  It is well-known that the excursions of a one-dimensional diffusion process
can be studied by considering a certain Riccati equation associated with the
process. We show that, in many cases of interest, the Riccati equation can be
solved in terms of an infinite continued fraction. We examine the probabilistic
significance of the expansion. To illustrate our results, we discuss some
examples of diffusions in deterministic and in random environments.
