On the one-sided exit problem for stable processes in random scenery

Fabienne Castell,Nadine Guillotin-Plantard,Bruno Schapira,Françoise Pène

doi:10.1214/ecp.v18-2444

Abstract

We consider the one-sided exit problem for stable Levy process in random scenery, that is the asymptotic behaviour for $T$ large of the probability $$\mathbb{P}\Big[ \sup_{t\in[0,T]} \Delta_t \leq 1\Big] $$ where $$\Delta_t = \int_{\mathbb{R}} L_t(x) \, dW(x).$$ Here $W=(W(x))_{x\in\mathbb{R}}$ is a two-sided standard real Brownian motion and $(L_t(x))_{x\in\mathbb{R},t\geq 0}$ the local time of a stable Levy process with index $\alpha\in (1,2]$, independent from the process $ W$. Our result confirms some physicists prediction by Redner and Majumdar.

Highlights

Random processes in random scenery are simple models of processes in disordered media with long-range correlations. These processes have been used in a wide variety of models in physics to study anomalous dispersion in layered random flows [13, 5], diffusion with random sources, or spin depolarization in random fields
Let us mention the fact that these processes are functional limits of random walks in random scenery [9, 6, 7, 4, 8]
We choose β > c/(2α) and p such that p(1 −

Summary

Introduction

Random processes in random scenery are simple models of processes in disordered media with long-range correlations. These processes have been used in a wide variety of models in physics to study anomalous dispersion in layered random flows [13, 5], diffusion with random sources, or spin depolarization in random fields (we refer the reader to Le Doussal’s review paper [11] for a discussion of these models). Let {Xt; t ≥ 0} be a continuous process, self-similar with index H ∈ (0, 1), with stationary increments s.t. for every θ > 0,. By applying this result to our random process ∆ we get Proposition 2.1.

The process

Hence we have

Remark that