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.
All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Mathematical Physics
- Homological equation
- Levy excursion
- Markov chain
- Stable law