no. 17-18
Probability Theory
McKean–Vlasov diffusions: From the asynchronization to the synchronization
[Diffusions de McKean–Vlasov : De lʼasynchronisation à la synchronisation]

Présenté par : the Editorial Board

Tugaut, Julian1
1 Fakultät für Mathematik, Universität Bielefeld, 33615 Bielefeld, Germany
Comptes Rendus. Mathématique, Tome 349 (2011) no. 17-18, pp. 983-986.

On procède à lʼanalyse asymptotique de lʼunique mesure stationnaire symétrique pour un processus auto-stabilisant à petit bruit. Il a été prouvé dans des travaux antécédents que cette mesure converge avec un taux linéaire dans le cas asynchrone et dans le cas strictement synchrone mais la convergence est moins rapide dans le cas intermédiaire. Le but de cette Note est de zoomer autour de cette transition de phase.

We make the asymptotic analysis of the unique symmetric stationary measure of a self-stabilizing process in the small-noise limit. It has been proved in previous works that this measure converges with a linear rate in the asynchronized case and in the strictly synchronized case but it is slower in the intermediate case. The aim of this Note is to zoom around this phase transition.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.08.002
Tugaut, Julian 1

1 Fakultät für Mathematik, Universität Bielefeld, 33615 Bielefeld, Germany 
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Tugaut, Julian. McKean–Vlasov diffusions: From the asynchronization to the synchronization. Comptes Rendus. Mathématique, Tome 349 (2011) no. 17-18, pp. 983-986. doi : 10.1016/j.crma.2011.08.002. http://www.numdam.org/articles/10.1016/j.crma.2011.08.002/

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[5] Herrmann, S.; Tugaut, J. Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small noise limit (2009, accepted in ESAIM P&S) | HAL

[6] Tugaut, J. Convergence to the equilibria for self-stabilizing processes in double-well landscape http://www.math.uni-bielefeld.de/sfb701/preprints/view/507 (Preprint, Bielefeld Universität, 2010, accepted in Annals of Probability)

[7] Tugaut, J. Phase transitions of McKean–Vlasov processes in symmetric and asymmetric multi-wells landscape http://www.math.uni-bielefeld.de/sfb701/preprints/view/520 (Preprint, Bielefeld Universität, 2011)

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☆ Supported by the DFG-funded CRC 701, Spectral Structures and Topological Methods in Mathematics, at the University of Bielefeld.