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2010 Stationary measures for self-stabilizing processes: asymptotic analysis in the small noise limit
Samuel Herrmann, Julian Tugaut
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Electron. J. Probab. 15: 2087-2116 (2010). DOI: 10.1214/EJP.v15-842

Abstract

Self-stabilizing diffusions are stochastic processes, solutions of nonlinear stochastic differential equation, which are attracted by their own law. This specific self-interaction leads to singular phenomenons like non uniqueness of associated stationary measures when the diffusion moves in some non convex environment (see [5]). The aim of this paper is to describe these invariant measures and especially their asymptotic behavior as the noise intensity in the nonlinear SDE becomes small. We prove in particular that the limit measures are discrete measures and point out some properties of their support which permit in several situations to describe explicitly the whole set of limit measures. This study requires essentially generalized Laplace's method approximations.

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Samuel Herrmann. Julian Tugaut. "Stationary measures for self-stabilizing processes: asymptotic analysis in the small noise limit." Electron. J. Probab. 15 2087 - 2116, 2010. https://doi.org/10.1214/EJP.v15-842

Information

Accepted: 6 July 2010; Published: 2010
First available in Project Euclid: 1 June 2016

zbMATH: 1225.60095
MathSciNet: MR2745727
Digital Object Identifier: 10.1214/EJP.v15-842

Subjects:
Primary: 60H10
Secondary: 41A60, 60G10, 60J60

JOURNAL ARTICLE
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Vol.15 • 2010
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