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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)].

##### MSC:
 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
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