One-point reflection

We examine symmetric extensions of symmetric Markov processes with one boundary point. Relationship among various normalizations of local times, entrance laws and excursion laws is studied. Dirichlet form characterization of elastic one-point reflection of symmetric Markov processes is derived. We give a direct construction of Walsh’s Brownian motion as a one-point reflection together with its Dirichlet form characterization. This yields directly the analytic characterization of harmonic and subharmonic functions for Walsh’s Brownian motion, recently obtained by Fitzsimmons and Kuter (2014) using a different method. We further study as a one-point reflection two-dimensional Brownian motion with darning (BMD).