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

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