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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9947-99-02341-7
MathSciNet review: 1615955
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Abstract | References | Similar Articles | Additional Information

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
MR Author ID: 64505
Email: evans@stat.berkeley.edu

Ruth J. Williams
Affiliation: Department of Mathematics, University of California at San Diego, 9500 Gilman Drive La Jolla, California 92093-0112
Email: williams@russel.ucsd.edu

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