Volume 16, 2012
|Page(s)||277 - 305|
|Published online||11 July 2012|
Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small-noise limit
Institut de Mathématiques de Bourgogne, Université de
Bourgogne, 9, rue Alain
2 Fakultät für Mathematik, Universität Bielefeld, 33615 Bielefeld, Germany
Revised: 6 May 2011
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.
Mathematics Subject Classification: 60J60 / 60H10 / 41A60
Key words: Self-interacting diffusion / McKean–Vlasov equation / stationary measures / double-well potential / perturbed dynamical system / Laplace’s method / fixed point theorem / uniqueness problem
© EDP Sciences, SMAI, 2012
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