Uniform propagation of chaos and creation of chaos for a class of nonlinear diffusions

References

• McKean, H. P. Jr. (1966). A class of Markov processes associated with nonlinear parabolic equations. Proc. Nat. Acad. Sci. U.S.A. 56(6):1907–1911. DOI: 10.1073/pnas.56.6.1907.
   PubMed Web of Science ®Google Scholar
• McKean, H. P. Jr. (1967). Propagation of chaos for a class of non-linear parabolic equations. In Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967). Arlington, VA: Air Force Office Sci. Res., pp. 41–57.
   Google Scholar
• Herrmann, S., Tugaut, J. (2010). Non-uniqueness of stationary measures for self-stabilizing processes. Stochastic Process. Appl. 120(7):1215–1246. DOI: 10.1016/j.spa.2010.03.009.
   Web of Science ®Google Scholar
• Tugaut, J. (2013). Phase transitions of McKean-Vlasov processes in double-wells landscape. Stochastics 86(2):1–28.
   Google Scholar
• Tugaut, J. (2014). Self-stabilizing processes in multi-wells landscape in Rd - Invariant probabilities. J. Theor. Probab. 27(1):57–79.
   Web of Science ®Google Scholar
• Bolley, F., Gentil, I., Guillin, A. (2013). Uniform convergence to equilibrium for granular media. Arch. Rational Mech. Anal. 208(2):429–445. DOI: 10.1007/s00205-012-0599-z.
   Web of Science ®Google Scholar
• Del Moral, P., Miclo, L. (2000). Branching and interacting particle systems approximations of Feynman-Kac formulae with applications to non-linear filtering. In: Azéma, J., Emery, M., Ledoux, M., Yor, M., eds. Séminaire de Probabilités XXXIV. Lecture Notes in Mathematics, Vol. 1729, Berlin: Springer-Verlag, pp. 1–145
   Google Scholar
• Brenier, Y. (1991). Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 44(4):375–417. DOI: 10.1002/cpa.3160440402.
   Web of Science ®Google Scholar
• Herrmann, S., Imkeller, P., Peithmann, D. (2008). Large deviations and a Kramers’ type law for self-stabilizing diffusions. Ann. Appl. Probab. 18(4):1379–1423. DOI: 10.1214/07-AAP489.
   Web of Science ®Google Scholar
• Bolley, F., Gentil, I., Guillin, A. (2012). Convergence to equilibrium in Wasserstein distance for Fokker-Planck equations. J. Funct. Anal. 263(8):2430–2457. DOI: 10.1016/j.jfa.2012.07.007.
   Web of Science ®Google Scholar
• Cattiaux, P., Guillin, A., Malrieu, F. (2007). Probabilistic approach for granular media equations in the non-uniformly convex case. Probab. Theory Relat. Fields 140(1–2):19–40. DOI: 10.1007/s00440-007-0056-3.
   Web of Science ®Google Scholar
• Ambrosio, L., Gigli, N., Savaré, G. (2008). Gradient flows in metric spaces and in the space of probability measures. Lectures in Math. ETH Zürich. Birkhäuser, Bassel.
   Google Scholar
• Tugaut, J. (2013). Convergence to the equilibria for self-stabilizing processes in double-well landscape. Ann. Probab. 41(3A):1427–1460. no.
   Web of Science ®Google Scholar
• Carrillo, J. A., McCann, R. J., Villani, C. (2003). Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates. Rev. Mat. Iberoam. 19(3):971–1018.
   Web of Science ®Google Scholar
• Stroock, D. W., Varadhan, S. R. S. (1979). Multidimensional Diffusion Processes, Volume 233 of Grundlehren Der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Berlin: Springer-Verlag.
   Google Scholar
• Benachour, S., Roynette, B., Talay, D., Vallois, P. (1998). Nonlinear self-stabilizing processes. I. Existence, invariant probability, propagation of chaos. Stochastic Process. Appl. 75(2):173–201. DOI: 10.1016/S0304-4149(98)00018-0.
   Web of Science ®Google Scholar
• Dyda, B., Tugaut, J. (2017). Exponential rate of convergence independent of the dimension in a mean-field system of particles. Probab. Math. Stat. 37(1):145–161.
   Web of Science ®Google Scholar
• Sznitman, A.-S. (1991). Topics in propagation of chaos. In École D’Été de Probabilités de Saint-Flour XIX—1989, Volume 1464 of Lecture Notes in Math. Berlin: Springer, pp. 165–251.
   Google Scholar
• Benedetto, D., Caglioti, E., Carrillo, J. A., Pulvirenti, M. (1998). A non-Maxwellian steady distribution for one-dimensional granular media. J. Stat. Phys. 91(5/6):979–990. DOI: 10.1023/A:1023032000560.
   Web of Science ®Google Scholar
• Benachour, S., Roynette, B., Vallois, P. (1998). Nonlinear self-stabilizing processes. II. Convergence to invariant probability. Stochastic Process Appl. 75(2):203–224. DOI: 10.1016/S0304-4149(98)00019-2.
   Web of Science ®Google Scholar
• Tugaut, J. Self-stabilizing processes in multi-wells landscape in Rd - Convergence. Stochastic Processes Appl. 123(5):1780–1801.
   Google Scholar