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

Abstract

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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.

How to cite

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Herrmann, 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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  10. 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.  
  11. Y. Tamura, on asymptotic behaviors of the solution of a nonlinear diffusion equation. J. Fac. Sci. Univ. Tokyo Sect. IA Math.31 (1984) 195–221.  
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