A one-dimensional diffusion hits points fast

TL;DR

This paper investigates the rapid hitting times of points by a one-dimensional Markov process, demonstrating that such processes can reach any point in the state space quickly with positive probability.

Contribution

It establishes that one-dimensional regular strong Markov processes hit points in the state space rapidly with positive probability, providing insights into their hitting time behavior.

Findings

Processes hit points quickly with positive probability.

Hitting times can be arbitrarily small with positive probability.

Results apply to regular, strong Markov processes in one dimension.

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 $\tau_z = \inf \{t \geq 0:X_{t} = z\}$, then $P_\xi({ \tau}_z<\varepsilon)>0$ for all $\xi \in E$ and $\varepsilon>0$.