Simple Random Walks on Tori
Abstract
We consider a Markov chain whose phase space is a d-dimensional torus. A point x jumps to x+ ω with probability p( x) and to x- ω with probability 1- p( x). For Diophantine ω and smooth p we prove that this Markov chain has an absolutely continuous invariant measure and the distribution of any point after n steps converges to this measure.
- Publication:
-
Journal of Statistical Physics
- Pub Date:
- February 1999
- DOI:
- 10.1023/A:1004564824697
- Bibcode:
- 1999JSP....94..695S
- Keywords:
-
- Markov chain;
- homological equation;
- Levy excursion;
- stable law