Snakes and Spiders: Brownian Motion on R{trees

We consider di usion processes on a class of R{trees. The processes are de ned in a manner similar to that of Le Gall's Brownian snake. Each point in the tree has a real{valued \height" or \generation", and the height of the di usion process evolves as a Brownian motion. When the height process decreases the di usion retreats back along a lineage, whereas when the height process increases the di usion chooses among branching lineages according to relative weights given by a possibly in nite measure on the family of lineages. The class of R{trees we consider can have branch points with countably in nite branching and lineages along which the branch points have points of accumulation. We give a rigorous construction of the di usion process, identify its Dirichlet form, and obtain a necessary and su cient condition for it to be transient. We show that the tail { eld of the di usion is always trivial and draw the usual conclusion that bounded space{time harmonic functions are constant. In the transient case, we identify the Martin compacti cation and obtain the corresponding integral representations of excessive and harmonic functions. Using Ray{Knight methods, we show that the only entrance laws for the di usion are the trivial ones that arise from starting the process inside the state{space. Finally, we use the Dirichlet form stochastic calculus to obtain a semimartingale description of the di usion that involves local time additive functionals associated with each branch point of the tree. Date: April 19, 1999. 1991 Mathematics Subject Classi cation. 60J60, 60J65, 60J50, 60J35.