On the one-sided exit problem for stable processes in random scenery
Nadine Guillotin-Plantard (Université de Lyon)
Françoise Pène (Université de Brest)
Bruno Schapira (Aix-Marseille Université)
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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Pages: 1-7
Publication Date: May 14, 2013
DOI: 10.1214/ECP.v18-2444
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