Moments of Brownian Motions on Lie Groups

摘要

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 $(int_{GL(n,{Bbb R})} x^{otimes k} dmu_t(x))_{t ge 0}$ 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.