In this paper, we provide a survey of recent distributional results obtained for Brownian type processes observed up to some random times. We focus on the case of hitting times and inverse local times and consider the situation where the processes are randomly sampled through a uniform random variable. We present various explicit formulas, some of them being quite remarkable.
"Random scaling and sampling of Brownian motion." J. Math. Soc. Japan 67 (4) 1771 - 1784, October, 2015. https://doi.org/10.2969/jmsj/06741771