General Prerequisites: Good command of Part A Integration, Probability and Differential Equations 1 are essential; the main concepts which will be used are the convergence theorems and the theorems of Fubini and Tonelli, and the notions of measurable functions, integrable functions, null sets and L^p spaces. The Cauchy-Lipschitz theory and Picard´s theorem proofs will be used. Basic knowledge of random variables, laws, expectations, and independence are needed. A good working knowledge of Part A Core Analysis (metric spaces) is expected. Knowledge of B8.1 Probability, Measure and Martingales will certainly help but it is not essential.
Course Overview: This course will serve as an introduction to optimal transportation theory, its application in the analysis of PDE, and its connections to the macroscopic description of interacting particle systems.
Lecturer(s):
Prof. Jose Carrillo de la Plata
Assessment Type:
2-hour written examination paper
Learning Outcomes: Getting familar with the Monge-Kantorovich problem and transport distances. Derivation of macroscopic models via the mean-field limit and their analysis based on contractivity of transport distances. Dynamic Interpretation and Geodesic convexity. A brief introduction to gradient flows and examples.
Course Synopsis:
- Interacting Particle Systems & PDE
- Granular Flow Models and McKean-Vlasov Equations.
- Nonlinear Diffusion and Aggregation-Diffusion Equations.
Optimal Transportation: The metric side
- Functional Analysis tools: weak convergence of measures. Prokhorov’s Theorem. Direct Method of Calculus of Variations.
- Monge Problem. Kantorovich Duality.
- Transport distances between measures: properties. The real line. Probabilistic Interpretation: couplings.
Mean Field Limit & Couplings
- Continuity Equation: measures sliding down a convex valley.
- Dobrushin approach: derivation of the Aggregation Equation.
- Boltzmann Equation for Maxwellian molecules: Tanaka Theorem.
An Introduction to Gradient Flows
- Dynamic Interpretation of optimal tranport.
- McCann’s Displacement Convexity: Internal, Interaction and Confinement Energies.
- Gradient Flows: Differential and metric viewpoints.