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

Abstract

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

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

Dates
First available in Project Euclid: 3 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1336049980

Digital Object Identifier
doi:10.1215/ijm/1336049980

Mathematical Reviews number (MathSciNet)
MR2928340

Zentralblatt MATH identifier
1266.60163

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

Citation

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. https://projecteuclid.org/euclid.ijm/1336049980


Export citation

References

  • 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  • D. Coppersmith and P. Diaconis, Random walks with reinforcement, unpublished manuscript, 1987.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.