On unattainable boundaries of a diffusion process range of values: semi-Markov approach

One-dimensional homogeneous semi-Markov processes of diffusion type are considered. A transition function of such a process satisfy an ordinary second order differential equation. It is supposed that the process does not break and has no any interval of constancy. Under these conditions the Dirihlet problem has a solution on any finite interval. This solution is presented in explicit form in terms of solutions having values 1, and 0 on the boundaries of the interval. A criterion for the left boundary of the interval to be unattainable is derived, and for corresponding values 0, and 1 a criterion for the right boundary of the interval to be unattainable is derived. This criterion being applied to a diffusion process follows from known formulas which are derived by considerably complex methods of the stochastic differential equations theory.