Introduction to Brownian snakes
Topic
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.
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.
Speakers
Additional Information
Probability Seminar 2006
Mathieu Merle (U. British Columbia)
Mathieu Merle (U. British Columbia)