Some uniform estimates for the transition density of a Brownian motion on a Carnot group and their application to local times

For a specific Brownian motion on a Carnot group several estimates for its transition density are established, which are uniform w.r.t. external parameter. These estimates can be used for studying functionals of any Brownian motion on a Carnot group. As an application we show the existence of the renormalized local time for the increments of Levy area. This result has a lot in common with the well-known existence of the renormalized self-intersection local time for two-dimensional Brownian motion.