Moments of Brownian Motions on Lie Groups

Abstract

Let (Bt)t ≥ 0 be a Brownian motion on \(GL(n,{\Bbb R})\) with the corresponding Gaussian convolution semigroup (μt)t ≥ 0 and generator L. We show that algebraic relations between L and the generators of the matrix semigroups \((\int_{GL(n,{\Bbb R})} x^{\otimes k}\ d\mu_t(x))_{t \ge 0}\) lead to \(E((B_t-B_s)_{i,j}^{2k}) =O((t-s)^k)\) for t → s, k ≥ 1, and all coordinates i,j. These relations will form the basis for a martingale characterization of (Bt)t ≥ 0 in terms of generalized heat polynomials. This characterization generalizes a corresponding result for the Brownian motion on \({\Bbb R}\) in terms of Hermite polynomials due to J. Wesolowski and may be regarded as a variant of the Levy characterization without continuity assumptions.