On delay and asymmetry points of one-dimensional diffusion processes

A homogeneous second order linear differential equation is considered. On an open interval where the equation has sense, it generates a family of operators of the Dirichlet problem on the set of all subintervals; this family is a generalized semi-group. Let the equation be defined on two disjoint intervals with a common boundary point z. It is shown that an extension of the corresponding two semi-groups of operators to the semi-group of operators corresponding to the union of the intervals and the point z is not unique and depends on two abritrary constants. To give an interpretation of 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 the 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 values of the latter constant, 0 and ∞, determine the reflection of the process from the point z when the process approaches the point from the left and right, respectively. Bibliography: 4 titles.