Noise reinforcement for L{\'e}vy processes
Abstract
In a step reinforced random walk, at each integer time and with a fixed probability p (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.
Cite
@article{arxiv.1810.08364,
title = {Noise reinforcement for L{\'e}vy processes},
author = {Jean Bertoin},
journal= {arXiv preprint arXiv:1810.08364},
year = {2018}
}