Bouchaud walks with variable drift

TL;DR

This paper investigates the scaling limits of Bouchaud trap models with drift on the integer lattice, revealing three regimes depending on the decay rate of the drift, including stable, diffusion, and a new critical process.

Contribution

It introduces a detailed analysis of how varying decay rates of drift influence the scaling limits of Bouchaud trap models, identifying a new critical process at a specific decay speed.

Findings

Slow decay of drift leads to an inverse alpha-stable subordinator as limit.

Fast decay results in the F.I.N. diffusion as limit.

A critical decay speed separates the two regimes, revealing a new process.

Abstract

In this paper we study a sequence of Bouchaud trap models on $\mathbb{Z}$ with drift. We analyze the possible scaling limits for a sequence of walks, where we make the drift decay to 0 as we rescale the walks. Depending on the speed of the decay of the drift we obtain three different scaling limits. If the drift decays slowly as we rescale the walks we obtain the inverse of an \alpha$-stable subordinator as scaling limit. If the drift decays quickly as we rescale the walks, we obtain the F.I.N. diffusion as scaling limit. There is a critical speed of decay separating these two main regimes, where a new process appears as scaling limit. This critical speed is related to the index $\alpha$ of the inhomogeneity of the environment.