One-dimensional homogeneous semi-Markov processes of diffusion type are considered. The transition function of such a process satisfies an ordinary second-order differential equation. It is supposed that the process does not break and has no intervals of constancy. Under these conditions, the Dirichlet problem has a solution on any finite interval. This solution is presented in explicit form in terms of solutions having values 1 and 0 at the boundaries of the interval. A criterion for the left-hand boundary of the interval to be unattainable is derived, and for the corresponding values 0 and 1, a criterion for the right-hand boundary of the interval to be unattainable is derived. This criterion applied to a diffusion process follows from known formulas which are derived by considerably complex methods of the stochastic differential equations theory.
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Harlamov, B.P. On Unattainable Boundary of a Diffusion Process Range: Semi-Markov Approach. J Math Sci 244, 912–924 (2020) doi:10.1007/s10958-020-04663-x