- Lecturer: Jose Carrillo De La Plata
General Prerequisites:
Course Term: Michaelmas
Course Lecture Information: 16 lectures
Course Weight: 1
Course Level: M
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.
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:
1. Interacting Particle Systems & PDE
Granular Flow Models and McKean-Vlasov Equations.
Nonlinear Diffusion and Aggregation-Diffusion Equations.
2. 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. Probabilistic Interpretation: couplings.
The real line.
3. Mean Field Limit & Couplings
Continuity Equation: Measures sliding down a convex potential.
Dobrushin approach: derivation of the Aggregation Equation.
Boltzmann Equation for Maxwellian molecules: Tanaka Theorem.
4. Gradient Flows: Aggregation-Diffusion Equations
Brenier’s Theorem and Dynamic Interpretation of optimal tranport. Otto’s calculus.
McCann’s Displacement Convexity: Internal, Interaction and Confinement Energies.
Gradient Flow approach: Minimizing movements for the (McKean)-Vlasov equation. Properties of the variational scheme. Connection to mean-field limits.
Granular Flow Models and McKean-Vlasov Equations.
Nonlinear Diffusion and Aggregation-Diffusion Equations.
2. 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. Probabilistic Interpretation: couplings.
The real line.
3. Mean Field Limit & Couplings
Continuity Equation: Measures sliding down a convex potential.
Dobrushin approach: derivation of the Aggregation Equation.
Boltzmann Equation for Maxwellian molecules: Tanaka Theorem.
4. Gradient Flows: Aggregation-Diffusion Equations
Brenier’s Theorem and Dynamic Interpretation of optimal tranport. Otto’s calculus.
McCann’s Displacement Convexity: Internal, Interaction and Confinement Energies.
Gradient Flow approach: Minimizing movements for the (McKean)-Vlasov equation. Properties of the variational scheme. Connection to mean-field limits.