main-content

## Swipe to navigate through the chapters of this book

Published in:

2021 | OriginalPaper | Chapter

# Universality of Noise Reinforced Brownian Motions

Author : Jean Bertoin

Publisher:

## 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.

Literature
1.
Baur, E.: Baur, E.: On a class of random walks with reinforced memory. J. Stat. Phys. 181, 772–802 (2020). https://​doi.​org/​10.​1007/​s10955-020-02602-3
2.
Baur, E., Bertoin, J.: Elephant random walks and their connection to Pólya-type urns. Phys. Rev. E 94, 052134 (2016) CrossRef
3.
Bercu, B.: A martingale approach for the elephant random walk. J. Phys. A 51(1), 015201, 16 (2018)
4.
Bercu, B., Laulin, L.: On the multi-dimensional elephant random walk. J. Stat. Phys. 175(6), 1146–1163 (2019)
5.
Bertoin, J.: Noise reinforcement for Lévy processes. Ann. Inst. Henri Poincaré B 56, 2236–2252 (2020). https://​doi.​org/​10.​1214/​19-AIHP1037 MATH
6.
Businger, S.: The shark random swim (Lévy flight with memory). J. Stat. Phys. 1723, 701–717 (2018)
7.
Coletti, C.F., Papageorgiou, I.: Asymptotic analysis of the elephant random walk (2020). arXiv:1910.03142
8.
Coletti, C.F., Gava, R., Schütz, G.M.: Central limit theorem and related results for the elephant random walk. J. Math. Phys. 58(5), 053303, 8 (2017)
9.
Coletti, C.F., Gava, R., Schütz, G.M.: A strong invariance principle for the elephant random walk. J. Stat. Mech. Theory Exp. 12, 123207, 8 (2017)
10.
González-Navarrete, M., Lambert, R.: Non-Markovian random walks with memory lapses. J. Math. Phys. 5911, 113301, 11 (2018)
11.
Gouet, R.: Martingale functional central limit theorems for a generalized Pólya urn. Ann. Probabil. 213, 1624–1639 (1993) MATH
12.
Gut, A., Stadtmueller, U.: Elephant random walks with delays (2019). arXiv:1906.04930
13.
Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes. Grundlehren der mathematischen Wissenschaften. Springer, Berlin (2002)
14.
Janson, S.: Functional limit theorems for multitype branching processes and generalized pólya urns. Stoch. Process. Their Appl. 110(2), 177–245 (2004) CrossRef
15.
Kallenberg, O.: Foundations of Modern Probability. Probability and Its Applications (New York), 2nd edn. Springer, New York (2002)
16.
Kious, D., Sidoravicius, V.: Phase transition for the once-reinforced random walk on $$\mathbb {Z}^{d}$$-like trees. Ann. Probab. 464, 2121–2133 (2018)
17.
Kubota, N., Takei, M.: Gaussian fluctuation for superdiffusive elephant random walks. J. Stat. Phys. 177(6), 1157–1171 (2019)
18.
Kürsten, R.: Random recursive trees and the elephant random walk. Phys. Rev. E 93(3), 032111, 11 (2016)
19.
Pemantle, R.: A survey of random processes with reinforcement. Probab. Surveys 4, 1–79 (2007)
20.
Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, 3rd ed. Springer, Berlin (1999)
21.
Schütz, G.M., Trimper, S.: Elephants can always remember: exact long-range memory effects in a non-markovian random walk. Phys. Rev. E 70, 045101 (2004) CrossRef
22.
Whitt, W.: Proofs of the martingale FCLT. Probab. Surv. 4, 268–302 (2007)