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Transactions of the American Mathematical Society

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Transition operators of diffusions
reduce zero-crossing

Authors: Steven N. Evans and Ruth J. Williams
Journal: Trans. Amer. Math. Soc. 351 (1999), 1377-1389
MSC (1991): Primary 60J60, 60J35; Secondary 35B05, 35K10, 60H30
MathSciNet review: 1615955
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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.

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Additional Information

Steven N. Evans
Affiliation: Department of Statistics #3860, University of California at Berkeley, 367 Evans Hall, Berkeley, California 94720-3860

Ruth J. Williams
Affiliation: Department of Mathematics, University of California at San Diego, 9500 Gilman Drive La Jolla, California 92093-0112

Keywords: Zero--crossing, variation diminishing, time--inhomogeneous diffusion, measure--valued process, martingale problem, partial differential equation
Received by editor(s): January 16, 1998
Additional Notes: Research of the first author supported in part by NSF grant DMS-9703845
Research of the second author supported in part by NSF grant DMS-9703891
Article copyright: © Copyright 1999 American Mathematical Society

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