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The probability law of the Brownian motion normalized by its range

   
@article{ECP2568,
	author = {Florin Spinu},
	title = {The probability law of the Brownian motion normalized by its range},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {18},
	year = {2013},
	keywords = {Brownian motion, Hurwitz zeta},
	abstract = {
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.
},
	pages = {no. 46, 1-8},
	issn = {1083-589X},
	doi = {10.1214/ECP.v18-2568},    
        url = {http://ecp.ejpecp.org/article/view/2568}}