Brownian Exit Distributions from Normal Balls in $S^3 \times H^3$

Abstract

Let $X_t$ be Brownian motion on a Riemannian manifold $M$ started at $m$ and let $T$ be the first time $X_t$ exits a normal ball about $m$. The first exit time $T$ for $M = S^3 \times H^3$ has the same distribution as the first exit time for $M = \mathbf{R}^6$. For $M = S^3 \times H^3, T$ and $X_T$ are independent random variables.