Electronic Journal of Probability

Large Deviations Asymptotics and the Spectral Theory of Multiplicatively Regular Markov Processes

Ioannis Kontoyiannis and Sean Meyn

Full-text: Open access


In this paper we continue the investigation of the spectral theory and exponential asymptotics of primarily discrete-time Markov processes, following Kontoyiannis and Meyn (2003). We introduce a new family of nonlinear Lyapunov drift criteria, which characterize distinct subclasses of geometrically ergodic Markov processes in terms of simple inequalities for the nonlinear generator. We concentrate primarily on the class of multiplicatively regular Markov processes, which are characterized via simple conditions similar to (but weaker than) those of Donsker-Varadhan. For any such process $\{\Phi(t)\}$ with transition kernel $P$ on a general state space $X$, the following are obtained. Spectral Theory: For a large class of (possibly unbounded) functionals $F$ on $X$, the kernel $\hat P(x,dy) = e^{F(x)} P(x,dy)$ has a discrete spectrum in an appropriately defined Banach space. It follows that there exists a "maximal," well-behaved solution to the "multiplicative Poisson equation," defined as an eigenvalue problem for $\hat P$. Multiplicative Mean Ergodic Theorem: Consider the partial sums of this process with respect to any one of the functionals $F$ considered above. The normalized mean of their moment generating function (and not the logarithm of the mean) converges to the above maximal eigenfunction exponentially fast. Multiplicative regularity: The Lyapunov drift criterion under which our results are derived is equivalent to the existence of regeneration times with finite exponential moments for the above partial sums. Large Deviations: The sequence of empirical measures of the process satisfies a large deviations principle in a topology finer that the usual tau-topology, generated by the above class of functionals. The rate function of this LDP is the convex dual of logarithm of the above maximal eigenvalue, and it is shown to coincide with the Donsker-Varadhan rate function in terms of relative entropy. Exact Large Deviations Asymptotics: The above partial sums are shown to satisfy an exact large deviations expansion, analogous to that obtained by Bahadur and Ranga Rao for independent random variables.

Article information

Electron. J. Probab., Volume 10 (2005), paper no. 3, 61-123.

Accepted: 24 February 2005
First available in Project Euclid: 1 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Markov process large deviations entropy Lyapunov function empirical measures nonlinear generator large deviations principle

This work is licensed under aCreative Commons Attribution 3.0 License.


Kontoyiannis, Ioannis; Meyn, Sean. Large Deviations Asymptotics and the Spectral Theory of Multiplicatively Regular Markov Processes. Electron. J. Probab. 10 (2005), paper no. 3, 61--123. doi:10.1214/EJP.v10-231. https://projecteuclid.org/euclid.ejp/1464816804

Export citation


  • Bahadur, R. R.; Ranga Rao, R. On deviations of the sample mean. Ann. Math. Statist. 31 1960 1015–1027.
  • Balaji, S.; Meyn, S. P. Multiplicative ergodicity and large deviations for an irreducible Markov chain. Stochastic Process. Appl. 90 (2000), no. 1, 123–144.
  • Bass, Richard F.; Hsu, Pei. Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains. Ann. Probab. 19 (1991), no. 2, 486–508.
  • Bolthausen, Erwin. Markov process large deviations in $\tau$-topology. Stochastic Process. Appl. 25 (1987), no. 1, 95–108.
  • Bovier, Anton; Eckhoff, Michael; Gayrard, Véronique; Klein, Markus. Metastability and small eigenvalues in Markov chains. J. Phys. A 33 (2000), no. 46, L447–L451.
  • Boyd, Stephen; Vandenberghe, Lieven. Convex optimization. Cambridge University Press, Cambridge, 2004. xiv+716 pp. ISBN: 0-521-83378-7
  • Chaganty, Narasinga Rao; Sethuraman, Jayaram. Strong large deviation and local limit theorems. Ann. Probab. 21 (1993), no. 3, 1671–1690.
  • Davies, E. Brian. Pseudospectra of differential operators. J. Operator Theory 43 (2000), no. 2, 243–262.
  • Dawson, Donald A.; Gärtner, Jürgen. Large deviations from the McKean-Vlasov limit for weakly interacting diffusions. Stochastics 20 (1987), no. 4, 247–308.
  • de Acosta, A. Large deviations for empirical measures of Markov chains. J. Theoret. Probab. 3 (1990), no. 3, 395–431.
  • Dellnitz, Michael; Junge, Oliver. On the approximation of complicated dynamical behavior. SIAM J. Numer. Anal. 36 (1999), no. 2, 491–515 (electronic).
  • Dembo, Amir; Zeitouni, Ofer. Large deviations techniques and applications. Second edition. Applications of Mathematics (New York), 38. Springer-Verlag, New York, 1998. xvi+396 pp. ISBN: 0-387-98406-2
  • Deuschel, Jean-Dominique; Stroock, Daniel W. Large deviations. Pure and Applied Mathematics, 137. Academic Press, Inc., Boston, MA, 1989. xiv+307 pp. ISBN: 0-12-213150-9
  • Donsker, M. D.; Varadhan, S. R. S. Asymptotic evaluation of certain Markov process expectations for large time. I. II. Comm. Pure Appl. Math. 28 (1975), 1–47; ibid. 28 (1975), 279–301.
  • Donsker, M. D.; Varadhan, S. R. S. Asymptotic evaluation of certain Markov process expectations for large time. III. Comm. Pure Appl. Math. 29 (1976), no. 4, 389–461.
  • Donsker, M. D.; Varadhan, S. R. S. Asymptotic evaluation of certain Markov process expectations for large time. IV. Comm. Pure Appl. Math. 36 (1983), no. 2, 183–212.
  • Down, D.; Meyn, S. P.; Tweedie, R. L. Exponential and uniform ergodicity of Markov processes. Ann. Probab. 23 (1995), no. 4, 1671–1691.
  • Dupuis, Paul; Zeitouni, Ofer. A nonstandard form of the rate function for the occupation measure of a Markov chain. Stochastic Process. Appl. 61 (1996), no. 2, 249–261.
  • Eichelsbacher, P.; Schmock, U. Large deviations for products of empirical measures of dependent sequences. Markov Process. Related Fields 7 (2001), no. 3, 435–468.
  • Feng, Jin. Martingale problems for large deviations of Markov processes. Stochastic Process. Appl. 81 (1999), no. 2, 165–216.
  • Feng, Jin. Large deviations for stochastic processes. Preprint. (2000)
  • Fleming, Wendell H. Exit probabilities and optimal stochastic control. Appl. Math. Optim. 4 (1977/78), no. 4, 329–346.
  • Fukushima, M.; Stroock, D. Reversibility of solutions to martingale problems. Probability, statistical mechanics, and number theory, 107–123, Adv. Math. Suppl. Stud., 9, Academic Press, Orlando, FL, 1986.
  • Groeneboom, P.; Oosterhoff, J.; Ruymgaart, F. H. Large deviation theorems for empirical probability measures. Ann. Probab. 7 (1979), no. 4, 553–586.
  • Hordijk, Arie; Spieksma, Flora. On ergodicity and recurrence properties of a Markov chain with an application to an open Jackson network. Adv. in Appl. Probab. 24 (1992), no. 2, 343–376.
  • Huisinga, Wilhelm; Meyn, Sean; Schütte, Christof. Phase transitions and metastability in Markovian and molecular systems. Ann. Appl. Probab. 14 (2004), no. 1, 419–458.
  • Iscoe, I.; Ney, P.; Nummelin, E. Large deviations of uniformly recurrent Markov additive processes. Adv. in Appl. Math. 6 (1985), no. 4, 373–412.
  • Jain, Naresh C. Large deviation lower bounds for additive functionals of Markov processes. Ann. Probab. 18 (1990), no. 3, 1071–1098.
  • Jensen, Jens Ledet. Saddlepoint expansions for sums of Markov dependent variables on a continuous state space. Probab. Theory Related Fields 89 (1991), no. 2, 181–199.
  • Kartashov, N. V. Criteria for uniform ergodicity and strong stability of Markov chains with a common phase space. (Russian) Teor. Veroyatnost. i Mat. Statist. No. 30, (1984), 65–81, 151.
  • Kartashov, N. V. Inequalities in theorems of ergodicity and stability for Markov chains with a common phase space. Theor. Probability Appl. no. 30, (1985), 247–259.
  • Kontoyiannis, I.; Meyn, S. P. Spectral theory and limit theorems for geometrically ergodic Markov processes. Ann. Appl. Probab. 13 (2003), no. 1, 304–362.
  • Kunita, Hiroshi. Stochastic flows and stochastic differential equations. Cambridge Studies in Advanced Mathematics, 24. Cambridge University Press, Cambridge, 1990. xiv+346 pp. ISBN: 0-521-35050-6
  • Meyn, S. P.; Tweedie, R. L. Markov chains and stochastic stability. Communications and Control Engineering Series. Springer-Verlag London, Ltd., London, 1993. xvi+ 548 pp. ISBN: 3-540-19832-6
  • Meyn, Sean P.; Tweedie, R. L. Computable bounds for geometric convergence rates of Markov chains. Ann. Appl. Probab. 4 (1994), no. 4, 981–1011.
  • Miller, H. D. A convexivity property in the theory of random variables defined on a finite Markov chain. Ann. Math. Statist. 32 1961 1260–1270. (23 #A4180)
  • Ney, P.; Nummelin, E. Markov additive processes. I. Eigenvalue properties and limit theorems. Ann. Probab. 15 (1987), no. 2, 561–592.
  • Ney, P.; Nummelin, E. Markov additive processes II. Large deviations. Ann. Probab. 15 (1987), no. 2, 593–609.
  • Nummelin, Esa. General irreducible Markov chains and nonnegative operators. Cambridge Tracts in Mathematics, 83. Cambridge University Press, Cambridge, 1984. xi+156 pp. ISBN: 0-521-25005-6
  • Pinsky, Ross G. Positive harmonic functions and diffusion. Cambridge Studies in Advanced Mathematics, 45. Cambridge University Press, Cambridge, 1995. xvi+474 pp. ISBN: 0-521-47014-5
  • Rey-Bellet, Luc; Thomas, Lawrence E. Asymptotic behavior of thermal nonequilibrium steady states for a driven chain of anharmonic oscillators. Comm. Math. Phys. 215 (2000), no. 1, 1–24.
  • Rey-Bellet, L.; Thomas, L. E. Fluctuations of the entropy production in anharmonic chains. Ann. Henri Poincaré 3 (2002), no. 3, 483–502.
  • Varadhan, S. R. S. Large deviations and applications. CBMS-NSF Regional Conference Series in Applied Mathematics, 46. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1984. v+75 pp. ISBN: 0-89871-189-4 (86h:60067b)
  • Wu, Liming. Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems. Stochastic Process. Appl. 91 (2001), no. 2, 205–238.