Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-08T08:15:49.127Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  11 July 2012

Samuel Herrmann  and
Julian Tugaut
Samuel Herrmann
Affiliation:
Institut de Mathématiques de Bourgogne, Université de Bourgogne, 9, rue Alain Savary, 21078 Dijon, France. [email protected]
Julian Tugaut
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, 33615 Bielefeld, Germany
Get access

Abstract

In the context of self-stabilizing processes, that is processes attracted by their ownlaw, living in a potential landscape, we investigate different properties of the invariantmeasures. The interaction between the process and its law leads to nonlinear stochasticdifferential equations. In [S. Herrmann and J. Tugaut. Electron. J. Probab.15 (2010) 2087–2116], the authors proved that, for linearinteraction and under suitable conditions, there exists a unique symmetric limit measureassociated to the set of invariant measures in the small-noise limit. The aim of thisstudy is essentially to point out that this statement leads to the existence, as the noiseintensity is small, of one unique symmetric invariant measure for the self-stabilizingprocess. Informations about the asymmetric measures shall be presented too. The main keyconsists in estimating the convergence rate for sequences of stationary measures usinggeneralized Laplace’s method approximations.

Keywords

Self-interacting diffusionMcKean–Vlasov equationstationary measuresdouble-well potentialperturbed dynamical systemLaplace’s methodfixed point theoremuniqueness problem
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 16 , 2012 , pp. 277 - 305
Copyright
© EDP Sciences, SMAI, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Benachour, S., Roynette, B., Talay, D. and Vallois, P., Nonlinear self-stabilizing processes. I. Existence, invariant probability, propagation of chaos. Stoc. Proc. Appl. 75 (1998) 173–201. Google Scholar
Benachour, S., Roynette, B. and Vallois, P., Nonlinear self-stabilizing processes. II. Convergence to invariant probability. Stoc. Proc. Appl. 75 (1998) 203–224. Google Scholar
Cattiaux, P., Guillin, A. and Malrieu, F., Probabilistic approach for granular media equations in the non-uniformly convex case. Probab. Theory Relat. Fields 140 (2008) 19–40. Google Scholar
Funaki, T., A certain class of diffusion processes associated with nonlinear parabolic equations. Z. Wahrsch. Verw. Gebiete 67 (1984) 331–348. Google Scholar
Herrmann, S. and Tugaut, J., Non-uniqueness of stationary measures for self-stabilizing processes. Stoc. Proc. Appl. 120 (2010) 1215–1246. Google Scholar
Herrmann, S. and Tugaut, J., Stationary measures for self-stabilizing processes : asymptotic analysis in the small noise limit. Electron. J. Probab. 15 (2010) 2087–2116. Google Scholar
Herrmann, S., Imkeller, P. and Peithmann, D., Large deviations and a Kramers’ type law for self-stabilizing diffusions. Ann. Appl. Probab. 18 (2008) 1379–1423. Google Scholar
Malrieu, F., Logarithmic Sobolev inequalities for some nonlinear PDE’s. Stoc. Proc. Appl. 95 (2001) 109–132. Google Scholar
McKean, H.P. Jr., A class of Markov processes associated with nonlinear parabolic equations. Proc. Natl. Acad. Sci. USA 56 (1966) 1907–1911. Google ScholarPubMed
Sznitman, A.-S., 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. Google Scholar
Tamura, Y., on asymptotic behaviors of the solution of a nonlinear diffusion equation. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31 (1984) 195–221. Google Scholar
Tamura, Y., Free energy and the convergence of distributions of diffusion processes of McKean type. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987) 443–484. Google Scholar
Veretennikov, A.Yu., On ergodic measures for McKean–Vlasov stochastic equations. Monte Carlo and Quasi-Monte Carlo Methods 2004 (2006) 471–486. Google Scholar