  If $u(t,x)$ is a solution of a one--dimensional, parabolic, second--order,
linear partial differential equation (PDE), then it is known that, under
suitable conditions, the number of zero--crossings of the function $u(t,\cdot)$
decreases (that is, does not increase) as time $t$ increases. Such theorems
have applications to the study of blow--up of solutions of semilinear PDE, time
dependent Sturm Liouville theory, curve shrinking problems and control theory.
We generalise the PDE results by showing that the transition operator of a
(possibly time--inhomogenous) one--dimensional diffusion reduces the number of
zero--crossings of a function or even, suitably interpreted, a signed measure.
Our proof is completely probabilistic and depends in a transparent manner on
little more than the sample--path continuity of diffusion processes.
