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.