Probability and Stochastic Analysis - University of Bonn

# Advanced Topics in Stochastic Analysis 2014

## Anton Bovier, winter term 2014/15

Time and place: Tue. 10-12, Fri 14-16, Room 0.011 Mathematics Centre

Description: Branching Brownian motion (BBM) is a classical process in probability. It combines purely random motion (Brownian motion) with random genealogies (Galton-Watson process). Thus BBM describes the spatial evolution of particles that reproduce randomly and perform independent Brownian motions. BBM is a basic stochastic process which is of interest in itself. It is also interesting because it is closely related to a class of non-linear partial differential equations, the Fischer-Kolmogorov-Petrovsky-Piscounov (F-KPP) equations. Solutions of these equations can be represented as functionals of BBM, and interesting quantities related to BBM can be expressed a solutions of the F-KPP equation. Questions of particular interest concern the distribution of particles that are at the front of the cloud of BBM particles, and these questions can be related to properties of travelling wave solutions of the F-KPP equations. The groundbreaking work in this area was done by Maury Bramson in his Ph.D. thesis around 1980. His work presents a detailed anlysis of the travelling wave solutions and of the convergence to solutions to travelling waves. As a corollary, he obtained the precise asymptotics of the distribution of the maximum of BBM as time tendes to infinity. Rather recently, this was extended to the construction of the so-called "extremal process", i.e. the joint distribution of all particles "near" the maximum.

The lecture will present BBM in the larger context of extreme value theory of stochastic processes, and notably Gaussian processes. We will start with a brief exposition of  extreme value theory, point processses, and some facts about Gaussian processes. We then turn to BBM (which we see as a Gaussian process indexed by a (random) tree) and present some of Bramson's analysis of the F-KPP equations. Then we show how this analysis allows to show convergence of the extremal process to limit and how this can be characterised as a Poisson cluster process. If time permits we may look at variants of the model, in particular the "variable speed" BBM.

Prerequisits: Good background in probability theory, roughly on the level of the "introduction to stochastic analysis" lecture. Some basic knowledge about partial differential equations will also be helpful.

Literature:  I am writing detailed lecture notes for this course. They are still in a preliminary state and can be downloaded   here.

• Maury Bramson, Convergence of Solutions of the Kolmogorov Equation to Travelling Waves, Memoirs of the AMS,  Vol. 44, 1983
• Henry P. McKean, Application of Brownian motion to the equation of Komogorov-Petrovskii-Piscounov, CPAM 28, 323-331, 1975
• Steve Lalley and Thomas Sellke, A conditional limit theorem for the frontier of a branching Brownian motion. Ann. Probab. 15, 1052–1061, 1987
• Louis-Pierre Arguin, Anton Bovier, Nicola Kistler, The extremal process of branching Brownian motion, Probab. Theor. Rel. Fields 157, 535-574, 2013