## 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.

Probability Seminar 2006