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.

Additional Metadata
Keywords Feynman-Kac, stopping times, Scorer's function, Brownian motion, parabolic drift, parabolic partial differential equations
MSC Brownian motion (msc 60J65)
THEME Other (theme 6)
Groeneboom, P, Lalley, S.P, & Temme, N.M. (2013). Chernoff's distribution and parabolic partial differential equations.