On the reduction to one dimensional of first–passage–time problems for diffusion processes

A sufficient condition is given such that the marginal conditional p.d.f. of an n-dimensional diffusion process (n>1) originating at an assigned point in the diffusion region reduces to the transition p.d.f. of a one-dimensional diffusion process. Under such condition, the first-passage-time p.d.f. through a curve in the diffusion region can be seen to identify with the first-passage-time p.d.f. for a one-dimensional diffusion process through a suitable boundary. The first-passage-time p.d.f. through either of two curves for the n-dimensional process is then also reduced to a first-passage-time p.d.f. for a one-dimensional diffusion process through a pair of suitable boundaries.