Invariant measure for random walks on ergodic environments on a strip

Abstract

Environment viewed from the particle is a powerful method of analyzing random walks (RW) in random environment (RE). It is well known that in this setting the environment process is a Markov chain on the set of environments. We study the fundamental question of existence of the density of the invariant measure of this Markov chain with respect to the measure on the set of environments for RW on a strip. We first describe all positive sub-exponentially growing solutions of the corresponding invariant density equation in the deterministic setting and then derive necessary and sufficient conditions for the existence of the density when the environment is ergodic in both the transient and the recurrent regimes. We also provide applications of our analysis to the question of positive and null recurrence, the study of the Green functions and to random walks on orbits of a dynamical system. MSC 2010 subject classifications: Primary 60K37; secondary 60J05, 82C44 Keywords and phrases: RWRE, random walks on a strip, invariant measure 1 imsart-aop ver.

Constructive approach to limit theorems for recurrent diffusive random walks on a strip

“…Second, in the case when we deal with stationary environment the solution to (2.27) provides invariant densities for the environment viewed from the particle process. We refer the reader to [20] for a comprehensive analysis of this equation on the strip. The invariant measure equation for the stationary walks on Z with bounded jumps was studied in [10].…”

“…The first one is the asymptotic formula for the Green function in a large finite domain obtained in Section 4. The derivation of this formula relies on an entirely new approach to the analysis of the martingale and the invariant measure equations which was recently discovered in [20]. This approach is further developed in this work and leads to new algebraic properties of the solutions to these equations.…”

On First-Passage Times and Sojourn Times in Finite QBD Processes and Their Applications in Epidemics

“…These would lead to QBD processes with infinitely many possible values of the level variable and/or the phase variable, and piecewise-deterministic Markov processes, respectively. In these frameworks, the study of the stochastic descriptors analyzed in Sections 2-4 should require an analytical treatment and related numerical procedures that will be substantially different from those used in this paper; in particular, the analysis of first-passage times and sojourn times could benefit from the approach described by Dolgopyat and Goldsheid [45] , who obtain necessary and sufficient conditions for the existence of the density of the invariant measure for random walks when the environment is ergodic in both the transient and recurrent regimes. Another aspect that deserves further exploration is how the finite-dimensional linear algebra ideas in Sections 2 and 3 could be replaced by some suitable extension to the QBD case of the Karlin-McGregor orthogonal polynomial/spectral representation by translating the arguments in Reference [46][47][48] for discrete time processes to continuous time.…”

Erratic behavior for 1-dimensional random walks in a Liouville quasi-periodic environment

“…The results of Theorems 2.1-2.3 have been extended to random walks driven by rotations of T d , for arbitrary d ∈ N, to random walks with bounded jumps where the walker can move from x to x + jα with |j| ≤ L for some L > 1 and to quasi-periodic walks on the strip, see [3,6,7,8].…”