Let τ be the first exit time of a diffusion x(t) from a bounded domain Ω ⊂ ⁿ. This paper demonstrates that certain integral functionals ϕ↦E[ϕ(t)dt | x(0) = x], ϕ:[0, ∞) → , may be characterized as solutions to elliptic boundary value problems. The result is established using probabilistic arguments together with results from the theory of partial differential equations. One particular functional, a stochastic analogue of the Fourier transform, is analyzed carefully. Its basic computational properties, including an inversion formula, are developed.