Characteristic operator of a diffusion process

Semi-Markov processes of diffusion type in the $d$-dimensional space ($d\geq1$) are considered. The transition generating function of such a process is assumed to satisfy the second order differential equation of elliptical type. Using methods of differential equation theory, especially that of Dirichlet problem, the transition generating function for a small neighborhood of the initial point of the process is investigated. The asymptotic expansions on a small scale parameter are obtained both for the first exit point distribution density, and for the first exit time expectation, when the trajectory of the process leaves a small neighborhood of the initial point. The characteristic operator of E. B. Dynkin determined by a decreasing sequence of neighborhoods is proved to exist.