The Annals of Probability

Carne–Varopoulos bounds for centered random walks

Pierre Mathieu

Full-text: Open access


We extend the Carne–Varopoulos upper bound on the probability transitions of a Markov chain to a certain class of nonreversible processes by introducing the definition of a “centering measure.” In the case of random walks on a group, we study the connections between different notions of centering.

Article information

Ann. Probab., Volume 34, Number 3 (2006), 987-1011.

First available in Project Euclid: 27 June 2006

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Centered Markov chains random walks Carne–Varopoulos bounds Poisson boundary


Mathieu, Pierre. Carne–Varopoulos bounds for centered random walks. Ann. Probab. 34 (2006), no. 3, 987--1011. doi:10.1214/009117906000000052.

Export citation


  • Alexopoulos, G. K. (2002). Random walks on discrete groups of polynomial volume growth. Ann. Probab. 30 723--801.
  • Baldi, P., Lohoué, N. and Peyrière, J. (1977). Sur la classification des groupes récurrents. C. R. Acad. Sci. Paris Ser. A--B 285 1103--1104.
  • Carne, T. K. (1985). A transmutation formula for Markov chains. Bull. Sci. Math. 109 399--405.
  • Chen, M. F. (1991). Comparison theorems for Green functions of Markov chains. Chinese Ann. Math. Ser. A 12 237--242.
  • Derriennic, Y. (1980). Quelques applications du théorème ergodique sous-additif. Astérisque 74 183--201.
  • Hebisch, W. and Saloff-Coste, L. (1993). Gaussian estimates for Markov chains and random walks on groups. Ann. Probab. 21 673--709.
  • Jiang, D. Q., Qian, M. and Qian, M. P. (2004). Mathematical Theory of Nonequilibrium Steady States. On the Frontier of Probability and Dynamical Systems. Lecture Notes in Math. 1833. Springer, Berlin.
  • Kaimanovich, V. A. (1991). Poisson boundaries of random walks on discrete solvable groups. In Probability Measures on Groups X (H. Heyer, ed.) 205--238. Plenum, New York.
  • Kaimanovich, V. A. (2001). Poisson boundary of discrete groups. Available at
  • Kalpazidou, S. L. (1995). Cycle Representations of Markov Processes. Springer, New York.
  • Varopoulos, N. Th. (1985). Long range estimates for Markov chains. Bull. Sci. Math. 109 225--252.
  • Vershik, A. M. (2000). Dynamic theory of growth in groups: Entropy, boundaries, examples. Russ. Math. Surveys 55 667--733.
  • Woess, W. (2000). Random Walks on Infinite Graphs and Groups. Cambridge Univ. Press.