Probability and real trees. Ecole d’Eté de Probabilités de Saint-Flour XXXV – 2005. Lecture given at the Saint-Flour probability summer school, July 6–23, 2005. (English) Zbl 1139.60006

Evans, Steven N.

Lecture Notes in Mathematics 1920. Berlin: Springer (ISBN 978-3-540-74797-0/pbk). xi, 193 p. (2008).

These Lecture Notes are from the Probability Summer School given at Saint-Flour in July 2005. Steven Evans presents his and his collaborators recent research on the asymptotic properties of random trees embedded into various spaces of ‘tree-like objects’. The classical bijection between rooted planar trees and excursion lattice paths of walk processes can be used to define weak convergence of random trees to limiting objects called Brownian continuum random trees. More general tree-like objects are identified as metric spaces with certain special properties, and the Gromov-Hausdorff distance can be used for assigning a distance between two such objects. The notes review these concepts and discuss various works on pruning operators, Galton-Watson processes, coalescing Brownian motions, diffusions on tree-like objects and more. It is an important and inspiring text reflecting the research front in this area.

Reviewer: Ove Frank (Stockholm)

MSC:

60B10	Convergence of probability measures
60B11	Probability theory on linear topological spaces
60G17	Sample path properties
60J65	Brownian motion
60J80	Branching processes (Galton-Watson, birth-and-death, etc.)
28C10	Set functions and measures on topological groups or semigroups, Haar measures, invariant measures
28C20	Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)

Keywords:

Continuum random tree; geodesic metric space; Gromov-Hausdorff distance; Dirichlet form theory