We consider diffusion processes on a class of ℝ-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 ℝ-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 σ-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.
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