Electronic Communications in Probability

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

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

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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.

Article information

Electron. Commun. Probab., Volume 18 (2013), paper no. 33, 7 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60F17: Functional limit theorems; invariance principles 60G15: Gaussian processes 60G18: Self-similar processes 60K37: Processes in random environments

Stable process Random scenery First passage time One-sided barrier problem One-sided exit problem Survival exponent

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Castell, Fabienne; Guillotin-Plantard, Nadine; Pène, Françoise; Schapira, Bruno. On the one-sided exit problem for stable processes in random scenery. Electron. Commun. Probab. 18 (2013), paper no. 33, 7 pp. doi:10.1214/ECP.v18-2444. https://projecteuclid.org/euclid.ecp/1465315572

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