6533b829fe1ef96bd1289b16
RESEARCH PRODUCT
Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small-noise limit
Julian TugautSamuel Herrmannsubject
Statistics and ProbabilityMcKean-Vlasov equationLaplace transformdouble-well potential010102 general mathematicsMathematical analysisFixed-point theoremfixed point theoremDouble-well potentialInvariant (physics)01 natural sciencesself-interacting diffusionuniqueness problem[MATH.MATH-PR]Mathematics [math]/Probability [math.PR]010104 statistics & probabilityRate of convergenceLaplace's methodUniquenessInvariant measureperturbed dynamical systemstationary measures0101 mathematicsLaplace's methodprimary 60G10; secondary: 60J60 60H10 41A60Mathematicsdescription
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.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2011-05-04 |