Brownian motion on the continuum tree
The author constructs a so-called Brownian motion on the continuum tree, that is a continuous symmetric Markov process with further special properties of hitting times of points. The elegant construction relies on the theory of Dirichlet forms. Typically, a continuous Markov process \(X^ 0\) is first obtained as the diffusion associated to some regular local Dirichlet form; and then the Brownian motion \(X\) is defined by a time-change of \(X^ 0\) based on an adequate additive functional. Moreover, one can establish the existence of a jointly continuous version of the local times of \(X^ 0\), using recent results of \textit{M. B. Marcus} and \textit{J. Rosen} [Ann. Probab. 20, No. 4, 1603-1684 (1992; Zbl 0762.60068)].
scientific article; zbMATH DE number 751358
scientific article