On Differentiability Preserving Properties of Semigroups Associated with One-Dimensional Singular Diffusions

Abstract

In this paper we investigate the differentiability preserving properties of the semigroup $\{T_t: t \geq 0\}$ whose infinitesimal generator is a closed extension of the one-dimensional diffusion operator $L = a(x)d^2/dx^2 + b(x)d/dx$ acting on $C^2(I)$, where $I$ is a closed and bounded interval. Especially we treat the case in which the smoothness of the diffusion coefficient fails at the boundary. We get that $\{T_t: t \geq 0\}$ preserves the one and two-times differentiabilities but does not the three-times one of sufficiently many initial data.