zbMATH — the first resource for mathematics

On the one-sided exit problem for stable processes in random scenery. (English) Zbl 1329.60356
Summary: We consider the one-sided exit problem for a stable Lévy process in random scenery, that is, the asymptotic behaviour for large \(T\) 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 S. Redner [“Superdiffusion in random velocity fields”, Physica A 168, No. 1, 551–560 (1990; doi:10.1016/0378-4371(90)90408-K); “Survival probability in a random velocity field”, Phys. Rev. E 56, No. 5, Article ID 4967 (1997; doi:10.1103/PhysRevE.56.4967)] and S. N. Majumdar [“Persistence of a particle in the Matheron-de Marsily velocity field”, Phys. Rev. E 68, No. 5, Article ID 050101(R) (2003; doi:10.1103/PhysRevE.68.050101)].

60K37 Processes in random environments
60G51 Processes with independent increments; Lévy processes
60G52 Stable stochastic processes
60F05 Central limit and other weak theorems
60F17 Functional limit theorems; invariance principles
60G15 Gaussian processes
60G18 Self-similar stochastic processes
Full Text: DOI