Grigoris Pavliotis
Prof. Grigoris Pavliotis

180 Queen’s Gate
Department of Mathematics
Imperial College London
London SW7 2AZ
UK

Homepage: Grigoris Pavliotis

Linear and Nonlinear Markov Semigroups

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The goal of these lectures is to present some aspects of the theory of Markov processes, with particular emphasis to Ito diffusion processes, both linear and nonlinear. In the first part of the course we will present some elements of the theory of Markov diffusion semigroups: infinitesimal generators, ergodic theory for Markov processes, convergence to equilibrium, functional inequalities, Bakry-Emery theory/Gamma calculus. In the second part of the course we will discuss about nonlinear diffusion processes of Mc Kean type, i.e. stochastic differential equations whose coefficients depend on the law of the process: we will derive the Mc Kean SDE and the corresponding forward Kolmogorov equation, the so-called Mc Kean-Vlasov equation, as the mean field limit of a system of weakly interacting diffusions. We will then develop a basic existence and uniqueness theory for the Mc Kean SDE and for the Mc Kean-Vlasov equation. Finally we will study the long time behaviour of solutions to the Mc Kean-Vlasov equation and we will study the possible non-uniqueness of invariant measures for the dynamics. The connection between properties of solutions to the stationary Mc Kean-Vlasov equation and the theory of phase transitions will be elucidated.

Contents:

  • 1. Chapter:
    Markov semigroups for diffusion processes (stochastic differential equations)

  • 2. Chapter:
    Nonlinear Markov processes and McKean SDEs

  • 3. Chapter:
    Linear and Nonlinear (McKean-Vlasov) Fokker-Planck equations

  • 4. Chapter:
    Characterization of stationary states and convergence to equilibrium

    • Bibliography:

    Pavliotis, Grigorios A. Stochastic processes and applications. Diffusion processes, the Fokker-Planck and Langevin equations. Texts in Applied Mathematics, 60. Springer, New York, 2014.

    Bakry, Dominique; Gentil, Ivan; Ledoux, Michel Analysis and geometry of Markov diffusion operators. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 348. Springer, Cham, 2014

    Bogachev, Vladimir I.; Krylov, Nicolai V.; Röckner, Michael; Shaposhnikov, Stanislav V. Fokker-Planck-Kolmogorov equations.Mathematical Surveys and Monographs, 207. American Mathematical Society, Providence, RI, 2015

    Dawson, Donald A. Critical dynamics and fluctuations for a mean-field model of cooperative behavior. J. Statist. Phys. 31 (1983), no. 1,29–85.

    Chayes, L.; Panferov, V. The Mc Kean-Vlasov equation in finite volume. J. Stat. Phys. 138 (2010), no. 1-3, 351-380.

    Chazelle, Bernard; Jiu, Quansen; Li, Qianxiao; Wang, Chu Well-posedness of the limiting equation of a noisy consensus model in opinion dynamics. J. Differential Equations 263 (2017), no. 1, 365–397.

    Long-time behaviour and phase transitions for the Mc Kean--Vlasov equation on the torus J. A. Carrillo, R. S. Gvalani, G. A. Pavliotis, A. Schlichting arXiv:1806.01719