TRANSITION OPERATORS OF DIFFUSIONS REDUCE ZERO-CROSSING
S. N. EVANS AND R. J. WILLIAMS
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.
Appears in Transactions of the American Mathematical Society,
351 (1999), 1377--1389.