Large deviations and a Kramers’ type law for self-stabilizing diffusions
Samuel Herrmann, Peter Imkeller, and Dierk Peithmann
Source: Ann. Appl. Probab.
Volume 18, Number 4
(2008), 1379-1423.
Abstract
We investigate exit times from domains of attraction for the motion of a self-stabilized particle traveling in a geometric (potential type) landscape and perturbed by Brownian noise of small amplitude. Self-stabilization is the effect of including an ensemble-average attraction in addition to the usual state-dependent drift, where the particle is supposed to be suspended in a large population of identical ones. A Kramers’ type law for the particle’s exit from the potential’s domains of attraction and a large deviations principle for the self-stabilizing diffusion are proved. It turns out that the exit law for the self-stabilizing diffusion coincides with the exit law of a potential diffusion without self-stabilization and a drift component perturbed by average attraction. We show that self-stabilization may substantially delay the exit from domains of attraction, and that the exit location may be completely different.
Primary Subjects: 60F10, 60H10
Secondary Subjects: 60K35, 37H10, 82C22
Keywords: Self-stabilization; diffusion; exit time; exit law; large deviations; interacting particle systems; domain of attraction; propagation of chaos
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aoap/1216677126
Digital Object Identifier: doi:10.1214/07-AAP489
Mathematical Reviews number (MathSciNet):
MR2434175
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