Extensions of reflecting Markov processes on unbounded domains

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1. The time changed reflecting Brownian motion on a 3-space consisting of two infinite cones is shown to admit 4 and only 4 possible symmetric Markovian extensions.2. The Komatu-Loewner differential equation for a multiply connected planar domain is derived by the Brownian motion with darning (BMD). The notion of stochastic Loewner evolution (SLE) for simply connected domains is extended to multiply connected domains as stochastic Komatu-Loewner evolution (SKLE).3. The minimal diffusion on an interval is shown to be symmetric for the canonical measure and all its symmetric diffusion extensions are identified in relation to Ito-McKean’ s theory. As a result, the one-dimensional diffusions with general boundary conditions can be directly constructed by means of regular Dirichlet forms.