Prépa agreg

Vous trouverez ci-dessous des documents relatifs a la préparation a l'agrégation de mathématiques.

M2 probabilites-statistiques

Vous trouverez ci-dessous des notes pour un cours de M2 sur les chemins rugueux / rough paths que j'ai donné plusieurs années dans le Master 2 de probabilités de Rennes 1. Lecture notes are available, with separate Files for the different parts of the course below. Here are also some slides for the course.

  • The first part of the course introduces the machinery of approximate flows as a tool for constructing flows.
  • The second part of the course introduces the set of p-rough paths and their topology.
  • Rough differential equations on flows are considered in the third part, using the machinery of approximate flows developped in the first part.
  • The fourth part develops the now classical applications Lyons' universal limit theorem (that is the continuity of Ito solution map associated with a rough differential equation) to stochastic analysis, such as Stroock and Varadhan support theorem and the basics of Freidlin-Wentzell large deviation theory.

Cambridge Part III -- Advanced Probability

During my stay in Cambridge as a postdoc (2007-2011), I taught for two years the Part III course Advanced Probability. I wrote my own set of lecture notes for the occasion; you can find them below. The first part of the course deals with the construction of measures and processes, and ranges from Caratheodory and Kolmogorov's extension theorems to the use of weak convergence in separable Banach spaces. The second part of the course gives the basics of Doob's classic theory of discrete and continuous time martingales. The third part studies in some detail Brownian motion and Lévy processes. A number of complements related to the material treated in the course are included. Special attention has been paid to provide short and simple complete proofs of all the results.

Notes on potential theory

You can find below three lecture notes that I have written about different aspects of potential theory, at a basic level; they are written in french. The first paper is about the probabilistic treatment of Dirichlet problem, the second is about electrostatics and equilibrium measures (without probability), and the third one is about convexity, from the basics to Krein-Milman theorem and Choquet theory on integral representations in convex sets.

  • Dirichlet problem from a probabilistic point of view for beginners. pdf.
  • Equilibrium measures in electrostatics pdf.
  • Around the notion of convexity: from Krein-Milman to Choquet theory pdf.