Large deviations principle for a stochastic process with random reinforced relocations

TL;DR

This paper establishes a large deviations principle for a non-Markovian stochastic process with random relocations, modeling animal behavior, by overcoming challenges posed by its memory-dependent jumps and random environment.

Contribution

It proves a quenched large deviations principle for a process with random reinforced relocations, addressing the non-Markovian nature and environmental randomness.

Findings

Established a large deviations principle for the process at large times.

Addressed the non-Markovian challenge due to relocations.

Handled the randomness of inter-relocation times as environmental factors.

Abstract

Stochastic processes with random reinforced relocations have been introduced in the physics literature to model animal foraging behaviour. Such a process evolves as a Markov process, except at random relocation times, when it chooses a time at random in its whole past according to some ``memory kernel'', and jumps to its value at that random time. We prove a quenched large deviations principle for the value of the process at large times. The difficulty in proving this result comes from the fact that the process is not Markov because of the relocations. Furthermore, the random inter-relocation times act as a random environment.