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.