INTEGRAL REPRESENTATION OF SOLUTIONS OF HALF-SPACE HOMOGENEOUS DIRICHLET AND NEUMANN PROBLEMS FOR AN EQUATION OF FOKKER-PLANCK-KOLMOGOROV TYPE OF NORMAL MARKOV PROCESS

Abstract

Solutions of a homogeneous model equation of the Fokker--Planck--Kolmogorov type of a normal Markov process are consider. They are defined in $\{(t,x_1,\dots,x_n)\in\mathbb{R}^{n+1}|0<t\le T, -\infty<x_j<\infty, j\in\{1,\dots,n-1\}, x_n>0\}$ and for $x_n=0$ satisfy the homogeneous Dirichlet or Neumann conditions and relate to special weighted Lebesgue $L_p$-spaces $L_p^{k(\cdot,a)}$. The representation of such solutions in the form of Poisson integrals is established. The kernels of these integrals are the homogeneous Green's functions of the considered problems, and their densities belong to specially constructed sets $\Phi_p^a$ of functions or generalized measures. The results obtained will be used to describe solutions of the problems from spaces $L_p^{k(\cdot,a)}$. Thus, the well-known Eidelman-Ivasyshen approach will be implemented for the considered problems. According to this approach, if the initial data are taken from the set $\Phi_p^a$, then there is only one solution to the problem from the space $L_p^{k(\cdot,a)}$. It is represented as a Poisson integral. Conversely, for any solution from the space $L_p^{k(\cdot,a)}$ there is only one element $\varphi \in\Phi_p^a$ such that this solution can be represented as a Poisson integral with density $\varphi$. In this case, it becomes clear in what sense the initial condition is satisfied.