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

top
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

top

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/274338>.

@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; McKean-Vlasov equation, stationary measures; perturbed dynamical systems; Laplace method},
language = {eng},
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/274338},
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
PY - 2012
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; McKean-Vlasov equation, stationary measures; perturbed dynamical systems; Laplace method
UR - http://eudml.org/doc/274338
ER -

References

top
  1. [1] S. Benachour, B. Roynette, D. Talay and P. Vallois, Nonlinear self-stabilizing processes. I. Existence, invariant probability, propagation of chaos. Stoc. Proc. Appl. 75 (1998) 173–201. Zbl0932.60063MR1632193
  2. [2] S. Benachour, B. Roynette and P. Vallois, Nonlinear self-stabilizing processes. II. Convergence to invariant probability. Stoc. Proc. Appl. 75 (1998) 203–224. Zbl0932.60064MR1632197
  3. [3] P. Cattiaux, A. Guillin and F. Malrieu, Probabilistic approach for granular media equations in the non-uniformly convex case. Probab. Theory Relat. Fields140 (2008) 19–40. Zbl1169.35031MR2357669
  4. [4] T. Funaki, A certain class of diffusion processes associated with nonlinear parabolic equations. Z. Wahrsch. Verw. Gebiete67 (1984) 331–348. Zbl0546.60081MR762085
  5. [5] S. Herrmann and J. Tugaut, Non-uniqueness of stationary measures for self-stabilizing processes. Stoc. Proc. Appl.120 (2010) 1215–1246. Zbl1197.60052MR2639745
  6. [6] 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. Zbl1225.60095MR2745727
  7. [7] S. Herrmann, P. Imkeller and D. Peithmann, Large deviations and a Kramers’ type law for self-stabilizing diffusions. Ann. Appl. Probab.18 (2008) 1379–1423. Zbl1149.60020MR2434175
  8. [8] F. Malrieu, Logarithmic Sobolev inequalities for some nonlinear PDE’s. Stoc. Proc. Appl.95 (2001) 109–132. Zbl1059.60084MR1847094
  9. [9] H.P. McKean Jr., A class of Markov processes associated with nonlinear parabolic equations. Proc. Natl. Acad. Sci. USA56 (1966) 1907–1911. Zbl0149.13501MR221595
  10. [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. Zbl0732.60114MR1108185
  11. [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. Zbl0544.60058MR743525
  12. [12] Y. Tamura, 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. Zbl0638.60070MR914029
  13. [13] A.Yu. Veretennikov, On ergodic measures for McKean–Vlasov stochastic equations. Monte Carlo and Quasi-Monte Carlo Methods2004 (2006) 471–486. Zbl1098.60056MR2208726

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.