Transition operators of di usions reduce zero{crossing
نویسندگان
چکیده
If u(t; x) is a solution of a one{dimensional, parabolic, second{order, linear partial di erential 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 di usion 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 di usion processes.
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