Branching Brownian Motion
Branching Brownian motion is, on the one hand, a classical basic model in probability theory. It describes a particle system where particles do two things purely at random:
1) they move as Brownian motion, and
2) they split into several particles that move on independently at unpredictable random times (i.e. at exponential times of mean one) according to some branching mechanism.
We will usually be interested in the case when the branching is supercritical, and to be nice we may want to avoid death. If we normalise the expected offspring to be two, then this the number of particles grows with time exponentially with rate one.
One can naturally ask a lot of questions about such a process. We are predominantly interested in the way this process spreads out. This put the problem in the general context of extreme value theory of random processes.
Some of the particular interest in BBM comes from the fact that it is closely linked to a non-linear partial differential equation, the F-KPP equation, that was introduced by Fischer, and later by Kolmogorov, Petrovsky and Piscounov.
For more details, a review on recent results, and many references, see my lecture notes on this topic. I tought a graduate course on this topic in the fall term 2014/15 and there is a more extensive set of lecture notes, which will appear very soon as a book.
Here is a link to a talk I gave on BBM in Buenos Aires.