Moments of Brownian Motions on Lie Groups

Abstract.

Let (B _t )_{t ≥ 0} be a Brownian motion on $GL(n,{\Bbb R})$ with the corresponding Gaussian convolution semigroup (μ _t )_{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 (B _t )_{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 Lévy characterization without continuity assumptions.