| Abstract: | Let (Bt)t ≥ 0 be a Brownian motion on GL(n,Bbb R)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 (òGL(n,Bbb R) x?k dmt(x))t 3 0(int_{GL(n,{Bbb R})} x^{otimes k} dmu_t(x))_{t ge 0} lead to E((Bt-Bs)i,j2k) = O((t-s)k)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{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. |
