Limit theorems for random walks with spatio-temporal drift

TL;DR

This paper analyzes the asymptotic behavior of a class of discrete-time random walks with polynomially decaying spatio-temporal drift, revealing phase transitions and different limit regimes based on the drift's linearity and moments.

Contribution

It characterizes the limiting distributions and phase transitions for random walks with polynomially decaying drift, including Gaussian, non-Gaussian, and localization regimes.

Findings

In the linear drift case, normalized processes converge to Gaussian limits.

In the nonlinear case, three regimes are identified with distinct behaviors.

The walk exhibits phase transitions depending on the drift's structure and moments.

Abstract

We study a class of discrete-time random walks in $\mathbb{R}^d$ whose conditional drift decays polynomially in time and grows polynomially with the distance from the origin to the current position. This class is related to several models of self-interacting random processes. We determine the asymptotic behavior of the walk under the assumption that its increments have moments of order $p$ for some $p>2$. In the linear case, where the drift depends linearly on the current position, we establish a phase transition in the convergence in distribution of the normalized process to Gaussian limits. In the nonlinear case, we identify three distinct regimes separated by a critical line and show that the normalized process exhibits qualitatively different behaviors in each regime, including convergence in distribution to a Gaussian law, convergence to a non-Gaussian limit given by the stationary distribution of a stochastic differential equation, and almost sure localization on a hypersphere.