The problem of pasting together two diffusion processes and classical potentials

The paper is a survey of some analytical methods for constructing a diffusion process in ${\mathbb R}^d$ that is a result of pasting together two diffusion processes. It is an exposition in written of a lecture that the authors delivered at one of the plenary sessions of the Conference “Stochastic analysis and random dynamics” which held in Lviv, June 14–20, 2009.