Calculation of Fourier transforms of a Brownian motion on the Heisenberg group using splitting formulas

摘要

If (xi(t))(t)>= 0 is a Brownian motion in the Heisenberg group H-n, and {pi +/-lambda : lambda > 0} are the Schrodinger representations of H-n on L-2(R-n), then the Fourier transforms (E pi +/-lambda (xi(t)))(t)>= 0 form a one-parameter semigroup of contractions on L-2(R-n). The infinitesimal generator N(pi +/-lambda) of this semigroup is a second order element of the universal enveloping algebra of the Lie algebra H-n of H-n which can be identified with an element of a subalgebra of sl (2n + 2, C). To find an explicit formula for E pi +/-lambda (xi(t)) = e(tN), a new method is presented based on the theory of analytic vectors developed by Nelson [E. Nelson, Analytic vectors, Ann. of Math. 70 (3) (1959) 572-615]. In order to calculate the action of e(tN)(pi +/-lambda), we show that this operator can be decomposed as a product of simpler operators on a dense subspace of analytic vectors of L-2 (R-n) and for sufficiently small t >= 0. The main idea is that an element in a sufficiently small neighbourhood of the identity of a Lie group can be decomposed as a product in terms of coordinates of the second kind (called splitting formula), and this carries over to the related operators by the Baker-Campbell-Hausdorff formula. (c) 2007 Elsevier Inc. All rights reserved.