## Transition operators of diffusions reduce zero-crossing

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- by Steven N. Evans and Ruth J. Williams PDF
- Trans. Amer. Math. Soc.
**351**(1999), 1377-1389 Request permission

## 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.## References

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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
- 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 1999 American Mathematical Society
- 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