Estimates for the Derivative of Diffusion Semigroups

Abstract

Let $\{P_t\}_{t\ge 0}$ be the transition semigroup of a diffusion process. It is known that $P_t$ sends continuous functions into differentiable functions so we can write $DP_tf$. But what happens with this derivative when $t\to 0$ and $P_0f=f$ is only continuous?. We give estimates for the supremum norm of the Frechet derivative of the semigroups associated with the operators ${\cal A}+V$ and ${\cal A}+Z\cdot\nabla$ where ${\cal A}$ is the generator of a diffusion process, $V$ is a potential and $Z$ is a vector field.