Limit Theorems for Generalized Excited Random Walks in time-inhomogeneous Bernoulli environment

TL;DR

This work analyzes limit theorems for generalized excited random walks in a time-inhomogeneous Bernoulli environment, focusing on the p_n-GERW and its special case p_n-ERW with $p_n = \mathcal{C}n^{-\beta} \wedge 1$. It establishes a strong law of large numbers for the range in dimensions where the excitation process dominates, and a sub-ballistic SLLN for the ERW when $\beta<1/2$, contingent on range bounds. Diffusive scaling results yield a Functional Central Limit Theorem (FCLT) for $\beta>1/2$ (and a Brownian limit for $\beta=1/2$ in certain dimensions), with Brownian motion plus a time-sqrt correction appearing in the boundary cases, modulated by the dimension through parameters like $\pi_d$ and the drift component $\mathbb{E}[\gamma_1]$. The paper thus delineates the interplay between dimension, decay of excitation, and random-environment effects, revealing transitions between zero-mean-like behavior, diffusion-limited propagation, and ballistic-like drift in various regimes. These results advance understanding of non-Markovian random walks under time-inhomogeneous excitation and contribute to the broader study of intermittent driving in high-dimensional stochastic processes.

Abstract

We study a variant of the Generalized Excited Random Walk (GERW) on $\mathbb{Z}^d$ introduced by Menshikov, Popov, Ramírez and Vachkovskaia in [Ann. Probab. 40 (5), 2012]. It consists in a particular version of the model studied in [arXiv preprint arXiv:2211.05715, 2022] where excitation may or may not occur according to a time-dependent probability. Specifically, given $\{p_n\}_{n \ge 1}$, $p_n \in (0, 1]$ for all $n \ge 1$, whenever the process visits a site at time $n$ for the first time, with probability $p_n$ it gains a drift in a fixed direction. Otherwise, it behaves as a $d$-martingale with zero-mean vector. We refer to the model as a GERW in time-inhomogeneous Bernoulli environment, in short, $p_n$-GERW. Assuming bounded jumps and $p_n \approx n^{-β}$, we show a series of results for the $p_n$-\Name{} depending on the value of $β$ and on the dimension $d$. Specifically, for every $β\in(0,1]$ and $d=2$ or $d>h(β)$, with $h$ a decreasing function of $β$, we prove a SLLN for the range, while for $β<1/2$ we prove a sub-ballistic SLLN for the process whenever the SLLN for the range holds. We also study the $p_n$-\Name{} under diffusive scaling and we obtain a Functional Central Limit Theorem for $β> 1/2$ and $d\geq 2$, or $β=1/2$ and $d=2$. Finally, for $β=1/2$ and $d>22$ we show that the diffusively rescaled $p_n$-\Name{} converges in distribution to a Brownian Motion plus a multiple of the square root of time.

Theorems & Definitions (43)

• Remark 2.1
• Proposition 2.1
• Proposition 2.2
• Theorem 2.1: SLLN for the range
• Theorem 2.2
• Proposition 2.3
• Corollary 2.1
• Remark 2.2
• Theorem 2.3
• Corollary 2.2