Abstract
We provide some sharp criteria for studying the ergodicity and asymptotic stability of general Feller semigroups on Polish metric spaces. As an application, the 2D Navier-Stokes equations with degenerate stochastic forcing will be simply revisited.
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References
Billingsley P. Convergence of Probability Measures, 2nd ed. New York: John Wiley & Sons, 1999
Bouleau N. Une structure uniforme sur un espace F(E; F). Cah Topol Géom Différ Catég, 1969, 11: 207–214
Bouleau N. On the coarsest topology preserving continuity. ArXiv: 0610373, 2006
Chen M F. From Markov Chains to Non-Equilibrium Particle Systems, 2nd ed. River Edge, NJ: World Scientific, 2004
Cotar C, Deuschel J D. Decay of covariances, uniqueness of ergodic component and scaling limit for a class of tbφ systems with non-convex potential. Ann Inst H Poincaré Probab Statist, 2012, 48: 819–853
Cotar C, Deuschel J D, Müller S. Strict convexity of the free energy for a class of non-convex gradient models. Comm Math Phys, 2009, 286: 359–376
E W N, Mattingly J C. Ergodicity for the Navier-Stokes equation with degenerate random forcing: Finite-dimensional approximation. Comm Pure Appl Math, 2001, 54: 1386–1402
Funaki T, Spohn H. Motion by mean curvature from the Ginzburg-Landau interface model. Comm Math Phys, 1997, 185: 1–36
Hairer M, Mattingly J C. Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing. Ann of Math, 2006, 164: 993–1032
Hairer M, Mattingly J C. A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs. Electron J Probab, 2011, 16: 658–738
Lasota A, Szarek T. Lower bound technique in the theory of a stochastic differential equation. J Differential Equations, 2006, 231: 513–533
Meyn S P, Tweedie R L. Markov Chains and Stochastic Stability, 2nd ed. Cambridge: Cambridge University Press, 2009
Röckner M, Zhu R C, Zhu X C. Sub- and supercritical stochastic quasi-geostrophic equation. ArXiv:1110. 1984, 2011
Szarek T. Feller processes on nonlocally compact spaces. Ann Probab, 2006, 34: 1849–1863
Szarek T, Ślęczka M, Urbański M. On stability of velocity vectors for some passive tracer models. Bull London Math Soc, 2010, 42: 923–936
Worm D T H, Hille S C. Ergodic decompositions associated with regular Markov operators on Polish spaces. Ergodic Theory Dynam Systems, 2011, 31: 571–597
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Gong, F., Liu, Y. Ergodicity and asymptotic stability of Feller semigroups on Polish metric spaces. Sci. China Math. 58, 1235–1250 (2015). https://doi.org/10.1007/s11425-015-4971-y
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DOI: https://doi.org/10.1007/s11425-015-4971-y