Probability Theory and Related Fields

, Volume 76, Issue 3, pp 311–323 | Cite as

On an estimate of Cranston and McConnell for elliptic diffusions in uniform domains

  • Rodrigro Bañuelos
Article

Summary

We show that ifD⊂∝ n ,n≧3,n≽3, is a bounded uniform domain, then the lifetime of the Doobh-paths inD for elliptic diffusions in divergence form is finite. This result holds for any bounded domainD in the plane.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aizenman, M., Simon, B.: Brownian motion and Harnach inequality for Schrödinger operators. Commun. Pure Appl. Math.35, 209–273 (1982)Google Scholar
  2. 2.
    Aronson, D.G.: Bounds for the fundamental solution of a parabolic equation. Bull. Am. Math. Soc.73, 890–896 (1967)Google Scholar
  3. 3.
    Bandle, C.: On symmetrization in parabolic equations. J. Anal. Math.30, 98–112 (1976)Google Scholar
  4. 4.
    Bauman, P.: Equivalence of the Green's functions for diffusion operators in ∝n: a counterexample. Proc. Am. Math. Soc.91, 64–68 (1984)Google Scholar
  5. 5.
    Bauman, P.: Properties of nonnegative solutions of second-order elliptic equations and their adjoints, Ph. D. Thesis, University of Minnesota, Minneapolis, Minnesota (1982)Google Scholar
  6. 6.
    Burkholder, D.L.: Distribution function inequalities for martingales. Ann. Probab.1, 19–42 (1973)Google Scholar
  7. 7.
    Caffarelli, L., Fabes, E., Mortola, S., Salsa, S.: Boundary behaviour of nonnegative solutions of elliptic operators in divergence form. Indian J. Math.30, 21–640 (1981)Google Scholar
  8. 8.
    Chavel, I.: Eigenvalues in Riemannian geometry. New York: Academic Press 1984Google Scholar
  9. 9.
    Chung, K.L.: The lifetime of condition Brownian motion in the plane. Ann. Inst. Henri Poincaré20, 349–351 (1984)Google Scholar
  10. 10.
    Cranston, M.: Lifetime of conditioned Brownian motion in Lipschitz domains. Z. Wahrscheinlichkeitstheor. Verw. Geb.70, 335–340 (1985)Google Scholar
  11. 11.
    Cranston, M., Fabes, E., Zhao, Z.: Potential theory for the Schrödinger equation. PreprintGoogle Scholar
  12. 12.
    Cranston, M., McConnell, T.: The lifetime of conditioned Brownian motion. Z. Wahrscheinlichkeitstheor. Verw. Geb.65, 1–11 (1983)Google Scholar
  13. 13.
    Fabes, E., Stroock, D.: TheL p-integrability of Green's functions and fundamental solutions for elliplic and parabolic equations. Duke Math. J.51, 997–1016 (1984)Google Scholar
  14. 14.
    Falkner, N.: Conditional Brownian motion in rapidly exhaustible domains. PreprintGoogle Scholar
  15. 15.
    Federer, H.: Geometric measure theory. Berlin Heidelberg New York: Springer 1969Google Scholar
  16. 16.
    Fukusima, M.: Dirichlet forms and Markov process. Amsterdam Oxford New York: North-Holland/Kodansha 1980Google Scholar
  17. 17.
    Garabedian, P.R.: Partial differential equations. New York: Wiley 1964Google Scholar
  18. 18.
    Garnett, J.B.: Bounded analytic functions. New York London: Academic Press 1980Google Scholar
  19. 19.
    Gehring, F.W.: Characteristic properties of quasidisk. University of Montreal Lecture Notes, 1982Google Scholar
  20. 20.
    Gehring, F.W., Osgood, B.G.: Uniform domains and the quasi-hyperbolic metric. J. Anal. Math.36, 50–74 (1979)Google Scholar
  21. 21.
    Gehring, F.W., Väisälä, J.: Hausdorff dimension and quasiconformal mappings. J. London Math. Soc.6, 504–512 (1973)Google Scholar
  22. 22.
    Hunt, R.A., Wheeden, R.L.: Positive harmonic functions and Lipschitz domains. Trans. Am. Math. Soc.132, 307–322 (1968)Google Scholar
  23. 23.
    Jerison, D.S., Kening, C.E.: Boundary behavior of harmonic functions in non-tangentially accessible domains. Adv. Math.46, 80–147 (1982)Google Scholar
  24. 24.
    Jones, P.W.: Extension theorems for BMO. Indian. J. Math.29, 41–66 (1980)Google Scholar
  25. 25.
    Jones, P. W.: Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math.147, 71–88 (1981)Google Scholar
  26. 26.
    Kanda, M.: Regular points and Green functions in Markov processes. J. Math. Soc. Japan19, 246–269 (1967)Google Scholar
  27. 27.
    Kunita, H.: General boundary conditions for multidimensional diffusion processes. J. Math. Kyoto Univ.10, 273–335 (1970)Google Scholar
  28. 28.
    Krylov, N.V., Safanov, M.V.: An estimate of the probability that a diffusion process hits a set of positive. Dokl. Acad. Nauk SSSR245, 253–255 (1979) [English transl., Soviet Math. Dokl.20, 253–255 (1979)]Google Scholar
  29. 29.
    Littman, W., Stampacchia, G., Weinberger, H.F.: Regular points for elliptic equations with discontinuous coefficients. Ann. Scuola Norm. Sup. Pisa17, 45–79 (1963)Google Scholar
  30. 30.
    Moser, J.: On Harnack's theorem for elliptic differential equations. Commun. Pure Appl. Math.14, 577–591 (1961)Google Scholar
  31. 31.
    Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton, N. J.: Princeton Univ. Press 1970Google Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • Rodrigro Bañuelos
    • 1
  1. 1.Department of Mathematics 253-37California Institute of TechnologyPasadenaUSA

Personalised recommendations