  In this paper we establish relationships between four important concepts: (a)
hitting time problems of Brownian motion, (b) 3-dimensional Bessel bridges, (c)
Schr\"odinger's equation with linear potential, and (d) heat equation problems
with moving boundary. We relate (a) and (b) by means of Girsanov's theorem,
which suggests a strategy to extend our ideas to problems in $\mathbb{R}^n$ and
general diffusions. This approach also leads to (c) because we may relate,
through a Feynman-Kac representation, functionals of a Bessel bridge with two
Schr\"odinger-type problems. In particular, we also find a fundamental solution
to a class of parabolic partial differential equations with linear potential.
Finally, the relationship between (c) and (d) suggests a possible link between
Generalized Airy processes and their hitting times.
