The Annals of Probability

On ergodicity of some Markov processes

Tomasz Komorowski, Szymon Peszat, and Tomasz Szarek

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Abstract

We formulate a criterion for the existence and uniqueness of an invariant measure for a Markov process taking values in a Polish phase space. In addition, weak-* ergodicity, that is, the weak convergence of the ergodic averages of the laws of the process starting from any initial distribution, is established. The principal assumptions are the existence of a lower bound for the ergodic averages of the transition probability function and its local uniform continuity. The latter is called the e-property. The general result is applied to solutions of some stochastic evolution equations in Hilbert spaces. As an example, we consider an evolution equation whose solution describes the Lagrangian observations of the velocity field in the passive tracer model. The weak-* mean ergodicity of the corresponding invariant measure is used to derive the law of large numbers for the trajectory of a tracer.

Article information

Source
Ann. Probab. Volume 38, Number 4 (2010), 1401-1443.

Dates
First available in Project Euclid: 8 July 2010

Permanent link to this document
http://projecteuclid.org/euclid.aop/1278593955

Digital Object Identifier
doi:10.1214/09-AOP513

Zentralblatt MATH identifier
05776085

Mathematical Reviews number (MathSciNet)
MR2663632

Subjects
Primary: 60J25: Continuous-time Markov processes on general state spaces 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 76N10: Existence, uniqueness, and regularity theory [See also 35L60, 35L65, 35Q30]

Keywords
Ergodicity of Markov families invariant measures stochastic evolution equations passive tracer dynamics

Citation

Komorowski, Tomasz; Peszat, Szymon; Szarek, Tomasz. On ergodicity of some Markov processes. Ann. Probab. 38 (2010), no. 4, 1401--1443. doi:10.1214/09-AOP513. http://projecteuclid.org/euclid.aop/1278593955.


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References

  • [1] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
  • [2] Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press, Cambridge.
  • [3] Da Prato, G. and Zabczyk, J. (1996). Ergodicity for Infinite-Dimensional Systems. London Mathematical Society Lecture Note Series 229. Cambridge Univ. Press, Cambridge.
  • [4] Doeblin, W. (1940). Éléments d’une théorie générale des chaines simples constantes de Markov. Ann. École Norm. 57 61–111.
  • [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.
  • [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.
  • [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.
  • [8] Furstenberg, H. (1961). Strict ergodicity and transformation of the torus. Amer. J. Math. 83 573–601.
  • [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.
  • [10] Hairer, M. (2002). Exponential mixing properties of stochastic PDEs through asymptotic coupling. Probab. Theory Related Fields 124 345–380.
  • [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.
  • [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.
  • [13] Komorowski, T. and Peszat, S. (2004). Transport of a passive tracer by an irregular velocity field. J. Stat. Phys. 115 1361–1388.
  • [14] Kuksin, S. and Shirikyan, A. (2001). Ergodicity for the randomly forced 2D Navier–Stokes equations. Math. Phys. Anal. Geom. 4 147–195.
  • [15] Lasota, A. and Mackey, M. C. (1985). Probabilistic Properties of Deterministic Systems. Cambridge Univ. Press, Cambridge.
  • [16] Lasota, A. and Szarek, T. (2006). Lower bound technique in the theory of a stochastic differential equation. J. Differential Equations 231 513–533.
  • [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).
  • [18] Lasota, A. and Yorke, J. A. (1994). Lower bound technique for Markov operators and iterated function systems. Random Comput. Dynam. 2 41–77.
  • [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.
  • [20] Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.
  • [21] Nualart, D. (1995). The Malliavin Calculus and Related Topics. Springer, New York.
  • [22] Peszat, S. and Zabczyk, J. (1995). Strong Feller property and irreducibility for diffusions on Hilbert spaces. Ann. Probab. 23 157–172.
  • [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.
  • [24] Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York.
  • [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.
  • [26] Szarek, T. (2006). Feller processes on nonlocally compact spaces. Ann. Probab. 34 1849–1863.
  • [27] Szarek, T., Ślȩczka, 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.
  • [29] Zaharopol, R. (2005). Invariant Probabilities of Markov–Feller Operators and Their Supports. Birkhäuser, Basel.