Характеризація розв’язків крайових задач для модельного рівняння Фоккера−Планка−Колмогорова нормального марковського процесу

Abstract

Background. The known Е-І approach by S.D. Eidelman and S.D. Ivasyshen allowed to characterize wide classes of solutions for parabolic equations with limited coefficients, in particular, to describe their ranges of initial values, to get the integral representation and to find out in what sense the initial conditions are met; if the coefficients of equations, as functions of x, can grow without limit at infinity, then these results do not almost exist.

Objective. The purpose is to consider Dirichlet and Neumann problems in the half-space, in which the boundary conditions are homogeneous, as initial values belong to a special weighted \[\Phi _{p}^{a}\] paces of functions or generalized measures for a homogeneous model Fokker-Planck-Kolmogorov equation of normal markovian process, that contains growing coefficients and to implement Е-І approach for such problems solutions.

Methods. The method modification, which was used for equations with limited coefficients, based on a detailed studying of the Poisson operators properties, generated by Green's functions of corresponding problems.

Results. The correct solvability and the integral representation are established by using Poisson operators of the solutions of such problems. Boundary value problems are investigated in the families of weight L_p-spaces of functions increasing exponentially as \[\left | x \right |\rightarrow \infty.\] These functions have the maximum exponent of growth 2 and the type depend on t. It was also clarified in what sense the solutions satisfy those initial conditions. Considered classes of solutions are characterized as the range of the Poisson operators defined on the spaces \[\Phi _{p}^{a}.\]

Conclusions. The article results show, how the presence in the equation of members with growing coefficients is manifested. They allow predicting similar results for more general boundary value problems and identify ways to obtain them.