A one-dimensional diffusion hits points fast

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 $τ_z = \inf \{t \geq 0:X_{t} = z\}$, then $P_ξ({ τ}_z<\varepsilon)>0$ for all $ξ\in E$ and $\varepsilon>0$.

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