Random Brownian scaling identities and splicing of Bessel processes

An identity in distribution due to Knight for Brownian motion is extended i n two different ways: first by replacing the supremum of a reflecting Brown ian motion by the range of an unreflected Brownian motion and second by rep lacing the reflecting Brownian motion by a recurrent Bessel process. Both e xtensions are explained in terms of random Brownian scaling transformations and Brownian excursions. The first extension is related to two different c onstructions of Ito's law of Brownian excursions, due to Williams and Bismu t, each involving back-to-back splicing of fragments of two independent thr ee-dimensional Bessel processes. Generalizations of both splicing construct ions are described, which involve Bessel processes and Bessel bridges of ar bitrary positive real dimension.