Introduction to Brownian snakes

  • Date: 12/06/2006

Mathieu Merle (U. British Columbia)


University of British Columbia


Discrete models for an evolving population -such as branching random
walks- arise in a variety of different contexts. In such models,
individuals undergo both a branching phenomenon and a spatial
displacement. Superprocesses are obtained as the weak continuous limits
of such discrete models. Hence, it is not surprising that their
genealogical evolution should be coded by some kind of continuous
branching structure.

In a first part of the talk, we will define this structure by
introducing real trees. We will then see that the random real tree
underlying a superprocess descended from one single individual can be
coded by a Brownian excursion under the Ito measure.

We will then attach to this random continuous branching structure a
random spatial displacement. This will lead to the definition of the
Brownian snake. We will finally see that superprocesses (and also, as a
consequence, the ISE) can be described in terms of the excursion
measure of the Brownian snake.

Other Information: 

Probability Seminar 2006