Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Persistence of some additive functionals of Sinai’s walk

Alexis Devulder

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Abstract

We are interested in Sinai’s walk $(S_{n})_{n\in\mathbb{N}}$. We prove that the annealed probability that $\sum_{k=0}^{n}f(S_{k})$ is strictly positive for all $n\in[1,N]$ is equal to $1/(\log N)^{(3-\sqrt{5})/2+o(1)}$, for a large class of functions $f$, and in particular for $f(x)=x$. The persistence exponent $\frac{3-\sqrt{5}}{2}$ first appears in a nonrigorous paper of Le Doussal, Monthus and Fischer, with motivations coming from physics. The proof relies on techniques of localization for Sinai’s walk and uses results of Cheliotis about the sign changes of the bottom of valleys of a two-sided Brownian motion.

Résumé

Nous nous intéressons à la marche de Sinai $(S_{n})_{n\in\mathbb{N}}$. Nous prouvons que la probabilité annealed que $\sum_{k=0}^{n}f(S_{k})$ soit strictement positive pour tout $n\in[1,N]$ est égale à $1/(\log N)^{(3-\sqrt{5})/2+o(1)}$, pour une large classe de fonctions $f$, et en particulier pour $f(x)=x$. L’exposant de persistance $\frac{3-\sqrt{5}}{2}$ est d’abord apparu dans un article non rigoureux de Le Doussal, Monthus et Fischer, avec des motivations venant de la physique. La preuve est basée sur des techniques de localisation pour la marche de Sinai et utilise des résultats de Cheliotis sur les changements de signe des fonds de vallées d’un mouvement Brownien indexé par $\mathbb{R}$.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 3 (2016), 1076-1105.

Dates
Received: 19 February 2014
Revised: 10 March 2015
Accepted: 7 April 2015
First available in Project Euclid: 28 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1469723512

Digital Object Identifier
doi:10.1214/15-AIHP679

Mathematical Reviews number (MathSciNet)
MR3531701

Zentralblatt MATH identifier
1350.60110

Subjects
Primary: 60K37: Processes in random environments 60J55: Local time and additive functionals

Keywords
Random walk in random environment Sinai’s walk Integrated random walk One-sided exit problem Persistence Survival exponent

Citation

Devulder, Alexis. Persistence of some additive functionals of Sinai’s walk. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 3, 1076--1105. doi:10.1214/15-AIHP679. https://projecteuclid.org/euclid.aihp/1469723512


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