%0 Journal Article
%T On the one-sided exit problem for stable processes in random scenery
%+ Laboratoire d'Analyse, Topologie, Probabilités (LATP)
%+ Institut Camille Jordan (ICJ)
%+ Laboratoire de mathématiques de Brest (LM)
%A Castell, Fabienne
%A Guillotin-Plantard, Nadine
%A Pene, Françoise
%A Schapira, Bruno
%< avec comité de lecture
%@ 1083-589X
%J Electronic Communications in Probability
%I Institute of Mathematical Statistics (IMS)
%P Vol. 18, No 33, 1--7
%8 2013
%D 2013
%R 10.1214/ECP.v18-2444
%Z Mathematics [math]/Probability [math.PR]Journal articles
%X We consider the one-sided exit problem for stable LÈvy 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.
%G English
%2 https://hal.science/hal-00753026/document
%2 https://hal.science/hal-00753026/file/exit-problem5.pdf
%L hal-00753026
%U https://hal.science/hal-00753026
%~ UNIV-ST-ETIENNE
%~ UNIV-BREST
%~ LATP
%~ CNRS
%~ ICJ
%~ UNIV-AMU
%~ UNIV-LYON1
%~ INSA-LYON
%~ EC-LYON
%~ MATHBREST
%~ EC-MARSEILLE
%~ LMBA
%~ UBS
%~ I2M
%~ INSA-GROUPE
%~ UDL
%~ UNIV-LYON