Minimal tubes of the sphere as the dimension grows: Volume and explosion of the mean exit time

The mean exit time of a Brownian motion from a tube around a closed, minimal embedded $n$-dimen-sional submanifold of the $(n+m)$-dimen-sional sphere is studied, establishing two-sided bounds via radial transplant. In accordance with known results when $n=0$ or $m=1,$ the mean exit time goes to infinity as $n\to \mathrm{\infty }$ and to $0$ as $m\to \mathrm{\infty }\mathrm{.}$ Simultaneously, bounds for the volume of such tubes are derived and compared to Weyl’s tube formula.