Abstract.
We prove uniqueness, ergodicity and strongly mixing property of the invariant measure for a class of stochastic reaction-diffusion equations with multiplicative noise, in which the diffusion term in front of the noise may vanish and the deterministic part of the equation is not necessary asymptotically stable. To this purpose, we show that the L1-norm of the difference of two solutions starting from any two different initial data converges ℙ-a.s. to zero, as time goes to infinity.
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References
Arnold, L., Crauel, H., Wihstutz, V.: Stabilization of linear systems by noise. SIAM Journal on Control and Optimization 21, 451–461 (1983)
Arnold, L.: Stabilization by noise revisited. Z. Angew. Math. Mech. 70, 235–246 (1990)
Caraballo, T., Liu, K., Mao, X.: On stabilization of partial differential equations by noise. Nagoya Mathematical Journal 161, 155–170 (2001)
Caraballo, T., Kloeden, P.E., Schmalfuss, B.: Exponentially stable stationary solutions for stochastic evolution equations and their perturbation. Preprint 2003
Cerrai, S.: Second Order PDE’s in Finite and Infinite Dimension. A Probabilistic Approach, Lecture Notes in Mathematics Series 1762, Springer Verlag, 2001
Cerrai, S.: Stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term. Probab. Theory Relat. Fields 125, 271–304 (2003)
Cerrai, S., Röckner, M.: Large deviations for stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term. Annals of Probability 32, 1–40 (2004)
Cerrai, S., Röckner, M.: Large deviations for invariant measures of general stochastic reaction-diffusion systems. Comptes Rendus Acad. Sci. Paris Sér. I Math. 337, 597–602 (2003)
Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite dimensions. Cambridge University Press, Cambridge, 1992
Da Prato, G., Zabczyk, J.: Ergodicity for Infinite Dimensional Systems. London Mathematical Society, Lecture Notes Series 229, Cambridge University Press, Cambridge, 1996
Hairer, M.: Exponential mixing properties of stochastic PDEs throught asymptotic coupling. Probab. Theory Relat. Fields 124, 345–380 (2002)
Ichikawa, A.: Semilinear stochastic evolution equations: boundedness, stability and invariant measures. Stochastics and Stochastics Reports 12, 1–39 (1984)
Kuksin, S., Shirikyan, A.: A coupling approach to randomly forced PDEs I. Communications in Mathematical Physics 221, 351–366 (2001)
Donati-Martin, C., Pardoux, E.: White noise driven SPDEs with reflection. Probab. Theory Relat. Fields 95, 1–24 (1993)
Kwiecińska, A.A.: Stabilization of partial differential equations by noise. Stochastic Processes and their Applications 79, 179–184 (1999)
Mattingly, J.: Exponential convergence for the stochastically forced Navier-Stokes equations and other partially dissipative dynamics. Communications in Mathematical Physics 230, 421–462 (2002)
Mueller, C.: Coupling and invariant measures for the heat equation with noise. Annals of Probability 21, 2189–2199 (1993)
Pardoux, E., Wihstutz, V.: Lyapunov exponents and rotation number of two dimensional linear stochastic systems with small diffusion. SIAM Journal on Applied Mathematics 48, 442–457 (1988)
Pardoux, E., Wihstutz, V.: Lyapunov exponents of linear stochastic systems with large diffusion term. Stochastic Processes and their Applications 40, 289–308 (1992)
Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Third edition, Springer Verlag, 1999
Sowers, R.: Large deviations for the invariant measure of a reaction-diffusion equation with non-Gaussian perturbations. Probab. Theory Relat. Fields 92, 393–421 (1992)
Walsh, J.B.: An introduction to stochastic partial differential equations. Ecole d’Eté de Probabilité de Saint-Flour XIV (1984), P.L. Hennequin editor, Lectures Notes in Mathematics 1180, pp. 265–439
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This paper was written while the author was visiting the Scuola Normale Superiore, Pisa
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Cerrai, S. Stabilization by noise for a class of stochastic reaction-diffusion equations. Probab. Theory Relat. Fields 133, 190–214 (2005). https://doi.org/10.1007/s00440-004-0421-4
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DOI: https://doi.org/10.1007/s00440-004-0421-4