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

We studied the random variable Vt = vol(S)(2) (g(t)B n B), where B is a disc on the sphere S-2 centered at the north pole and (g(t))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 V-t for 0 = t = t, where t is the first time when V-t vanishes. We obtained an integral formula using the heat equation on some Riemannian submanifold G(B) seen as the support of the function f(g) = vol(S)(2) (gB n B) immersed in SO(3). The integral formula depends on the mean curvature of G(B) and the diameter of B.