The first boundary-value problem for the Fokker–Planck equation with one spatial variable

The Fokker–Planck equation with one spatial variable without the lowest term is considered. The diffusion coefficient is assumed to be measurable, bounded, and separated from zero. The existence of a weak fundamental solution of the Fokker–Planck equation is proved and some properties of this solution are established. Under the additional assumption that the leading coefficient is a Hölder function, we consider the first boundary-value problem in a semi-bounded domain. We assume that the right-hand side of the equation and the initial function are zero and the boundary function is continuous. We prove the solvability of this problem in the class of bounded functions.