The Heat Equation on Submanifolds in Lie Groups and Random Motions on Spheres

Abstract

We studied the random variable Vt=volS2(gtB∩B), where B is a disc on the sphere S2 centered at the north pole and (gt)t≥0 is the Brownian motion on the special orthogonal group SO(3) starting at the identity. We applied the results of the theory of compact Lie groups to evaluate the expectation of Vt for 0≤t≤τ, where τ is the first time when Vt vanishes. We obtained an integral formula using the heat equation on some Riemannian submanifold ΓB seen as the support of the function f(g)=volS2(gB∩B) immersed in SO(3). The integral formula depends on the mean curvature of ΓB and the diameter of B.