Two recursive decompositions of Brownian bridge related to the asymptotics of random mappings

Aldous and Pitman (1994) studied asymptotic distributions as n!1, of various functionals of a uniform random mapping of the set f1; : : : ; ng, by constructing amapping-walk and showing these random walks converge weakly to a re ecting Brownian bridge. Two di erent ways to encode a mapping as a walk lead to two di erent decompositions of the Brownian bridge, each de ned by cutting the path of the bridge at an increasing sequence of recursively de ned random times in the zero set of the bridge. The random mapping asymptotics entail some remarkable identities involving the random occupation measures of the bridge fragments de ned by these decompositions. We derive various extensions of these identities for Brownian and Bessel bridges, and characterize the distributions of various path fragments Research supported in part by N.S.F. Grants DMS-9970901 and DMS-0071448