zbMATH — the first resource for mathematics

On ergodicity of some Markov processes. (English) Zbl 1214.60035
The article proves some ergodicity results for Feller processes on general (i.e., not necessarily locally compact) Polish state spaces. These are then applied to stochastic evolution equations with additive noise in Hilbert spaces.
\((\chi, \rho)\) is supposed to be a metric Polish space. \(C_b(\chi)\) denotes the space of bounded continuous functions on \(\chi\), and Lip\(_b(\chi)\) denotes bounded Lipschitz continuous functions. The ball of size \(\delta\) with center \(x\) is denoted by \(B(x,\delta)\). \(((Z^x_t)_{t \geq 0}, x \in \chi)\) is supposed to be a stochastically continuous Markov family with transition semigroup \((P_t)_{t \geq 0}\). \((P_t)\) is supposed to be a Feller semigroup, i.e., \(P_t(C_b(\chi)) \subseteq C_b(\chi)\).
The authors introduce the “e-property”: The semigroup \((P_t)_{t\geq 0}\) has the e-property if for any \(\psi \in\) Lip\(_b(\chi)\), for any \(x \in \chi\), and for any \(\varepsilon > 0\) there exists \(\delta > 0\) such that for all \(z\) with \(d(x,z) < \delta\) and for all \(t \geq 0\): \[ |P_t \psi(x) - P_t \psi(z)| < \varepsilon \] i.e., if the family of functions \((P_t\psi)_{t \geq 0}\) is equicontinuous at every \(x \in \chi\).
They also define for every \(T > 0\) and any probability measure \(\mu\) on \(\chi\) \[ Q^T \mu = \frac{1}{T} \int_0^T P^*_s \mu ds \] where \(P^*_s \mu\) is the usual dual of \(P_s\) acting on the space of finite positive measures: \[ P^*_s \mu(B) = \int_\chi P_s 1_B(x) \mu(dx) \] The set \(\tau\) is defined as the set of all \(x \in \chi\) for which \((Q^T \delta_x)_{T > 0}\) is a tight family of probability measures.
The first main theorem states that if \((P_t)_{t \geq 0}\) has the e-property, and if there exists \(z \in \chi\) such that for every \(\delta > 0\) and for every \(x \in \chi\) \[ \liminf_{T\rightarrow \infty} (Q^T\delta_x)(B(z,\delta)) > 0, \] then there exists a unique invariant probability measure \(\mu^*\) for \((P_t)_{t \geq 0}\). Further \(Q^T\nu\) converges weakly to \(\mu^*\) for every probability measure \(\nu\) whose support is included in \(\tau\).
The authors proceed by giving an example where the set \(\tau\) does not equal \(\chi\). This serves as a motivation for the next result. A semigroup \((P_t)_{t \geq 0}\) is called weak-\(^*\) mean ergodic if there exists a measure \(\mu^*\) on \(\chi\) such that for every probability measure \(\nu\) on \(\chi\), \(Q^T\nu\) converges weakly to \(\mu^*\) as \(T\) goes to infinity.
Theorem 2 gives a refinement of Theorem 1 in the case where the lower bound is satisfied locally uniformly: Let \((P_t)_{t \geq 0}\) again be a Feller semigroup satisfying the e-property. Now assume that there exists \(z \in \chi\) such that for every bounded set \(A\) and every \(\delta > 0\) \[ \inf_{x\in A} \liminf_{T\rightarrow \infty} (Q^T \delta_x)(B(z,\delta)) > 0 \] Further assume that for every \(\varepsilon > 0\) and every \(x \in \chi\) there exists a bounded Borel set \(D \subset \chi\) such that \[ \liminf_{T\rightarrow \infty} (Q^T\delta_x)(D) > 1 - \varepsilon \] Then there exists a unique invariant probability measure \(\mu^*\), the semigroup \((P_t)_{t \geq 0}\) is weak-\(^*\) mean ergodic, and for every \(\psi \in \) Lip\(_b(\chi)\) and every probability measure \(\nu\) on \(\chi\) the weak law of large numbers holds: \[ \lim_{T\rightarrow \infty} \frac{1}{T} \int_0^T \psi(Z_s) ds = \int_\chi \psi(x) \mu^*(dx) \] where the convergence holds in \(P_\nu\)-probability.
These results are then applied to stochastic evolution equations in Hilbert spaces.

60J25 Continuous-time Markov processes on general state spaces
37A30 Ergodic theorems, spectral theory, Markov operators
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
Full Text: DOI arXiv
[1] Billingsley, P. (1999). Convergence of Probability Measures , 2nd ed. Wiley, New York. · Zbl 0944.60003
[2] Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44 . Cambridge Univ. Press, Cambridge. · Zbl 0761.60052 · doi:10.1017/CBO9780511666223
[3] Da Prato, G. and Zabczyk, J. (1996). Ergodicity for Infinite-Dimensional Systems. London Mathematical Society Lecture Note Series 229 . Cambridge Univ. Press, Cambridge. · Zbl 0849.60052 · doi:10.1017/CBO9780511662829
[4] Doeblin, W. (1940). Éléments d’une théorie générale des chaines simples constantes de Markov. Ann. École Norm. 57 61-111. · Zbl 0024.26503 · numdam:ASENS_1940_3_57__61_0 · eudml:81554
[5] E, W. and Mattingly, J. C. (2001). Ergodicity for the Navier-Stokes equation with degenerate random forcing: Finite-dimensional approximation. Comm. Pure Appl. Math. 54 1386-1402. · Zbl 1024.76012 · doi:10.1002/cpa.10007
[6] Eckmann, J. P. and Hairer, M. (2001). Uniqueness of the invariant measure for a stochastic PDE driven by degenerate noise. Comm. Math. Phys. 219 523-565. · Zbl 0983.60058 · doi:10.1007/s002200100424
[7] Fannjiang, A., Komorowski, T. and Peszat, S. (2002). Lagrangian dynamics for a passive tracer in a class of Gaussian Markovian flows. Stochastic Process. Appl. 97 171-198. · Zbl 1058.60048 · doi:10.1016/S0304-4149(01)00129-6
[8] Furstenberg, H. (1961). Strict ergodicity and transformation of the torus. Amer. J. Math. 83 573-601. JSTOR: · Zbl 0178.38404 · doi:10.2307/2372899 · links.jstor.org
[9] Gilbarg, D. and Trudinger, N. S. (1983). Elliptic Partial Differential Equations of Second Order , 2nd ed. Grundlehren der Mathematischen Wissenschaften [ Fundamental Principles of Mathematical Sciences ] 224 . Springer, Berlin. · Zbl 0562.35001
[10] Hairer, M. (2002). Exponential mixing properties of stochastic PDEs through asymptotic coupling. Probab. Theory Related Fields 124 345-380. · Zbl 1032.60056 · doi:10.1007/s004400200216
[11] Hairer, M. and Mattingly, J. C. (2006). Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing. Ann. of Math. (2) 164 993-1032. · Zbl 1130.37038 · doi:10.4007/annals.2006.164.993 · euclid:annm/1172614618
[12] Hairer, M. and Mattingly, J. C. (2008). Spectral gaps in Wasserstein distances and the 2D stochastic Navier-Stokes equations. Ann. Probab. 36 2050-2091. · Zbl 1173.37005 · doi:10.1214/08-AOP392
[13] Komorowski, T. and Peszat, S. (2004). Transport of a passive tracer by an irregular velocity field. J. Stat. Phys. 115 1361-1388. · Zbl 1157.76335 · doi:10.1023/B:JOSS.0000028063.58764.68
[14] Kuksin, S. and Shirikyan, A. (2001). Ergodicity for the randomly forced 2D Navier-Stokes equations. Math. Phys. Anal. Geom. 4 147-195. · Zbl 1013.37046 · doi:10.1023/A:1011989910997
[15] Lasota, A. and Mackey, M. C. (1985). Probabilistic Properties of Deterministic Systems . Cambridge Univ. Press, Cambridge. · Zbl 0606.58002 · doi:10.1017/CBO9780511897474
[16] Lasota, A. and Szarek, T. (2006). Lower bound technique in the theory of a stochastic differential equation. J. Differential Equations 231 513-533. · Zbl 1387.60100 · doi:10.1016/j.jde.2006.04.018
[17] Lasota, A. and Yorke, J. A. (1973). On the existence of invariant measures for piecewise monotonic transformations. Trans. Amer. Math. Soc. 186 481-488 (1974). · Zbl 0298.28015 · doi:10.2307/1996575
[18] Lasota, A. and Yorke, J. A. (1994). Lower bound technique for Markov operators and iterated function systems. Random Comput. Dynam. 2 41-77. · Zbl 0804.47033
[19] Mattingly, J. C. (2002). Exponential convergence for the stochastically forced Navier-Stokes equations and other partially dissipative dynamics. Comm. Math. Phys. 230 421-462. · Zbl 1054.76020 · doi:10.1007/s00220-002-0688-1
[20] Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability . Springer, London. · Zbl 0925.60001
[21] Nualart, D. (1995). The Malliavin Calculus and Related Topics . Springer, New York. · Zbl 0837.60050
[22] Peszat, S. and Zabczyk, J. (1995). Strong Feller property and irreducibility for diffusions on Hilbert spaces. Ann. Probab. 23 157-172. · Zbl 0831.60083 · doi:10.1214/aop/1176988381
[23] Peszat, S. and Zabczyk, J. (2007). Stochastic Partial Differential Equations with Lévy Noise. Encyclopedia of Mathematics and Its Applications 113 . Cambridge Univ. Press, Cambridge. · Zbl 1205.60122
[24] Pollard, D. (1984). Convergence of Stochastic Processes . Springer, New York. · Zbl 0544.60045
[25] Port, S. C. and Stone, C. (1976). Random measures and their application to motion in an incompressible fluid. J. Appl. Probab. 13 499-506. JSTOR: · Zbl 0374.60012 · doi:10.2307/3212469 · links.jstor.org
[26] Szarek, T. (2006). Feller processes on nonlocally compact spaces. Ann. Probab. 34 1849-1863. · Zbl 1108.60064 · doi:10.1214/009117906000000313
[27] Szarek, T., Śle\ogonek czka, M. and Urbański, M. (2009). On stability of velocity vectors for some passive tracer models. Submitted for publication. Available at http://www.math.unt.edu/ urbanski/papers/pt.pdf.
[28] Vakhania, N. N. (1975). The topological support of Gaussian measure in Banach space. Nagoya Math. J. 57 59-63. · Zbl 0301.60006
[29] Zaharopol, R. (2005). Invariant Probabilities of Markov-Feller Operators and Their Supports . Birkhäuser, Basel. · Zbl 1072.37007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.