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

¹ Institut de Mathématiques de Bourgogne, Université de Bourgogne, 9, rue Alain Savary, 21078 Dijon, France
samuel.herrmann@u-bourgogne.fr
² Fakultät für Mathematik, Universität Bielefeld, 33615 Bielefeld, Germany

Received: 8 January 2010
Revised: 6 May 2011

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.