Illinois Journal of Mathematics

Recurrence and transience preservation for vertex reinforced jump processes in one dimension

Burgess Davis and Noah Dean

Full-text: Open access


We show that the application of linear vertex reinforcement to one dimensional nearest neighbor Markov processes, yielding associated vertex reinforced jump processes, preserves both recurrence and transience. The analog for discrete time linear bond reinforcement is due to Takeshima. This together with another result we prove adds to the numerous known parallels between these two reinforcements. Martingales are the primary tool used to study vertex reinforced jump processes.

Article information

Illinois J. Math., Volume 54, Number 3 (2010), 869-893.

First available in Project Euclid: 3 May 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K37: Processes in random environments 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)


Davis, Burgess; Dean, Noah. Recurrence and transience preservation for vertex reinforced jump processes in one dimension. Illinois J. Math. 54 (2010), no. 3, 869--893. doi:10.1215/ijm/1336049980.

Export citation


  • A. Collevecchio, Limit theorems for reinforced random walks on certain trees, Probab. Theory Related Fields 136 (2006), 81–101.
  • A. Collevecchio, On the transience of processes defined on Galton–Watson trees, Ann. Probab. 34 (2006), 870–878.
  • A. Collevecchio, Limit theorems for vertex-reinforced jump processes on regular trees, Electron. J. Probab. 14 (2009), 1936–1962.
  • D. Coppersmith and P. Diaconis, Random walks with reinforcement, unpublished manuscript, 1987.
  • B. Davis and S. Volkov, Continuous time vertex-reinforced jump processes, Probab. Theory Related Fields 123 (2002), 281–300.
  • B. Davis and S. Volkov, Vertex-reinforced jump processes on trees and finite graphs, Probab. Theory Related Fields 128 (2004), 42–62.
  • P. Hoel, S. Port, and C. Stone, Introduction to stochastic processes, Houghton Mifflin, Boston, 1972.
  • R. Pemantle, Phase transition in reinforced random walk and RWRE on trees, Ann. Probab. 16 (1988), 1229–1241.
  • R. Pemantle, Vertex-reinforced random walk, Probab. Theory Related Fields 92 (1992), 117–136.
  • R. Pemantle and S. Volkov, Vertex-reinforced random walk on $Z$ has finite range, Ann. Probab. 27 (1999), 1368–1388.
  • M. Takeshima, Behavior of 1-dimensional reinforced random walk, Osaka J. Math. 37 (2000), 355–372.
  • P. Tarrès, Vertex-reinforced random walk on $\mathbb{Z}$ eventually gets stuck on five points, Ann. Probab. 32 (2004), 2650–2701.