Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Brownian motion and random walk above quenched random wall

Bastien Mallein and Piotr Miłoś

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We study the persistence exponent for the first passage time of a random walk below the trajectory of another random walk. More precisely, let $\{B_{n}\}$ and $\{W_{n}\}$ be two centered, weakly dependent random walks. We establish that $\mathbb{P}(\forall_{n\leq N}B_{n}\geq W_{n}\vert W)=N^{-\gamma+o(1)}$ for a non-random $\gamma\geq1/2$. In the classical setting, $W_{n}\equiv0$, it is well-known that $\gamma=1/2$. We prove that for any non-trivial $W$ one has $\gamma>1/2$ and the exponent $\gamma$ depends only on $\operatorname{Var}(B_{1})/\operatorname{Var}(W_{1})$. Our result holds also in the continuous setting, when $B$ and $W$ are independent and possibly perturbed Brownian motions or Ornstein–Uhlenbeck processes. In the latter case the probability decays at exponential rate.

Résumé

On s’intéresse à l’exposant de persistance du temps de premier passage d’une marche aléatoire en-dessous de la trajectoire d’une autre marche aléatoire. Plus précisément, étant données deux marches aléatoires $\{B_{n}\}$ et $\{W_{n}\}$, centrées et faiblement corrélées, on établit que $\mathbb{P}(\forall_{n\leq N}B_{n}\geq W_{n}\vert W)=N^{-\gamma+o(1)}$ pour un certain exposant $\gamma\geq1/2$ déterministe. Il est bien connu que lorsque $W_{n}\equiv0$, on a $\gamma=1/2$. On prouve ici que lorsque $W$ n’est pas la marche nulle, alors $\gamma>1/2$, et dépend seulement du rapport $\operatorname{Var}(B_{1})/\operatorname{Var}(W_{1})$. Notre résultat est également valable en temps continu, lorsque $B$ et $W$ sont des mouvements browniens ou des processus d’Ornstein–Uhlenbeck indépendants. Dans ce dernier cas cependant, la queue du temps de premier passage décroit à taux exponentiel.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 4 (2018), 1877-1916.

Dates
Received: 1 February 2016
Revised: 3 August 2017
Accepted: 15 August 2017
First available in Project Euclid: 18 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1539849787

Digital Object Identifier
doi:10.1214/17-AIHP859

Mathematical Reviews number (MathSciNet)
MR3865661

Zentralblatt MATH identifier
06996553

Subjects
Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 60G10: Stationary processes 60G15: Gaussian processes 60G50: Sums of independent random variables; random walks 60K37: Processes in random environments

Keywords
Brownian motion Persistence exponent Quenched environment First passage time Ornstein–Uhlenbeck process

Citation

Mallein, Bastien; Miłoś, Piotr. Brownian motion and random walk above quenched random wall. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 4, 1877--1916. doi:10.1214/17-AIHP859. https://projecteuclid.org/euclid.aihp/1539849787


Export citation

References

  • [1] F. Aurzada and T. Simon Persistence probabilities and exponents. In Lévy Matters V: Functionals of Lévy Processes 183–224. Springer, Cham, 2015.
  • [2] D. Barbato. FKG inequality for Brownian motion and stochastic differential equations. Electron. Commun. Probab. 10 (2005) 7–16.
  • [3] D. Bertacchi and G. Giacomin. Enhanced interface repulsion from quenched hard-wall randomness. Probab. Theory Related Fields 124 (4) (2002) 487–516.
  • [4] D. Bertacchi and G. Giacomin. On the repulsion of an interface above a correlated substrate. Bull. Braz. Math. Soc. (N.S.) 34 (3) (2003) 401–415.
  • [5] R. J. Gardner. The Brunn-Minkowski inequality. Bull. Amer. Math. Soc. (N.S.) 39 (3) (2002) 355–405.
  • [6] G. Giacomin. Aspects of Statistical Mechanics of Random Surfaces. Notes of the Lectures Given at IHP, 2001. Available at www.proba.jussieu.fr/pageperso/giacomin/pub/IHP.ps.
  • [7] O. Kallenberg. Foundations of Modern Probability. Probability and Its Applications. Springer, New York, 1997.
  • [8] M. Lifshits. Lecture Notes on Strong Approximation, Publications IRMA Lille, 2000. Available at https://sites.google.com/site/mlprobability/home10/ml09.
  • [9] B. Mallein and P. Miłoś. Maximal displacement of a supercritical branching random walk in a time-inhomogeneous random environment. To appear. DOI:10.1016/j.spa.2018.09.008.
  • [10] G. Metafune, J. Prüss, A. Rhandi and R. Schnaubelt. The domain of the Ornstein–Uhlenbeck operator on an $L^{p}$-space with invariant measure. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1 (2) (2002) 471–485.
  • [11] A. G. Nobile, L. M. Ricciardi and L. Sacerdote. Exponential trends of Ornstein–Uhlenbeck first-passage-time densities. J. Appl. Probab. 22 (2) (1985) 360–369.
  • [12] W. Rudin. Real and Complex Analysis, 3rd edition. McGraw-Hill, New York, 1987.