A one-dimensional diffusion hits points fast

Abstract

A one-dimensional, continuous, regular, and strong Markov process $X$ with state space $E$ hits any point $z\in E$ fast with positive probability. To wit, if ${\tau }_z=\inf \{t\ge 0:X_t=z\}$, then $P_{\xi }({\tau }_z<\epsilon )>0$ for all $\xi \in E$ and $\epsilon >0$.