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

Samuel Herrmann; Julian Tugaut

ESAIM: Probability and Statistics (2012)

- Volume: 16, page 277-305
- ISSN: 1292-8100

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topHerrmann, Samuel, and Tugaut, Julian. "Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small-noise limit." ESAIM: Probability and Statistics 16 (2012): 277-305. <http://eudml.org/doc/222477>.

@article{Herrmann2012,

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.},

author = {Herrmann, Samuel, Tugaut, Julian},

journal = {ESAIM: Probability and Statistics},

keywords = {Self-interacting diffusion; McKean–Vlasov equation; stationary measures; double-well potential; perturbed dynamical system; Laplace’s method; fixed point theorem; uniqueness problem; self-interacting diffusion; McKean-Vlasov equation, stationary measures; perturbed dynamical systems; Laplace method},

language = {eng},

month = {7},

pages = {277-305},

publisher = {EDP Sciences},

title = {Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small-noise limit},

url = {http://eudml.org/doc/222477},

volume = {16},

year = {2012},

}

TY - JOUR

AU - Herrmann, Samuel

AU - Tugaut, Julian

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

JO - ESAIM: Probability and Statistics

DA - 2012/7//

PB - EDP Sciences

VL - 16

SP - 277

EP - 305

AB - 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.

LA - eng

KW - Self-interacting diffusion; McKean–Vlasov equation; stationary measures; double-well potential; perturbed dynamical system; Laplace’s method; fixed point theorem; uniqueness problem; self-interacting diffusion; McKean-Vlasov equation, stationary measures; perturbed dynamical systems; Laplace method

UR - http://eudml.org/doc/222477

ER -

## References

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- S. Herrmann and J. Tugaut, Stationary measures for self-stabilizing processes : asymptotic analysis in the small noise limit. Electron. J. Probab.15 (2010) 2087–2116.
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- A.-S. Sznitman, Topics in propagation of chaos, in École d’Été de Probabilités de Saint-Flour XIX–1989, Springer, Berlin. Lect. Notes Math.1464 (1991) 165–251.
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