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Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small-noise limit

Published online by Cambridge University Press:  11 July 2012

Samuel Herrmann  and
Julian Tugaut
Samuel Herrmann
Affiliation:
Institut de Mathématiques de Bourgogne, Université de Bourgogne, 9, rue Alain Savary, 21078 Dijon, France. samuel.herrmann@u-bourgogne.fr
Julian Tugaut
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, 33615 Bielefeld, Germany
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Abstract

In the context of self-stabilizing processes, that is processes attracted by their own law, living in a potential landscape, we investigate different properties of the invariant measures. The interaction between the process and its law leads to nonlinear stochastic differential equations. In [S. Herrmann and J. Tugaut. Electron. J. Probab. 15 (2010) 2087–2116], the authors proved that, for linear interaction and under suitable conditions, there exists a unique symmetric limit measure associated to the set of invariant measures in the small-noise limit. The aim of this study is essentially to point out that this statement leads to the existence, as the noise intensity is small, of one unique symmetric invariant measure for the self-stabilizing process. Informations about the asymmetric measures shall be presented too. The main key consists in estimating the convergence rate for sequences of stationary measures using generalized Laplace’s method approximations.

Keywords

Self-interacting diffusionMcKean–Vlasov equationstationary measuresdouble-well potentialperturbed dynamical systemLaplace’s methodfixed point theoremuniqueness problem
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 16 , 2012 , pp. 277 - 305
Copyright
© EDP Sciences, SMAI, 2012

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