×

On an estimate of Cranston and McConnell for elliptic diffusions in uniform domains. (English) Zbl 0611.60071

We show that if \(D\subset {\mathbb{R}}^ n\), \(n\geq 3\), is a bounded uniform domain, then the lifetime of the Doob h-paths in D for elliptic diffusions in divergence form is finite. This result holds for any bounded domain D in the plane.

MSC:

60J60 Diffusion processes
60J45 Probabilistic potential theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aizenman, M.; Simon, B., Brownian motion and Harnach inequality for Schrödinger operators, Commun. Pure Appl. Math., 35, 209-273 (1982) · Zbl 0459.60069
[2] Aronson, D. G., Bounds for the fundamental solution of a parabolic equation, Bull. Am. Math. Soc., 73, 890-896 (1967) · Zbl 0153.42002
[3] Bandle, C., On symmetrization in parabolic equations, J. Anal. Math., 30, 98-112 (1976) · Zbl 0331.35036
[4] Bauman, P., Equivalence of the Green’s functions for diffusion operators in ∝^n: a counterexample, Proc. Am. Math. Soc., 91, 64-68 (1984) · Zbl 0574.35026
[5] Bauman, P.: Properties of nonnegative solutions of second-order elliptic equations and their adjoints, Ph. D. Thesis, University of Minnesota, Minneapolis, Minnesota (1982)
[6] Burkholder, D. L., Distribution function inequalities for martingales, Ann. Probab., 1, 19-42 (1973) · Zbl 0301.60035
[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) · Zbl 0512.35038
[8] Chavel, I., Eigenvalues in Riemannian geometry (1984), New York: Academic Press, New York · Zbl 0551.53001
[9] Chung, K. L., The lifetime of condition Brownian motion in the plane, Ann. Inst. Henri Poincaré, 20, 349-351 (1984) · Zbl 0573.60070
[10] Cranston, M., Lifetime of conditioned Brownian motion in Lipschitz domains, Z. Wahrscheinlichkeitstheor. Verw. Geb., 70, 335-340 (1985) · Zbl 0581.60062
[11] Cranston, M., Fabes, E., Zhao, Z.: Potential theory for the Schrödinger equation. Preprint · Zbl 0613.60068
[12] Cranston, M.; McConnell, T., The lifetime of conditioned Brownian motion, Z. Wahrscheinlichkeitstheor. Verw. Geb., 65, 1-11 (1983) · Zbl 0506.60071
[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) · Zbl 0567.35003
[14] Falkner, N.: Conditional Brownian motion in rapidly exhaustible domains. Preprint · Zbl 0627.60068
[15] Federer, H., Geometric measure theory (1969), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York · Zbl 0176.00801
[16] Fukusima, M., Dirichlet forms and Markov process (1980), Amsterdam Oxford New York: North-Holland/Kodansha, Amsterdam Oxford New York · Zbl 0422.31007
[17] Garabedian, P. R., Partial differential equations (1964), New York: Wiley, New York · Zbl 0124.30501
[18] Garnett, J. B., Bounded analytic functions (1980), New York London: Academic Press, New York London
[19] Gehring, F.W.: Characteristic properties of quasidisk. University of Montreal Lecture Notes, 1982 · Zbl 0495.30018
[20] Gehring, F. W.; Osgood, B. G., Uniform domains and the quasi-hyperbolic metric, J. Anal. Math., 36, 50-74 (1979) · Zbl 0449.30012
[21] Gehring, F. W.; Väisälä, J., Hausdorff dimension and quasiconformal mappings, J. London Math. Soc., 6, 504-512 (1973) · Zbl 0258.30020
[22] Hunt, R. A.; Wheeden, R. L., Positive harmonic functions and Lipschitz domains, Trans. Am. Math. Soc., 132, 307-322 (1968) · Zbl 0159.40501
[23] Jerison, D. S.; Kening, C. E., Boundary behavior of harmonic functions in non-tangentially accessible domains, Adv. Math., 46, 80-147 (1982) · Zbl 0514.31003
[24] Jones, P. W., Extension theorems for BMO, Indian. J. Math., 29, 41-66 (1980) · Zbl 0432.42017
[25] Jones, P. W., Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Math., 147, 71-88 (1981) · Zbl 0489.30017
[26] Kanda, M., Regular points and Green functions in Markov processes, J. Math. Soc. Japan, 19, 246-269 (1967) · Zbl 0178.20801
[27] Kunita, H., General boundary conditions for multidimensional diffusion processes, J. Math. Kyoto Univ., 10, 273-335 (1970) · Zbl 0204.41502
[28] Krylov, N. V.; Safanov, M. V., An estimate of the probability that a diffusion process hits a set of positive, Dokl. Acad. Nauk SSSR, 245, 253-255 (1979) · Zbl 0459.60067
[29] Littman, W.; Stampacchia, G.; Weinberger, H. F., Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa, 17, 45-79 (1963) · Zbl 0116.30302
[30] Moser, J., On Harnack’s theorem for elliptic differential equations, Commun. Pure Appl. Math., 14, 577-591 (1961) · Zbl 0111.09302
[31] Stein, E. M., Singular integrals and differentiability properties of functions (1970), Princeton, N. J.: Princeton Univ. Press, Princeton, N. J. · Zbl 0207.13501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.