We show that (i) Brownian motion conditioned to be positive is Brownian meander; (ii) tied-down Brownian meander is Brownian excursion; and (iii) Brownian bridge conditioned to be positive is Brownian excursion. Using these results we derive the distribution of the suprema of the meander and excursion.
Richard T. Durrett. Donald L. Iglehart. Douglas R. Miller. "Weak Convergence to Brownian Meander and Brownian Excursion." Ann. Probab. 5 (1) 117 - 129, February, 1977. https://doi.org/10.1214/aop/1176995895