THE NONLOCAL CONJUGATION PROBLEM FOR A LINEAR SECOND ORDER PARABOLIC EQUATION OF KOLMOGOROV'S TYPE WITH DISCONTINUOUS COEFFICIENTS

Abstract

In this paper, we construct the two-parameter Feller semigroup associated with a certain one-dimensional inhomogeneous Markov process. This process may be described as follows. At the interior points of the finite number of intervals $(-\infty,r_1(s)),\,(r_1(s),r_2(s)),\ldots,\,(r_{n}(s),\infty)$ separated by points $r_i(s)\,(i=1,\ldots,n)$, the positions of which depend on the time variable, this process coincides with the ordinary diffusions given there by their generating differential operators, and its behavior on the common boundaries of these intervals is determined by the Feller-Wentzell conjugation conditions of the integral type, each of which corresponds to the inward jump phenomenon from the boundary. The study of the problem is done using analytical methods. With such an approach, the problem of existence of the desired semigroup leads to the corresponding nonlocal conjugation problem for a second order linear parabolic equation of Kolmogorov’s type with discontinuous coefficients. The main part of the paper consists in the investigation of this parabolic conjugation problem, the peculiarity of which is that the domains on the plane, where the equations are given, are curvilinear and have non-smooth boundaries: the functions $r_i(s)\,(i=1,\ldots,n)$, which determine the boundaries of these domains satisfy only the Hölder condition with exponent greater than $\frac{1}{2}$. Its classical solvability in the space of continuous functions is established by the boundary integral equations method with the use of the fundamental solutions of the uniformly parabolic equations and the associated potentials. It is also proved that the solution of this problem has a semigroup property. The availability of the integral representation for the constructed semigroup allows us to prove relatively easily that this semigroup yields the Markov process.