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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
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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  • 1. D.J. Aldous, Stopping times and tightness, Ann. Probab. 6 (1978), 335-340. MR 57:14086
  • 2. S. Angenent, The zero set of the solution of a parabolic equation, J. Reine Angew. Math. 390 (1988), 79-96. MR 89j:35015
  • 3. -, Nodal properties of solutions of parabolic equations, Rocky Mountain J. Math. 21 (1991), 585-592. CMP 91:16
  • 4. R.F. Bass, Diffusions and Elliptic Operators, Springer, New York, 1998. CMP 98:05
  • 5. R.M. Blumenthal and R.K. Getoor, Markov Processes and Potential Theory, Academic Press, New York, 1968. MR 41:9348
  • 6. L.D. Brown, I.M. Johnstone, and K.B. MacGibbon, Variation diminishing transformations: a direct approach to total positivity and its statistical applications, J. Amer. Statist. Assoc. 76 (1981), 824-832. MR 83f:62028
  • 7. D. Dawson, Measure-valued Markov processes, Ecole d'Eté de Probabilités de Saint-Flour XXI - 1991, Lecture Notes in Math., vol. 1541, Springer, Berlin, 1993, pp. 1-260. MR 94m:60101
  • 8. R. Durrett, Brownian Motion and Martingales in Analysis, Wadsworth, Belmont, CA, 1984. MR 87a:60054
  • 9. S.N. Ethier and T.G. Kurtz, Markov Processes: Characterization and Convergence, Wiley, New York, 1986. MR 88a:60130
  • 10. A. Friedman, Partial Differential Equations of Parabolic Types, Prentice-Hall, Englewood Cliffs, NJ, 1964. MR 31:6062
  • 11. -, Stochastic Differential Equations and Applications, vol. 1, Academic Press, New York, 1975. MR 58:13350a
  • 12. I. Karatzas and S. Shreve, Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics, vol. 113, Springer, New York, 1988. MR 89c:60096
  • 13. S. Karlin, Total positivity, absorption probabilities and applications, Trans. Amer. Math. Soc. 111 (1964), 33-107. MR 29:5275
  • 14. -, Total Positivity, vol. 1, Stanford University Press, Stanford, CA, 1968. MR 37:5667
  • 15. S. Karlin and J. McGregor, Coincidence probabilities, Pacific J. Math. 9 (1959), 1141-1164. MR 22:5072
  • 16. -, Classical diffusion processes and total positivity, J. Math. Anal. Appl. 1 (1960), 163-183. MR 22:12574
  • 17. -, Total positivity of fundamental solutions of parabolic equations, Proc. Amer. Math. Soc 13 (1962), 136-139. MR 24:A2761
  • 18. L.C.G. Rogers and D. Williams, Diffusions, Markov Processes, and Martingales: Itô Calculus, vol. 2, Wiley, Chichester, England, 1987. MR 89k:60117
  • 19. -, Diffusions, Markov Processes, and Martingales: Foundations, 2nd ed., vol. 1, Wiley, Chichester, England, 1994. MR 96h:60116
  • 20. U. Rösler, The variation diminishing property applied to diffusions, Markov Processes and Control Theory (H. Langer and V. Nollau, eds.), Mathematical Research, vol. 54, Akademie-Verlag, Berlin, 1989, pp. 164-177. MR 91j:60135
  • 21. E.J.P.G. Schmidt, On the total- and strict total-positivity of the kernels associated with parabolic initial value problems, J. Differential Equations 62 (1986), 275-298. MR 88g:35099
  • 22. D.W. Stroock and S.R.S. Varadhan, Multidimensional Diffusion Processes, Grundlehren der mathematischen Wissenschaften, vol. 233, Springer, Berlin, 1979. MR 81f:60108

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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