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2021 | OriginalPaper | Chapter

Universality of Noise Reinforced Brownian Motions

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Abstract

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.
Footnotes
1
Beware that we assumed p < 1∕2 in the preceding sections. The first part of the present section (super-diffusive regime) does not involve any noise reinforced Brownian motion, and the case p ≥ 1∕2 is allowed. In the second part of this section (diffusive regime), we shall again focus on the case p < 1∕2 and noise reinforced Brownian motions will then re-appear.
 
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Metadata
Title
Universality of Noise Reinforced Brownian Motions
Author
Jean Bertoin
Copyright Year
2021
DOI
https://doi.org/10.1007/978-3-030-60754-8_7

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