The Annals of Probability

Conditional ergodicity in infinite dimension

Xin Thomson Tong and Ramon van Handel

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Abstract

The goal of this paper is to develop a general method to establish conditional ergodicity of infinite-dimensional Markov chains. Given a Markov chain in a product space, we aim to understand the ergodic properties of its conditional distributions given one of the components. Such questions play a fundamental role in the ergodic theory of nonlinear filters. In the setting of Harris chains, conditional ergodicity has been established under general nondegeneracy assumptions. Unfortunately, Markov chains in infinite-dimensional state spaces are rarely amenable to the classical theory of Harris chains due to the singularity of their transition probabilities, while topological and functional methods that have been developed in the ergodic theory of infinite-dimensional Markov chains are not well suited to the investigation of conditional distributions. We must therefore develop new measure-theoretic tools in the ergodic theory of Markov chains that enable the investigation of conditional ergodicity for infinite dimensional or weak-∗ ergodic processes. To this end, we first develop local counterparts of zero–two laws that arise in the theory of Harris chains. These results give rise to ergodic theorems for Markov chains that admit asymptotic couplings or that are locally mixing in the sense of H. Föllmer, and to a non-Markovian ergodic theorem for stationary absolutely regular sequences. We proceed to show that local ergodicity is inherited by conditioning on a nondegenerate observation process. This is used to prove stability and unique ergodicity of the nonlinear filter. Finally, we show that our abstract results can be applied to infinite-dimensional Markov processes that arise in several settings, including dissipative stochastic partial differential equations, stochastic spin systems and stochastic differential delay equations.

Article information

Source
Ann. Probab. Volume 42, Number 6 (2014), 2243-2313.

Dates
First available in Project Euclid: 30 September 2014

Permanent link to this document
https://projecteuclid.org/euclid.aop/1412083626

Digital Object Identifier
doi:10.1214/13-AOP879

Mathematical Reviews number (MathSciNet)
MR3265168

Zentralblatt MATH identifier
1351.37032

Subjects
Primary: 37A30: Ergodic theorems, spectral theory, Markov operators {For operator ergodic theory, see mainly 47A35} 37L55: Infinite-dimensional random dynamical systems; stochastic equations [See also 35R60, 60H10, 60H15] 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx] 60H15: Stochastic partial differential equations [See also 35R60] 60J05: Discrete-time Markov processes on general state spaces

Keywords
Ergodic theory of infinite-dimensional Markov processes and non-Markov processes zero–two laws conditional ergodicity nonlinear filtering

Citation

Tong, Xin Thomson; van Handel, Ramon. Conditional ergodicity in infinite dimension. Ann. Probab. 42 (2014), no. 6, 2243--2313. doi:10.1214/13-AOP879. https://projecteuclid.org/euclid.aop/1412083626


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References

  • [1] Berbee, H. (1986). Periodicity and absolute regularity. Israel J. Math. 55 289–304.
  • [2] Blackwell, D. and Dubins, L. (1962). Merging of opinions with increasing information. Ann. Math. Statist. 33 882–886.
  • [3] Budhiraja, A. (2003). Asymptotic stability, ergodicity and other asymptotic properties of the nonlinear filter. Ann. Inst. Henri Poincaré Probab. Stat. 39 919–941.
  • [4] Calzolari, A., Florchinger, P. and Nappo, G. (2007). Convergence in nonlinear filtering for stochastic delay systems. SIAM J. Control Optim. 46 1615–1636 (electronic).
  • [5] Chigansky, P. and van Handel, R. (2010). A complete solution to Blackwell’s unique ergodicity problem for hidden Markov chains. Ann. Appl. Probab. 20 2318–2345.
  • [6] Çinlar, E. (1972). Markov additive processes. I, II. Z. Wahrsch. Verw. Gebiete 24 85–93; ibid. 24 (1972), 95–121.
  • [7] Cogburn, R. (1984). The ergodic theory of Markov chains in random environments. Z. Wahrsch. Verw. Gebiete 66 109–128.
  • [8] Crisan, D. and Rozovskiĭ, B., eds. (2011). The Oxford Handbook of Nonlinear Filtering. Oxford Univ. Press, Oxford.
  • [9] Da Prato, G. and Zabczyk, J. (1996). Ergodicity for Infinite-Dimensional Systems. London Mathematical Society Lecture Note Series 229. Cambridge Univ. Press, Cambridge.
  • [10] Dellacherie, C. and Meyer, P.-A. (1982). Probabilities and Potential. B: Theory of Martingales. North-Holland Mathematics Studies 72. North-Holland, Amsterdam.
  • [11] Derriennic, Y. (1976). Lois “zéro ou deux” pour les processus de Markov. Applications aux marches aléatoires. Ann. Inst. H. Poincaré Sect. B (N.S.) 12 111–129.
  • [12] 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.
  • [13] E, W., Mattingly, J. C. and Sinai, Y. (2001). Gibbsian dynamics and ergodicity for the stochastically forced Navier–Stokes equation. Comm. Math. Phys. 224 83–106.
  • [14] Es-Sarhir, A., Scheutzow, M. and van Gaans, O. (2010). Invariant measures for stochastic functional differential equations with superlinear drift term. Differential Integral Equations 23 189–200.
  • [15] Föllmer, H. (1979). Tail structure of Markov chains on infinite product spaces. Z. Wahrsch. Verw. Gebiete 50 273–285.
  • [16] 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.
  • [17] Hairer, M., Mattingly, J. C. and Scheutzow, M. (2011). Asymptotic coupling and a general form of Harris’ theorem with applications to stochastic delay equations. Probab. Theory Related Fields 149 223–259.
  • [18] Hajnal, J. (1958). Weak ergodicity in non-homogeneous Markov chains. Proc. Cambridge Philos. Soc. 54 233–246.
  • [19] Holley, R. A. and Stroock, D. W. (1989). Uniform and $L^{2}$ convergence in one-dimensional stochastic Ising models. Comm. Math. Phys. 123 85–93.
  • [20] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
  • [21] Komorowski, T., Peszat, S. and Szarek, T. (2010). On ergodicity of some Markov processes. Ann. Probab. 38 1401–1443.
  • [22] Kuksin, S. and Shirikyan, A. (2012). Mathematics of Two-Dimensional Turbulence. Cambridge Tracts in Mathematics 194. Cambridge Univ. Press, Cambridge.
  • [23] Kunita, H. (1971). Asymptotic behavior of the nonlinear filtering errors of Markov processes. J. Multivariate Anal. 1 365–393.
  • [24] Kurtz, T. G. (1998). Martingale problems for conditional distributions of Markov processes. Electron. J. Probab. 3 29 pp. (electronic).
  • [25] Kwong, R. H. and Willsky, A. S. (1978). Estimation and filter stability of stochastic delay systems. SIAM J. Control Optim. 16 660–681.
  • [26] Liggett, T. M. (2005). Interacting Particle Systems. Springer, Berlin.
  • [27] Lindvall, T. (2002). Lectures on the Coupling Method. Dover, Mineola, NY.
  • [28] Liptser, R. S. and Shiryaev, A. N. (2001). Statistics of Random Processes. I: General Theory, expanded ed. Applications of Mathematics (New York) 5. Springer, Berlin.
  • [29] Martinelli, F. (2004). Relaxation times of Markov chains in statistical mechanics and combinatorial structures. In Probability on Discrete Structures. Encyclopaedia Math. Sci. 110 175–262. Springer, Berlin.
  • [30] Mattingly, J. C. (2002). Exponential convergence for the stochastically forced Navier–Stokes equations and other partially dissipative dynamics. Comm. Math. Phys. 230 421–462.
  • [31] Mattingly, J. C. (2007). Ergodicity of dissipative SPDEs. In Lecture notes, École d’été de Probabilités de Saint-Flour, July 8–21.
  • [32] Meyn, S. and Tweedie, R. L. (2009). Markov Chains and Stochastic Stability, 2nd ed. Cambridge Univ. Press, Cambridge.
  • [33] Nummelin, E. (1984). General Irreducible Markov Chains and Nonnegative Operators. Cambridge Tracts in Mathematics 83. Cambridge Univ. Press, Cambridge.
  • [34] Orey, S. (1991). Markov chains with stochastically stationary transition probabilities. Ann. Probab. 19 907–928.
  • [35] Orstein, D. and Sucheston, L. (1970). An operator theorem on $L_{1}$ convergence to zero with applications to Markov kernels. Ann. Math. Statist. 41 1631–1639.
  • [36] Pachl, J. K. (1979). Measures as functionals on uniformly continuous functions. Pacific J. Math. 82 515–521.
  • [37] Revuz, D. (1984). Markov Chains, 2nd ed. North-Holland Mathematical Library 11. North-Holland, Amsterdam.
  • [38] Rudolph, D. J. (2004). Pointwise and $L^{1}$ mixing relative to a sub-sigma algebra. Illinois J. Math. 48 505–517.
  • [39] Shiryaev, A. N. (1996). Probability, 2nd ed. Graduate Texts in Mathematics 95. Springer, New York.
  • [40] Stettner, Ł. (1989). On invariant measures of filtering processes. In Stochastic Differential Systems (Bad Honnef, 1988). Lecture Notes in Control and Inform. Sci. 126 279–292. Springer, Berlin.
  • [41] Stuart, A. M. (2010). Inverse problems: A Bayesian perspective. Acta Numer. 19 451–559.
  • [42] Tong, X. T. and van Handel, R. (2012). Ergodicity and stability of the conditional distributions of nondegenerate Markov chains. Ann. Appl. Probab. 22 1495–1540.
  • [43] van Handel, R. (2009). The stability of conditional Markov processes and Markov chains in random environments. Ann. Probab. 37 1876–1925.
  • [44] van Handel, R. (2009). Uniform observability of hidden Markov models and filter stability for unstable signals. Ann. Appl. Probab. 19 1172–1199.
  • [45] van Handel, R. (2009). Uniform time average consistency of Monte Carlo particle filters. Stochastic Process. Appl. 119 3835–3861.
  • [46] van Handel, R. (2012). On the exchange of intersection and supremum of $\sigma$-fields in filtering theory. Israel J. Math. 192 763–784.
  • [47] Vinter, R. B. (1977). Filter stability for stochastic evolution equations. SIAM J. Control Optim. 15 465–485.
  • [48] Volkonskiĭ, V. A. and Rozanov, Yu. A. (1959). Some limit theorems for random functions. I. Theory Probab. Appl. 4 178–197.
  • [49] von Weizsäcker, H. (1983). Exchanging the order of taking suprema and countable intersections of $\sigma$-algebras. Ann. Inst. H. Poincaré Sect. B (N.S.) 19 91–100.
  • [50] Walters, P. (1982). An Introduction to Ergodic Theory. Graduate Texts in Mathematics 79. Springer, New York.
  • [51] Yor, M. (1977). Sur les théories du filtrage et de la prédiction. In Séminaire de Probabilités, XI (Univ. Strasbourg, Strasbourg, 1975/1976). Lecture Notes in Math. 581 257–297. Springer, Berlin.