McKean-Vlasov processes and Parabolic PDEs

In this course we will study the links between non-linear processes (diffusions) of the McKean-Vlasov type and non-linear parabolic PDEs. The main objective is to introduce all the basic notions in order to be able to study the propagation of chaos in a system of interacting particles towards its mean-field limit identified as a McKean-Vlasov process. We will see how the techniques of PDE analysis and stochastic calculus are combined in this context. The course will begin by dealing with regular and Markovian interactions between particles. We will also present various models from biology, physics, finance, etc., where the interactions are singular and even non-Markovian. These models pose problems in research and we will introduce some recent techniques for tackling them.

Prerequisites: “Stochastic Calculus” course in the 1st semester

Bibliography

• [1] I. Karatzas, and S. E. Shreve, Brownian motion and stochastic calculus, second ed., vol. 113 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1991.
• [2] A-S. Sznitman, Topics in propagation of chaos, École d’Été de Probabilités de Saint-Flour XIX|1989", volume 1464 of Lecture Notes in Math., Springer, Berlin, 1991, 165-251.
• [3] D. Talay and M. Tomasevic, A new McKean-Vlasov stochastic interpretation of the parabolic Keller-Segel model: The one-dimensional case, Bernoulli 26 (2020), 1323-1353.
• [4] N. Fournier and B. Jourdain, Stochastic particle approximation of the Keller-Segel equation and two-dimensional generalization of Bessel processes, Ann. Appl. Probab. 27 (2017), 2807-2861.