Séminaire MODAL'X : Lucile Laulin (MODAL'X)

Résumé : A step-reinforced random walk is a random walk that, at each step, either repeats one of its former steps chosen uniformly or takes a new step independently from its past. We introduce a variation of the step-reinforced random walk with general memory, which can be interpreted as amnesia. Our main purpose is to establish a version of Donsker’s invariance principle for such amnesic step-reinforced random walks in the so-called diffusive regime. While for the standard step-reinforced walk the scaling limit is a noise-reinforced Brownian motion, we show that for the amnesic version, the limiting Gaussian process is actually the sum of a noise-reinforced Brownian motion and a Brownian motion, which are dependent. (Joint work with Marco Bertenghi)