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A slow transient diffusion in a drifted stable potential

By Arvind Singh

Abstract

We consider a diffusion process $X$ in a random potential $\V$ of the form $\V_x = \S_x -\delta x$ where $\delta$ is a positive drift and $\S$ is a strictly stable process of index $\alpha\in (1,2)$ with positive jumps. Then the diffusion is transient and $X_t / \log^\alpha t$ converges in law towards an exponential distribution. This behaviour contrasts with the case where $\V$ is a drifted Brownian motion and provides an example of a transient diffusion in a random potential which is as "slow" as in the recurrent setting

Topics: diffusion with random potential, stable processes, MSC 60K37, 60J60, 60F05, [MATH.MATH-PR] Mathematics [math]/Probability [math.PR]
Publisher: HAL CCSD
Year: 2006
OAI identifier: oai:HAL:hal-00119374v1
Provided by: Hal-Diderot
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