Final Distribution of a Diffusion Process: Semi-Markov Approach

Abstract

A one-dimensional diffusion process is considered. This process is proposed to have the homogeneous Markov property with respect to the first exit time from any open interval (semi-Markov property). A diffusion property of the process is defined as asymptotical equiprobable exit through each of two edges of any symmetric neighborhood interval of an initial point of a sample process trajectory while length of the neighborhood tends to zero. Such a process is proved to have a limit as $t\to\infty$ if probability of the process not leaving this neighborhood is decreased as square of its length. In particular this condition is satisfied for a diffusion Markov process with a break for which the nonexit condition is replaced by the break condition. A semi-Markov method is applied to a derivation of the formula of a conditional final distribution of the diffusion process with a limit at infinity.