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2013 On the one-sided exit problem for stable processes in random scenery
Fabienne Castell, Nadine Guillotin-Plantard, Françoise Pène, Bruno Schapira
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Electron. Commun. Probab. 18(none): 1-7 (2013). DOI: 10.1214/ECP.v18-2444

Abstract

We consider the one-sided exit problem for stable Lévy 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 Lévy process with index $\alpha\in (1,2]$, independent from the process $W$. Our result confirms some physicists prediction by Redner and Majumdar.

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Fabienne Castell. Nadine Guillotin-Plantard. Françoise Pène. Bruno Schapira. "On the one-sided exit problem for stable processes in random scenery." Electron. Commun. Probab. 18 1 - 7, 2013. https://doi.org/10.1214/ECP.v18-2444

Information

Accepted: 14 May 2013; Published: 2013
First available in Project Euclid: 7 June 2016

zbMATH: 1329.60356
MathSciNet: MR3064992
Digital Object Identifier: 10.1214/ECP.v18-2444

Subjects:
Primary: 60F05
Secondary: 60F17, 60G15, 60G18, 60K37

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