On Feller semigroup generated by solution of nonlocal parabolic conjugation problem

Abstract

The paper deals with the problem of construction of Feller semigroup for one-dimensional inhomogeneous diffusion processes with membrane placed at a point whose position on the real line is determined by a given function that depends on the time variable. It is assumed that in the inner points of the half-lines separated by a membrane the desired process must coincide with the ordinary diffusion processes given there, and its behavior on the common boundary of these regions is determined by the nonlocal conjugation condition of Feller-Wentzell's type. This problem is often called a problem of pasting together two diffusion processes on a line. In order to study the described problem we use analytical methods. Such an approach allows us to determine the desired operator family using the solution of the corresponding problem of conjugation for a linear parabolic equation of the second order (the Kolmogorov backward equation) with discontinuous coefficients. This solution is constructed by the boundary integral equations method under the assumption that the coefficients of the equation satisfy the Holder condition with a nonzero exponent, the initial function is bounded and continuous on the whole real line, and the parameters characterizing the Feller-Wentzell conjugation condition and the curve defining the common boundary of the domains, where the equation is given, satisfies the Holder condition with exponent greater than $\frac{1}{2}.$