Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small-noise limit
ESAIM: Probability and Statistics, Tome 16 (2012), pp. 277-305.

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.

DOI : https://doi.org/10.1051/ps/2011152
Classification : 60J60,  60H10,  41A60
Mots clés : self-interacting diffusion, McKean-Vlasov equation, stationary measures, double-well potential, perturbed dynamical system, Laplace's method, fixed point theorem, uniqueness problem
@article{PS_2012__16__277_0,
     author = {Herrmann, Samuel and Tugaut, Julian},
     title = {Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small-noise limit},
     journal = {ESAIM: Probability and Statistics},
     pages = {277--305},
     publisher = {EDP-Sciences},
     volume = {16},
     year = {2012},
     doi = {10.1051/ps/2011152},
     zbl = {1302.60112},
     mrnumber = {2956576},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps/2011152/}
}
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
DA  - 2012///
SP  - 277
EP  - 305
VL  - 16
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ps/2011152/
UR  - https://zbmath.org/?q=an%3A1302.60112
UR  - https://www.ams.org/mathscinet-getitem?mr=2956576
UR  - https://doi.org/10.1051/ps/2011152
DO  - 10.1051/ps/2011152
LA  - en
ID  - PS_2012__16__277_0
ER  - 
Herrmann, Samuel; Tugaut, Julian. Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small-noise limit. ESAIM: Probability and Statistics, Tome 16 (2012), pp. 277-305. doi : 10.1051/ps/2011152. http://archive.numdam.org/articles/10.1051/ps/2011152/

[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. | MR 1632193 | Zbl 0932.60063

[2] S. Benachour, B. Roynette and P. Vallois, Nonlinear self-stabilizing processes. II. Convergence to invariant probability. Stoc. Proc. Appl. 75 (1998) 203-224. | MR 1632197 | Zbl 0932.60064

[3] P. Cattiaux, A. Guillin and F. Malrieu, Probabilistic approach for granular media equations in the non-uniformly convex case. Probab. Theory Relat. Fields 140 (2008) 19-40. | MR 2357669 | Zbl 1169.35031

[4] T. Funaki, A certain class of diffusion processes associated with nonlinear parabolic equations. Z. Wahrsch. Verw. Gebiete 67 (1984) 331-348. | MR 762085 | Zbl 0546.60081

[5] S. Herrmann and J. Tugaut, Non-uniqueness of stationary measures for self-stabilizing processes. Stoc. Proc. Appl. 120 (2010) 1215-1246. | MR 2639745 | Zbl 1197.60052

[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. | MR 2745727 | Zbl 1225.60095

[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. | MR 2434175 | Zbl 1149.60020

[8] F. Malrieu, Logarithmic Sobolev inequalities for some nonlinear PDE's. Stoc. Proc. Appl. 95 (2001) 109-132. | MR 1847094 | Zbl 1059.60084

[9] H.P. Mckean Jr., A class of Markov processes associated with nonlinear parabolic equations. Proc. Natl. Acad. Sci. USA 56 (1966) 1907-1911. | MR 221595 | Zbl 0149.13501

[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. | MR 1108185 | Zbl 0732.60114

[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. | MR 743525 | Zbl 0544.60058

[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. | MR 914029 | Zbl 0638.60070

[13] A.Yu. Veretennikov, On ergodic measures for McKean-Vlasov stochastic equations. Monte Carlo and Quasi-Monte Carlo Methods 2004 (2006) 471-486. | MR 2208726 | Zbl 1098.60056

Cité par Sources :