Tugaut, Julian Convergence to the equilibria for self-stabilizing processes in double-well landscape. (English) Zbl 1292.60060 Ann. Probab. 41, No. 3A, 1427-1460 (2013). The author considers the weak convergence of the scalar McKean-Vlasov diffusion \[ X_t = X_0+\sqrt{\epsilon} B_t + \int_0^t V'(X_s) ds -\int_0^t F'\ast u_s^\epsilon(X_s) ds, \] where \(B\) is a real Brownian motion and where in the self-stabilizing term the convolution is given by \[ F'\ast u_s^\epsilon(X_s(\omega_0)) = \int_\Omega F'(X_s(\omega_0)-X_s(\omega)) d\operatorname{P}(\omega) \] with \(\operatorname{P}\) being the underlying probability measure. Regarding the coefficients, it is assumed that \(V\) is a double-well potential and the interaction term \(F\) is a convex, polynomial with even degree \(\geq 2\) and convex second derivative. Furthermore, specific assumptions for the (marginal) law of the equation’s initial condition are imposed. The author then presents sufficient conditions in order to prove the main result of the paper: The weak convergence of the one dimensional marginal, i.e., the laws of \(X_t\), to a stationary measure for \(t\to\infty\), where the difficulty lies in the fact that the assumptions imply that there does not exist a unique but at least three stationary measures. An extensive discussion of the assumptions imposed and the methods of proof and its difficulties is incorporated. Reviewer: Martin Riedler (Wien) Cited in 1 ReviewCited in 26 Documents MSC: 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60J60 Diffusion processes 60G10 Stationary stochastic processes 35B40 Asymptotic behavior of solutions to PDEs 35K55 Nonlinear parabolic equations Keywords:McKean-Vlasov diffusion; stationary measure; double-well potential; self-stabilizing process; weak convergence × Cite Format Result Cite Review PDF Full Text: DOI arXiv Euclid References: [1] Ané, C., Blachère, S., Chafaï, D., Fougères, P., Gentil, I., Malrieu, F., Roberto, C. and Scheffer, G. (2000). Sur les Inégalités de Sobolev Logarithmiques. Panoramas et Synthèses [ Panoramas and Syntheses ] 10 . Société Mathématique de France, Paris. · Zbl 0982.46026 [2] Arnold, A., Markowich, P., Toscani, G. and Unterreiter, A. (2001). On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations. Comm. 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