In these lectures, we give an account of certain recent developments of the theory of spatial branching processes. These developments lead to several fas cinating probabilistic objects, which combine spatial motion with a continuous branching phenomenon and are closely related to certain semilinear partial dif ferential equations. Our first objective is to give a short self-contained presentation of the measure valued branching processes called superprocesses, which have been studied extensively in the last twelve years. We then want to specialize to the important class of superprocesses with quadratic branching mechanism and to explain how a concrete and powerful representation of these processes can be given in terms of the path-valued process called the Brownian snake. To understand this representation as well as to apply it, one needs to derive some remarkable properties of branching trees embedded in linear Brownian motion, which are of independent interest. A nice application of these developments is a simple construction of the random measure called ISE, which was proposed by Aldous as a tree-based model for random distribution of mass and seems to play an important role in asymptotics of certain models of statistical mechanics. We use the Brownian snake approach to investigate connections between super processes and partial differential equations. These connections are remarkable in the sense that almost every important probabilistic question corresponds to a significant analytic problem.

Autorentext

Jean-François Le Gall est un spécialiste de théorie des probabilités, avec des travaux de recherche dans des domaines comme le mouvement brownien, les processus de branchement, les arbres et les graphes aléatoires. Il a été Professeur à l'Université Pierre et Marie Curie (Paris 6) et à l'Ecole normale supérieure de Paris, et depuis 2007 il est Professeur à l'Université Paris-Sud Orsay et à l'Institut universitaire de France. Parmi d'autres distinctions, il a obtenu le Prix Loève 1997 et le Prix Fermat 2005. Jean-François Le Gall is a specialist of probability theory, who has worked in areas such as Brownian motion, branching processes, random trees and random graphs. He occupied positions at University Pierre et Marie Curie (Paris 6) and at Ecole normale supérieure de Paris, and since 2007 he has been a Professor at University Paris-Sud Orsay and at the Institut universitaire de France. Among other distinctions, he was awarded the 1997 Loeve Prize in probability theory and the 2005 Fermat prize for mathematical research. Jean-François Le Gall is a specialist of probability theory, who has worked in areas such as Brownian motion, branching processes, random trees and random graphs. He occupied positions at University Pierre et Marie Curie (Paris 6) and at Ecole normale supérieure de Paris, and since 2007 he has been a Professor at University Paris-Sud Orsay and at the Institut universitaire de France. Among other distinctions, he was awarded the 1997 Loeve Prize in probability theory and the 2005 Fermat prize for mathematical research.

Inhalt
I An Overview.- I.1 Galton-Watson processes and continuous-state branching processes.- I.2 Spatial branching processes and superprocesses.- I.3 Quadratic branching and the Brownian snake.- I.4 Some connections with partial differential equations.- I.5 More general branching mechanisms.- I.6 Connections with statistical mechanics and interacting particle systems.- II Continuous-state Branching Processes and Superprocesses.- II.1 Continuous-state branching processes.- II.2 Superprocesses.- II.3 Some properties of superprocesses.- II.4 Calculations of moments.- III The Genealogy of Brownian Excursions.- III.1 The Itô excursion measure.- III.2 Binary trees.- III.3 The tree associated with an excursion.- III.4 The law of the tree associated with an excursion.- III.5 The normalized excursion and Aldous' continuum random tree.- IV The Brownian Snake and Quadratic Superprocesses.- IV.1 The Brownian snake.- IV.2 Finite-dimensional marginals of the Brownian snake.- IV.3 The connection with superprocesses.- IV.4 The case of continuous spatial motion.- IV.5 Some sample path properties.- IV.6 Integrated super-Brownian excursion.- V Exit Measures and the Nonlinear Dirichlet Problem.- V.1 The construction of the exit measure.- V.2 The Laplace functional of the exit measure.- V.3 The probabilistic solution of the nonlinear Dirichlet problem.- V.4 Moments of the exit measure.- VI Polar Sets and Solutions with Boundary Blow-up.- VI.1 Solutions with boundary blow-up.- VI.2 Polar sets.- VI.3 Wiener's test for the Brownian snake.- VI.4 Uniqueness of the solution with boundary blow-up.- VII The Probabilistic Representation of Positive Solutions.- VII.1 Singular solutions and boundary polar sets.- VII.2 Some properties of the exit measure from the unit disk.- VII.3 The representationtheorem.- VII.4 Further developments.- VIII Lévy Processes and the Genealogy of General Continuous-state Branching Processes.- VIII.1 The discrete setting.- VIII.2 Lévy processes.- VIII.3 The height process.- VIII.4 The exploration process.- VIII.5 Proof of Theorem 2.- Bibliographical Notes.