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Transition operators of diffusions reduce zero-crossing
Author(s):
Steven
N.
Evans;
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  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  decreases (that is, does not increase) as time  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
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
DOI:
10.1090/S0002-9947-99-02341-7
PII:
S 0002-9947(99)02341-7
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
Copyright of article:
Copyright
1999,
American Mathematical Society
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