Chernoff's distribution and differential equations of parabolic and Airy type
We give a direct derivation of the distribution of the maximum and the location of the maximum of one-sided and two-sided Brownian motion with a negative parabolic drift. The argument uses a relation between integrals of special functions, in particular involving integrals with respect to functions which can be called "incomplete Scorer functions". The relation is proved by showing that both integrals, as a function of two parameters, satisfy the same extended heat equation, and the maximum principle is used to show that these solution must therefore have the stated relation. Once this relation is established, a direct derivation of the distribution of the maximum and location of the maximum of Brownian motion minus a parabola is possible, leading to a considerable shortening of the original proofs.