Transition operators of diffusions reduce zero-crossing

Abstract

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.