On the Hoelder Property of Trajectories in a Set of Full Wiener Measure on the Heisenberg Group

摘要

This paper studies a certain measure on the set of trajectories in the Heisenberg group H_3(ℝ). This measure is constructed as the image of the Wiener measure under the transfer of the trajectories of the two-dimensional Brownian motion in ℝ~2 to the group H_3(ℝ). The solution of the Cauchy problem for the generating operator of the corresponding semigroup and its representation in the form of a Wiener integral were studied by the author in [1]. In this paper, we show that the trajectories satisfying the Holder condition with exponent 0 < α < 1/2 with respect to the natural norm on H_3(ℝ) form a set of full measure and we present an outline of the proof that the measure is concentrated on continuous trajectories. The Wiener measure on trajectories in Riemannian manifolds and, in particular, in Lie groups, was also studied in [2]—[5] and in many other papers.