Time distribution from zero up to beginning of the final stop of semi-Markov diffusion process on interval with unattainable boundaries

Abstract

A one-dimensional process with continuous trajectories on non-negative semi-axis is considered. The process has the Markov property with respect to the first exit time from any open interval (semi-Markov process). This process is called to be diffusion if probability for its first exit point from any symmetric neighborhood of its initial point across any boundary tends to 1/2 as length of this neighborhood tends to zero. Time distribution from zero up to beginning of the final interval of constancy is investigated. This distribution depends on semi-Markov transition generating functions of the process. Representation for Laplace transform of this distribution is obtained in an integral form. The integrand of this representation explains sense of quadratic members of Tailor decomposition of a semi-Markov transition generating function by powers of diameter of symmetric neighborhood of the process initial point. Namely trajectory of the process has no any final interval of constancy if and only if coefficient of such a quadratic member is equal to zero.