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.
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.
Accepted:
Published online:
Julian Tugaut 1
@article{CRMATH_2011__349_17-18_983_0, author = {Julian Tugaut}, title = {McKean{\textendash}Vlasov diffusions: {From} the asynchronization to the synchronization}, journal = {Comptes Rendus. Math\'ematique}, pages = {983--986}, publisher = {Elsevier}, volume = {349}, number = {17-18}, year = {2011}, doi = {10.1016/j.crma.2011.08.002}, language = {en}, }
Julian Tugaut. McKean–Vlasov diffusions: From the asynchronization to the synchronization. Comptes Rendus. Mathématique, Volume 349 (2011) no. 17-18, pp. 983-986. doi : 10.1016/j.crma.2011.08.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.08.002/
[1] The Eyring–Kramers law for potentials with nonquadratic saddles, Markov Processes Relat. Fields, Volume 16 (2010), pp. 549-598
[2] Large deviations and a Kramersʼ type law for self-stabilizing diffusions, Ann. Appl. Probab., Volume 18 (2008) no. 4, pp. 1379-1423
[3] Non-uniqueness of stationary measures for self-stabilizing processes, Stochastic Process. Appl., Volume 120 (2010) no. 7, pp. 1215-1246
[4] Stationary measures for self-stabilizing processes: asymptotic analysis in the small noise limit, Electron. J. Probab., Volume 15 (2010), pp. 2087-2116
[5] Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small noise limit (2009, accepted in ESAIM P&S) | HAL
[6] 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] 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)
Cited by Sources:
☆ Supported by the DFG-funded CRC 701, Spectral Structures and Topological Methods in Mathematics, at the University of Bielefeld.
Comments - Policy