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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
##### Keywords:
Brownian motion; probability law; Hurwitz zeta function
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