On a Problem of H. P. McKean: Independence of Brownian Hitting Times and Places

Abstract

We show that for bounded domains $A \subseteq \mathbb{R}^N$ with $0\in A,$ if the exit time $\tau_A$ and exit place $X(\tau_A)$ are independent for a Brownian motion starting at 0, then $A$ is essentially a ball centered at 0. Extensions are given when $X(t)$ is a Brownian motion with constant drift and when $A$ is unbounded.