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The probability law of the Brownian motion normalized by its range. (English) Zbl 1306.60116
Summary: In the present paper we deduce explicit formulas for the probability laws of the quotients \(X_t/R_t\) and \(m_t/R_t\), where \(X_t\) is the standard Brownian motion and \(m_t, M_t, R_t\) 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.
MSC:
60J65 Brownian motion
11M35 Hurwitz and Lerch zeta functions
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