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May 2013 Convergence to the equilibria for self-stabilizing processes in double-well landscape
Julian Tugaut
Ann. Probab. 41(3A): 1427-1460 (May 2013). DOI: 10.1214/12-AOP749

Abstract

We investigate the convergence of McKean–Vlasov diffusions in a nonconvex landscape. These processes are linked to nonlinear partial differential equations. According to our previous results, there are at least three stationary measures under simple assumptions. Hence, the convergence problem is not classical like in the convex case. By using the method in Benedetto et al. [J. Statist. Phys. 91 (1998) 1261–1271] about the monotonicity of the free-energy, and combining this with a complete description of the set of the stationary measures, we prove the global convergence of the self-stabilizing processes.

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Julian Tugaut. "Convergence to the equilibria for self-stabilizing processes in double-well landscape." Ann. Probab. 41 (3A) 1427 - 1460, May 2013. https://doi.org/10.1214/12-AOP749

Information

Published: May 2013
First available in Project Euclid: 29 April 2013

zbMATH: 1292.60060
MathSciNet: MR3098681
Digital Object Identifier: 10.1214/12-AOP749

Subjects:
Primary: 35B40 , 60H10
Secondary: 35K55 , 60G10 , 60J60

Keywords: double-well potential , free-energy , Granular media equation , McKean-Vlasov stochastic differential equations , self-interacting diffusion , Stationary measures

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 3A • May 2013
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