semilinear stochastic evolution equations. Stochastics Stochastics Rep. 45 (1993), 17–44. DOI 10.1080/17442509308833854 | MR 1277360 | Zbl 0792.60058
[20] B. Maslowski and J. Seidler: Probabilistic approach to the strong Feller property. Probab. Theory Related Fields 118 (2000), 187–210. DOI 10.1007/s440-000-8014-0 | MR 1790081
[21] B. Maslowski and J. Seidler: Invariant measures for nonlinear SPDE’s: Uniqueness and stability. Arch. Math. (Brno) 34 (1998), 153–172. MR 1629692
[22] S. P. Meyn and R. L. Tweedie: Markov Chains and Stochastic Stability. Springer-Verlag, London, 1993. MR 1287609
[23] S. Peszat and J. Seidler: Maximal inequalities and space-time regularity of stochastic convolutions. Math. Bohem. 123 (1998), 7–32. MR 1618707
[24] G. O. Roberts and J. S. Rosenthal: Geometric ergodicity and hybrid Markov chains. Electron. Comm. Probab. 2 (1997), 13–25. DOI 10.1214/ECP.v2-981 | MR 1448322
[25] M. Röckner and T. S. Zhang: Probabilistic representations and hyperbound estimates for semigroups. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 2 (1999), 337–358. MR 1810996
[26] J. Seidler: Ergodic behaviour of stochastic parabolic equations. Czechoslovak Math. J. 47 (122) (1997), 277–316. DOI 10.1023/A:1022821729545 | MR 1452421 | Zbl 0935.60041
[27] T. Shardlow: Geometric ergodicity for stochastic PDEs. Stochastic Anal.Appl. 17 (1999), 857–869. DOI 10.1080/07362999908809639 | MR 1714903 | Zbl 0933.60074
[28] E. Sinestrari: Accretive differential operators. Boll. Un. Mat. Ital B. (5) 13 (1976), 19–31. MR 0425682 | Zbl 0343.35016
[29] Wang Feng-Yu: Functional inequalities, semigroup properties and spectrum estimate. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3 (2000), 263–295. MR 1812701
[30] Wu Liming: Uniformly integrable operators and large deviations for Markov processes. J. Funct. Anal. 172 (2000), 301–376. MR 1753178 | Zbl 0957.60032
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