Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Brownian motion and random walk above quenched random wall

Bastien Mallein and Piotr Miłoś

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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.


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

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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