A short note on the mean exit time of the Brownian motion

摘要

We investigate the functional Ω??(Ω) where Ω runs through the set of compact domains of fixed volume v in any Riemannian manifold (M,g) and where ?(Ω) is the mean exit time from Ω of the Brownian motion. We give an alternative analytical proof of a well-known fact on its critical points proved by McDonald: the critical points of ?(Ω) are harmonic domains.