Universality of Noise Reinforced Brownian Motions

A noise reinforced Brownian motion is a centered Gaussian process \(\hat B=(\hat B(t))_{t\geq 0}\) with covariance $$\displaystyle \mathbb {E}(\hat B(t)\hat B(s))=(1-2p)^{-1}t^ps^{1-p} \quad \text{for} \quad 0\leq s \leq t, $$ 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 \(\hat B\) arises as the universal scaling limit of the former. This extends known results on the asymptotic behavior of the so-called elephant random walk.