Limit theorems for random walks with spatio-temporal drift

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.