Subordinators Related to the Exponential Functionals of Brownian Bridges and Explicit Formulae for the Semigroups of Hyperbolic Brownian Motions

Abstract We prove that, if {bt(s), 0 ^ s ^ t} denotes the standard Brownian bridge of length t and at = JjJ exp(26t(s))ds, then the subordinator {/Cs,s ^ 0} with no drift and with Levy measure Ko{x)e~xx~1dx satisfies (/Cs)_1 = a,i/s for fixed s. Variants and extensions of this result, in particular to general Brownian bridges and their exponential functionals are also discussed. Key Words Subordinator, exponential functional, Brownian bridge, Brownian motion.