A clean, closed form, joint density is derived for Brownian motion, its least concave majorant, and its derivative, all at the same fixed point. Some remarkable conditional and marginal distributions follow from this joint density. For example, it is shown that the height of the least concave majorant of Brownian motion at a fixed time point has the same distribution as the distance from the Brownian motion path to its least concave majorant at the same fixed time point. Also, it is shown that conditional on the height of the least concave majorant of Brownian motion at a fixed time point, the left-hand slope of the least concave majorant of Brownian motion at the same fixed time point is uniformly distributed.
"Marginal Densities of the Least Concave Majorant of Brownian Motion." Ann. Statist. 29 (6) 1732 - 1750, December 2001. https://doi.org/10.1214/aos/1015345960