This paper describes a set of stochastic processes that is useful for modeling and analyzing a new genetic mapping method. Some of the processes are Markov chains, and some are best described as functions of Markov chains. The central issue is boundary-crossing probabilities, which correspond to p-values for the existence of genes for particular traits. The methods elaborated by Aldous (1989) provide very accurate approximate p-values, as spot-checked against simulations.

In this paper we obtain a new crossing result for Brownian motion. The boundary studied is a piecewise linear function consisting of two lines. The expression obtained for the boundary crossing probability is of a simple directly computable form.

Under mild conditions an explicit expression is obtained for the first-passage density of sample paths of a continuous Gaussian process to a general boundary. Since this expression will usually be hard to compute, an approximation is given which is computationally simple and which is exact in the limit as the boundary becomes increasingly remote. The integral of this approximating density is itself approximated by a simple formula and this also is exact in the limit. A new integral equation is derived for the first-passage density of a continuous Gaussian Markov process. This is used to obtain further approximations.