Speaker: Olav Kallenberg

 

Title: Branching diffusion processes and their underlying tree structures

Abstract: Branching diffusions arise in the limit, under suitable scaling, from classical Bienaym\'e branching processes. They may be thought of as diffuse clouds evolving randomly in space and time. Without the branching, the process reduces to a deterministic heat flow, whereas without the spatial motion it becomes a Bessel process, solving a simple SDE. Originally only some analytic tools were available, often involving semi-linear PDEs. --- More recently, an underlying cluster structure was identified, enabling us to analyze the process in terms of some basic probabilistic tools, such as Brownian motion and suitable Poisson and Cox processes. In this talk, I plan to comment on the role of various tree structures arising naturally in the study of basic features. Owing to the extraordinary subtlety of the subject, I shall mostly try to keep my talk on a non-technical and heuristic level.