The probability law of the Brownian motion normalized by its range

Abstract

In the present paper we deduce explicit formulas for the probability laws of the quotients Xt/Rt and mt/Rt, where Xt is the standard Brownian motion and mt, Mt, Rt are its running minimum, maximum and range, respectively. The computation makes use of standard techniques from analytic number theory and the theory of the Hurwitz zeta function.