Large deviations and a Kramers' type law for self-stabilizing diffusions

Samuel Herrmann 1, 2 Peter Imkeller Dierk Peithmann
INRIA Lorraine, CRISAM - Inria Sophia Antipolis - Méditerranée , UHP - Université Henri Poincaré - Nancy 1, Université Nancy 2, INPL - Institut National Polytechnique de Lorraine, CNRS - Centre National de la Recherche Scientifique : UMR7502
Abstract : We investigate exit times from domains of attraction for the motion of a self-stabilized particle travelling in a geometric (potential type) landscape and perturbed by Brownian noise of small amplitude. Self-stabilization is mediated by an ensemble-average attraction adding on to the individual potential 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 with 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.
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Submitted on : Wednesday, April 4, 2007 - 11:01:22 AM
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Samuel Herrmann, Peter Imkeller, Dierk Peithmann. Large deviations and a Kramers' type law for self-stabilizing diffusions. Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2008, 18 (4), pp.1379-1423. ⟨10.1214/07-AAP489⟩. ⟨hal-00139965⟩



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