Transition operators of diffusions reduce zero-crossing

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; .) 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.