Snakes and spiders: Brownian motion on R-trees
We consider diffusion processes on a class of $\bR$--trees. The processes are defined 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 diffusion process evolves as a Brownian motion. When the height process decreases the diffusion retreats back along a lineage, whereas when the height process increases the diffusion chooses among branching lineages according to relative weights given by a possibly infinite measure on the family of lineages. The class of $\bR$--trees we consider can have branch points with countably infinite branching and lineages along which the branch points have points of accumulation.
We give a rigorous construction of the diffusion process, identify its Dirichlet form, and obtain a necessary and sufficient condition for it to be transient. We show that the tail $\sigma$--field of the diffusion is always trivial and draw the usual conclusion that bounded space--time harmonic functions are constant. In the transient case, we identify the Martin compactification and obtain the corresponding integral representations of excessive and harmonic functions. Using Ray--Knight methods, we show that the only entrance laws for the diffusion 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 diffusion that involves local time additive functionals associated with each branch point of the tree.