Universality of Noise Reinforced Brownian Motions
Résumé
A noise reinforced Brownian motion is a centered Gaussian procesŝ B = (B(t)) t≥0 with covariance E(B(t)B(s)) = (1 − 2p) −1 t p s 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. Specifically,B arises as the universal scaling limit for a large family of step-reinforced random walks in the diffusive regime. This extends known results on the asymptotic behavior of the so-called elephant random walk.
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