We give an alternative route to the 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, using the Feynman-Kac formula with stopping times. The derivation also uses an interesting relations 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.

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Groeneboom, P., Lalley, S. P., & Temme, N. (2013). Chernoff's distribution and parabolic partial differential equations.