Feller semigroup for diffusion process piesewise-constant generalized diffusion matrix and generalized drift vector

We obtain an integral representation of the classical solution of the conjugation problem for the second order parabolic equation with derivatives with respect to tangent variables at the conjugation conditions. Using this solution, we construct the Feller semigroup to which there corresponds a diffusion process with a piecewise-constant generalized diffusion matrix and a generalized drift vector.