Prof. Grigoris Pavliotis
180 Queen’s Gate
Department of Mathematics Imperial College London London SW7 2AZ UK |

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.