Geometric optics approach to first-passage distributions: Caustic boundaries and exponentially small eigenvalues

We develop singular perturbation methods for computing the first passage ti me distribution for one-dimensional diffusion processes. Detailed results a re given for an Ornstein-Uhlenbeck process, and the method is sketched for more general problems. For some parameter values, we find the presence of c austic boundaries; whereas, for other parameter values, there are exponenti ally small eigenvalues. We use the ray method of geometric optics and asymp totic matching.