Random trees and SPDE approximation

Stochastics Seminar
Thursday, January 29, 2009 - 15:00
1 hour (actually 50 minutes)
Skiles 269
School of Mathematics, Georgia Tech
This work began in collaboration with C.Heitsch. I will briefly discuss the biological motivation. Then I will introduce Gibbs random trees and study their asymptotics as the tree size grows to infinity. One of the results is a "thermodynamic limit" allowing to introduce a limiting infinite random tree which exhibits a few curious properties. Under appropriate scaling one can obtain a diffusion limit for the process of generation sizes of the infinite tree. It also turns out that one can approach the study the details of the geometry of the tree by tracing progenies of subpopulations. Under the same scaling the limiting continuum random tree can be described as a solution of an SPDE w.r.t. a Brownian sheet.