Mini-course "McKean-Vlasov SDEs, Particle Systems and Calculus on Wasserstein Space" Lukasz Szpruch

On April 02 - 06 Lukasz Szpruch, assistant professor of School of Mathematics, University of Edinburgh, The Alan Turing Institute (London) read the mini-course "McKean-Vlasov SDEs, Particle Systems and Calculus on Wasserstein Space"

Abstract 

In his lectures at the College de France Lions demonstrated that the key tool to study Mean-Field Games is a new class of PDEs that describe the evolution of functions of measures. The new concept of a derivative with respect to the measure opened a new research field that impacts a large landscape of mathematics beyond Mean-Field Games: from Stochastic Analysis to Optimal Transport. It turns out that similar concepts to calculus on Wasserstein space allowed for a breakthrough in Kac’s programme in Kinetic Theory. In this course I will present the concept of Lions derivative and will show its consequences on study of McKean-Vlasov SDEs.

Program

1 From Particle Systems to McKean-Vlasov SDEs

1.1 Examples and area of applications

1.2 Well-posedness results via Fixed point

1.3 Well-posedness results via propagation of chaos - martingale problem

1.4 General compactness argument

1.5 Associated PDEs

2 Calculus of Wasserstein Space

2.1 Lions Derivative

2.2 Basic examples

2.3 Itô formula for functions of measures

2.4 PDEs on measure space

2.5 Regularisation by noise and a fixed point

2.6 A new perspective on Martingale problem

3 Applications

3.1 Mean-Filed Games

3.2 Mimicking Theorem by Gyöngy