Resumen de Moments of Brownian Motions on Lie Groups

Michael Voit

• Let (Bt)t = 0 be a Brownian motion on 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 lead to 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 in terms of Hermite polynomials due to J. Wesolowski and may be regarded as a variant of the Lévy characterization without continuity assumptions.