Electronic Communications in Probability

On the $f$-norm ergodicity of Markov processes in continuous time

Ioannis Kontoyiannis and Sean P. Meyn

Full-text: Open access

Abstract

Consider a Markov process $\boldsymbol{\Phi } =\{ \Phi (t) : t\geq 0\}$ evolving on a Polish space $\mathsf{X} $. A version of the $f$-Norm Ergodic Theorem is obtained: Suppose that the process is $\psi $-irreducible and aperiodic. For a given function $f\colon \mathsf{X} \to [1,\infty )$, under suitable conditions on the process the following are equivalent:

(i) There is a unique invariant probability measure $\pi $ satisfying $\int f\,d\pi <\infty $.

(ii) There is a closed set $C$ satisfying $\psi (C)>0$ that is “self $f$-regular.”

(iii) There is a function $V\colon \mathsf{X} \to (0,\infty ]$ that is finite on at least one point in $\mathsf{X} $, for which the following Lyapunov drift condition is satisfied, \[ \mathcal{D} V\leq - f+b\mathbb{I}_C\, , \tag{V3} \] where $C$ is a closed small set and $\mathcal{D}$ is the extended generator of the process.

For discrete-time chains the result is well-known. Moreover, in that case, the ergodicity of $\boldsymbol{\Phi }$ under a suitable norm is also obtained: For each initial condition $x\in \mathsf{X} $ satisfying $V(x)<\infty $, and any function $g\colon \mathsf{X} \to \mathbb{R} $ for which $|g|$ is bounded by $f$, \[ \lim _{t\to \infty } \mathsf{E}_x[g(\Phi (t))] = \int g\,d\pi . \] Possible approaches are explored for establishing appropriate versions of corresponding results in continuous time, under appropriate assumptions on the process $\boldsymbol{\Phi } $ or on the function $g$.

Article information

Source
Electron. Commun. Probab., Volume 21 (2016), paper no. 77, 10 pp.

Dates
Received: 2 December 2015
Accepted: 14 November 2016
First available in Project Euclid: 29 November 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1480388670

Digital Object Identifier
doi:10.1214/16-ECP4737

Mathematical Reviews number (MathSciNet)
MR3580446

Zentralblatt MATH identifier
1360.60146

Subjects
Primary: 60J25: Continuous-time Markov processes on general state spaces 37A30: Ergodic theorems, spectral theory, Markov operators {For operator ergodic theory, see mainly 47A35} 47H99: None of the above, but in this section

Keywords
Markov process continuous time generator stochastic Lyapunov function ergodicity

Rights
Creative Commons Attribution 4.0 International License.

Citation

Kontoyiannis, Ioannis; Meyn, Sean P. On the $f$-norm ergodicity of Markov processes in continuous time. Electron. Commun. Probab. 21 (2016), paper no. 77, 10 pp. doi:10.1214/16-ECP4737. https://projecteuclid.org/euclid.ecp/1480388670


Export citation

References

  • [1] Ethier, S. N. and Kurtz, T. G. (1986). Markov processes. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York. Characterization and convergence.
  • [2] Halmos, P. R. (1950). Measure Theory. D. Van Nostrand Company, Inc., New York, N. Y.
  • [3] Kontoyiannis, I. and Meyn, S. P. (2003). Spectral theory and limit theorems for geometrically ergodic Markov processes. Ann. Appl. Probab. 13, 1, 304–362.
  • [4] Kontoyiannis, I. and Meyn, S. P. (2005). Large deviations asymptotics and the spectral theory of multiplicatively regular Markov processes. Electron. J. Probab. 10, no. 3, 61–123 (electronic).
  • [5] Löcherbach, E. and Loukianova, D. (2008). On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions. Stochastic Process. Appl. 118, 8, 1301–1321.
  • [6] Löcherbach, E., Loukianova, D., and Loukianov, O. (2011). Polynomial bounds in the ergodic theorem for one-dimensional diffusions and integrability of hitting times. Ann. Inst. Henri Poincaré Probab. Stat. 47, 2, 425–449.
  • [7] Meyn, S. P. and Tweedie, R. L. (1993). Generalized resolvents and Harris recurrence of Markov processes. In Doeblin and modern probability (Blaubeuren, 1991). Contemp. Math., Vol. 149. Amer. Math. Soc., Providence, RI, 227–250.
  • [8] Meyn, S. P. and Tweedie, R. L. (1993). Stability of Markovian processes. II. Continuous-time processes and sampled chains. Adv. in Appl. Probab. 25, 3, 487–517.
  • [9] Meyn, S. P. and Tweedie, R. L. (1993). Stability of Markovian processes. III. Foster-Lyapunov criteria for continuous-time processes. Adv. in Appl. Probab. 25, 3, 518–548.
  • [10] Meyn, S. P. and Tweedie, R. L. (1993). Markov chains and stochastic stability. Communications and Control Engineering Series. Springer-Verlag London, Ltd., London.
  • [11] Neveu, J. (1972). Potentiel Markovien récurrent des chaînes de Harris. Ann. Inst. Fourier (Grenoble) 22, 2, 85–130.
  • [12] Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov processes, and martingales. Vol. 2. Cambridge Mathematical Library. Cambridge University Press, Cambridge. Itô calculus, Reprint of the second (1994) edition.