The Annals of Applied Probability

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

Samuel Herrmann, Peter Imkeller, and Dierk Peithmann

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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.

Article information

Source
Ann. Appl. Probab., Volume 18, Number 4 (2008), 1379-1423.

Dates
First available in Project Euclid: 21 July 2008

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1216677126

Digital Object Identifier
doi:10.1214/07-AAP489

Mathematical Reviews number (MathSciNet)
MR2434175

Zentralblatt MATH identifier
1149.60020

Subjects
Primary: 60F10: Large deviations 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 37H10: Generation, random and stochastic difference and differential equations [See also 34F05, 34K50, 60H10, 60H15] 82C22: Interacting particle systems [See also 60K35]

Keywords
Self-stabilization diffusion exit time exit law large deviations interacting particle systems domain of attraction propagation of chaos

Citation

Herrmann, Samuel; Imkeller, Peter; Peithmann, Dierk. Large deviations and a Kramers’ type law for self-stabilizing diffusions. Ann. Appl. Probab. 18 (2008), no. 4, 1379--1423. doi:10.1214/07-AAP489. https://projecteuclid.org/euclid.aoap/1216677126


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