Some Identities in Law

Abstract

AbstractSeveral identities in law are shown to derive from excursion theory. The law of the total local times process of the sum of reflecting Brownian motion and a multiple of its local time at 0 is shown to be that of a BESQ process, thus completing the Ray–Knight theorems. The Lévy–Khintchine formula for the BESQ laws is given. The joint law of the maximum, minimum, and local time of a Brownian bridge is described, as well as the analogous law for Brownian motion. Knight’s identity in law for the ratio of inverse local time divided by the square of the maximum up to that time is obtained. Likewise, the Foldes–Revesz identity in law about the measure of levels spent by Brownian local times up to inverse local time is derived. Several identities in law involving Bessel processes up to last passage times are shown in the same manner. The law of the Cauchy principal value of Brownian local times, considered up to inverse local time, or to an independent exponential time, is obtained. It has some parenthood with Lévy stochastic area formula. The functional equation for the Riemann zeta function is shown to be closely related with a symmetry property of the law of the sum of two independent hitting times of BES(3). KeywordsLocal TimeBrownian BridgeSymmetric Stable ProcessLocal Time ProcessExcursion TheoryThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.