Noise reinforcement for L{\'e}vy processes

TL;DR

This paper introduces a noise reinforced Lévy process as a continuous-time analog of step reinforced random walks, demonstrating convergence of discrete skeletons to this new process under certain conditions related to the process's Blumenthal-Getoor index.

Contribution

It constructs a noise reinforced Lévy process and proves the weak convergence of discrete step reinforced Lévy walks to this process as the time mesh shrinks.

Findings

Weak convergence of discrete reinforced Lévy walks to the continuous process.

Construction of noise reinforced Lévy process for sub-critical memory parameters.

Connection between reinforcement parameters and Lévy process properties.

Abstract

In a step reinforced random walk, at each integer time and with a fixed probability p $\in$ (0, 1), the walker repeats one of his previous steps chosen uniformly at random, and with complementary probability 1 -- p, the walker makes an independent new step with a given distribution. Examples in the literature include the so-called elephant random walk and the shark random swim. We consider here a continuous time analog, when the random walk is replaced by a L{\'e}vy process. For sub-critical (or admissible) memory parameters p < p c , where p c is related to the Blumenthal-Getoor index of the L{\'e}vy process, we construct a noise reinforced L{\'e}vy process. Our main result shows that the step-reinforced random walks corresponding to discrete time skeletons of the L{\'e}vy process, converge weakly to the noise reinforced L{\'e}vy process as the time-mesh goes to 0.