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

Source abstract

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 $n \to \infty$ and to $0$ as $m \to \infty.$ Simultaneously, bounds for the volume of such tubes are derived and compared to Weyl's tube formula.