# In and Out of Equilibrium 3: Celebrating Vladas SidoraviciusUniversality of Noise Reinforced Brownian Motions

In and Out of Equilibrium 3: Celebrating Vladas Sidoravicius: Universality of Noise Reinforced... [A noise reinforced Brownian motion is a centered Gaussian process B̂=(B̂(t))t≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\hat B=(\hat B(t))_{t\geq 0}$$ \end{document} with covariance 𝔼(B̂(t)B̂(s))=(1−2p)−1tps1−pfor0≤s≤t,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\displaystyle \mathbb {E}(\hat B(t)\hat B(s))=(1-2p)^{-1}t^ps^{1-p} \quad \text{for} \quad 0\leq s \leq t,$$ \end{document} 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 B̂\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\hat B$$ \end{document} arises as the universal scaling limit of the former. This extends known results on the asymptotic behavior of the so-called elephant random walk.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# In and Out of Equilibrium 3: Celebrating Vladas SidoraviciusUniversality of Noise Reinforced Brownian Motions

Part of the Progress in Probability Book Series (volume 77)
Editors: Vares, Maria Eulália; Fernández, Roberto; Fontes, Luiz Renato; Newman, Charles M.
14 pages      