Brownian motion with darning (BMD) is a diffusion process obtained from Brownian motion by shorting each hole in the space into one point. In this talk, I will present a quick introduction to BMD and its basic properties including the zero period property of its harmonic functions. I will then explain how BMD can be used to study conformal mappings of multiply connected planar domains and the associated Komatu-Leowner differential equations.

Based on an on-going joint work with Masatoshi Fukushima and Steffen Rohde.