Shevchuk R. Diffusion processes and nonlocal boundary-value problems for parabolic equations

The thesis deals with the problem of construction of two-parameter operator semigroups describing the general classes of one-dimentional inhomogeneous diffusion processes in bounded and half-bounded domains with the general boundary conditions or conjugation conditions of Feller-Wentzell's type given at the boundary points of these domains. Three problems are considered. The first one is to construct the two-parameter semigroup of linear operators, which describes inhomogeneous Feller process on a half-line coinciding at its interior points with the diffusion process generated by the given second-order differential operator and the behavior of this process at the point zero is determined by the general Feller-Wentzell boundary condition. The second problem is the so-called problem of pasting together two inhomogeneous diffusion processes on a line. This problem is to describe the general class of inhomogeneous Feller processes on a line separated into two domains with the common boundary such that in these domains they coinside with the diffusion processes given there and their behavior at the point of boundary is determined by the general conjugation condition of Feller-Wentzell's type. The third problem investigated in thesis is the generalization of two previous problems: construct the inhomogeneous Feller process on a closed interval which parts at the interior points of the corresponding domains of the given closed interval separated by some fixed point coincide with the inhomogeneous diffusion processes given there and the behavior of the required process at this fixed point and at the endpoints of the closed interval is determined by the conjugation condition and boundary conditions of Feller-Wentzell's type respectively. The study of problems formulated above is performed by the analytical methods. With such an approach the question on existence of the operator families describing the required processes in fact is being reduced to the investigation of the corresponding boundary-value problems and conjugation problems for a linear parabolic equation of second order with variable coefficients. The classical solvability of theese problems is established by the boundary integral equations method with the use of the ordinary parabolic simple-layer potentials.