Brownian Excursion

A Brownian excursion is a Brownian path segment between two visits to zero that stays on one side of zero in between. The normalized positive Brownian excursion on is the limiting law of a Brownian bridge from zero to zero conditioned to remain positive in its interior. Its continuous path is zero at both endpoints and positive throughout almost surely. The convention that the underlying Brownian motion has variance fixes its scale. This normalized excursion codes the Brownian continuum random tree through the minimum of the excursion between two times.