On delay and asymmetry points of one-dimensional semi-Markov diffusion processes

B. P. Harlamov

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 16, Tome 384 (2010), pp. 291-309

Résumé

A homogeneous linear differential equation of the second order is considered. For an open interval where the equation is treated a family of operators of the Dirichlet problem on the set of all subintervals is said to be a generalized semi-group due to its special property. Let the equation has meaning on each of two disjoint intervals with a common boundary point $z$. The extension of the corresponding two semi-groups of operators to a semi-group of operators corresponding to the union of these intervals and the point $z$ is shown to be not unique. It is determined by two arbitrary constants. In order to interpret these arbitrary constants we use a one-dimensional locally Markov diffusion process with special properties of passage of the point $z$. One of these arbitrary constants determines a delay of the process at the point $z$, and the second one induces an asymmetry of the process with respect to $z$. The two extremal meanings of the latter constant, 0 and $\infty$, determine reflection of the process from the point $z$ while going to the point from the left and from the right, respectively. Bibl. 4 titles.