Reaction-diffusion problems : from front dynamics to genealogies (4)

The fourth lecture was devoted entirely to branching Brownian motion. After recalling Einstein's theory of Brownian motion, the lecture began with a discussion of branching processes such as the Galton-Watson process. The link between branching Brownian motion and the Fisher-KPP equation. The link (McKean 1975) between motion allows us to determine the distribution of the position of the rightmost particle. It also allows us to understand the distribution of distances between the rightmost particles, and to relate the averages of these distances to the shift in front position when the initial condition of the Fisher-KPP equation is modified. This distance distribution is given by a decorated Poisson process of exponential density. The lecture ended by showing how known results on branching Brownian motion can be extended to the case of branching random walks, and how questions about the extremes of branching Brownian motion generalize to the Gaussian free-field case and to the first-visit times of a random walk on a two-dimensional network.