Gaussian Processes on Trees

This book presents recent advances in branching Brownian motion from the perspective of extreme value theory and statistical physics, for graduates.

Branching Brownian motion is a key model at the crossroads of value statistics for Gaussian processes, statistical physics, and non-linear partial differential equations. This book gives a concise introduction for graduate students and researchers leading up to the most recent developments in this active area of research.Branching Brownian motion (BBM) is a classical object in probability theory with deep connections to partial differential equations. This book highlights the connection to classical extreme value theory and to the theory of mean-field spin glasses in statistical mechanics. Starting with a concise review of classical extreme value statistics and a basic introduction to mean-field spin glasses, the author then focuses on branching Brownian motion. Here, the classical results of Bramson on the asymptotics of solutions of the F-KPP equation are reviewed in detail and applied to the recent construction of the extremal process of BBM. The extension of these results to branching Brownian motion with variable speed are then explained. As a self-contained exposition that is accessible to graduate students with some background in probability theory, this book makes a good introduction for anyone interested in accessing this exciting field of mathematics.