Scaling limit theorems for the $\kappa$-transient random walk in random and non-random environment

TL;DR

This paper establishes the scaling limit of the $$-transient random walk in random and non-random environments, showing convergence to a diffusion process in a Brownian environment with drift, linking discrete and continuous models.

Contribution

It constructs a sequence of $$-transient RWREs and proves their convergence to a diffusion process, also providing a convergence result for non-random environments.

Findings

Convergence of $$-transient RWRE to diffusion in Brownian environment.

Establishment of a scaling limit linking discrete and continuous models.

New results on non-random environment case.

Abstract

Kesten et al.( 1975) proved the stable law for the transient RWRE (here we refer it as the $\kappa$-transient RWRE). After that, some similar interesting properties have also been revealed for its continuous counterpart, the diffusion proces in a Brownian environment with drift $\kappa$. In the present paper we will investigate the connections between these two kind of models, i.e., we will construct a sequence of the $\kappa$-transient RWREs and prove it convergence to the diffusion proces in a Brownian environment with drift $\kappa$ by proper scaling. To this end, we need a counterpart convergence for the $\kappa$-transient random walk in non-random environment, which is interesting itself.