Random walks in random environment with Markov dependence on time

Abstract

We consider a simple model of discrete-time random walk on Z ν , ν = 1, 2, . . . in a random environment independent in space and with Markov evolution in time. We focus on the application of methods based on the properties of the transfer matrix and on spectral analysis. In section 2 we give a new simple proof of the existence of invariant subspaces, with an explicit condition on the parameters. The remaining part is devoted to a review of the results obtained so far for the quenched random walk and the environment from the point of view of the random walk, with a brief discussion of the methods.

Random mass splitting and a quenched invariance principle

“…Besides these models, a large variety of RWRE in dynamic correlated environments have been studied in recent years, see for example [5] [6] [10] [11] [14] [29]. The methods used vary from Fourier series in [5] [6], to martingale approximation in [10] [11], to regeneration times in [29]. Also note that the a version of RWRE where the environment is defined by random walks is considered in [13].…”

“…(ii) The environment under consideration has high correlations between different points in spacetime. The correlated environment models (such as [6] and [14]) considered so far are rather specialized. As far as we know, the assumptions made in those works do not apply to our model.…”

Weak dependence for a class of local functionals of Markov chains on $\mathbb{Z}^d$

“…The field η t (x) = ξ t (X t + x), t ∈ Z + is the "environment from the point of view of the particle". {η t : t ∈ Z + } is also a Markov chain with state space Ω, and it can be shown [6,9] that it is equivalent to the full process (X t , ξ t ), i.e, for all T ∈ Z + , T ≥ 1, given the sequence η 0 , . .…”

“…In many models, notably in the work of R.A. Minlos and collaborators (see, e.g. [1,3,4,6,13,14,15] and references therein) one can prove, usually by perturbative arguments, the existence of an invariant measure Π on the state space Ω = S Z d , and of a subspace of local functions H M ⊂ L 2 (Ω, Π), invariant with respect to the stochastic operator T and such that for all F ∈ H M with zero average F Π = 0, we have, for some constant μ ∈ (0, 1),…”