Absolute continuity and weak uniform mixing of random walk in dynamic random environment. (English) Zbl 1355.82042

Bethuelsen, Stein Andreas; Völlering, Florian

Electron. J. Probab. 21, Paper No. 71, 32 p. (2016).

Summary: We prove results for random walks in dynamic random environments which do not require the strong uniform mixing assumptions present in the literature. We focus on the “environment seen from the walker”-process and in particular its invariant law. Under general conditions it exists and is mutually absolutely continuous to the environment law. With stronger assumptions we obtain for example uniform control on the density or a quenched CLT. The general conditions are made more explicit by looking at hidden Markov models or Markov chains as environment and by providing simple examples.

MSC:

82C41	Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
82C43	Time-dependent percolation in statistical mechanics
60F17	Functional limit theorems; invariance principles
60K37	Processes in random environments

Keywords:

random walk; dynamic random environment; absolute continuity; stability; central limit theorem; hidden Markov models; disagreement percolation