Universality of Noise Reinforced Brownian Motions

2020/02/21 by Bertoin, Jean · 7 citations
#60G50 (Primary) 60G51 #60K35 (Secondary) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2002.09166

Abstract

A noise reinforced Brownian motion is a centered Gaussian process B=( B(t))_{t≥ 0} with covariance E( B(t) B(s))=(1-2p)^-1t^ps^1-p for 0≤ s ≤ t, where p∈(0,1/2) is a reinforcement parameter. Our main purpose is to establish a version of Donsker's invariance principle for a large family of step-reinforced random walks in the diffusive regime, and more specifically, to show that B arises as the universal scaling limit of the former. This extends known results on the asymptotic behavior of the so-called elephant random walk.

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