Central limit theorems for the monkey walk with steep memory kernel

Abstract

The monkey walk is a stochastic process defined as the trajectory of a walker that moves on $\mathbb R^d$ according to a Markovian generator, except at some random "relocation" times at which it jumps back to its position at a time sampled randomly in its past, according to some "memory kernel". The relocations make the process non-Markovian and introduce a reinforcement effect (the walker is more likely to relocate in a Borel set in which it has spent a lot of time in the past). In this paper, we focus on "steep" memory kernels: in these cases, the time sampled in the past at each relocation time is likely to be quite recent. One can see this as a way to model the case when the walker quickly "forgets" its past. We prove limit theorems for the position of the walker at large times, which confirm and generalise the estimates available in the physics literature.