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Potential Theory in Lipschitz Domains

Published online by Cambridge University Press:  20 November 2018

N. Th. Varopoulos*
Affiliation:
Institut Universitaire de France, Université Paris VI, Département de mathématiques, 4, place Jussieu, 75005 Paris, France
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Abstract

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We prove comparison theorems for the probability of life in a Lipschitz domain between Brownian motion and random walks.

Résumé

Résumé

On donne des théorèmes de comparaison pour la probabilité de vie dans un domain Lipschitzien entre le Brownien et de marches aléatoires.

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2001

References

[1] Doob, J. L., Classical Potential Theory and its probabilistic counterpart. Springer-Verlag, New York, 1984.Google Scholar
[2] Fukushima, M., Dirichlet Forms and Markov Processes. North-Holland, Amsterdam-New York, 1980.Google Scholar
[3] Fabes, E., Garofalo, N. and Salsa, S., A backward Harnack inequality and Fatou theorem for nonnegative solutions of parabolic equations. Illinois J. Math. 30(1986), 536565.Google Scholar
[4] Varopoulos, N. Th., Potential theory in conical domains (II). Math. Proc. Camb. Phil. Soc. 129(2000), 301319.Google Scholar
[5] Varopoulos, N. Th., Potential theory in conical domains. Math. Proc. Camb. Phil. Soc. 125(1999), 335384.Google Scholar
[6] Davies, E. B., Gaussian upper bounds for the heat kernels of some second-order operators on Riemannian manifolds. J. Funct. Anal. 80(1988), 1632.Google Scholar
[7] Varopoulos, N. Th., Diffusion on Lie groups. III. Canad. J. Math. (3) 48(1996), 641672.Google Scholar
[8] Ancona, A., Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien. Ann. Inst. Fourier (4) 28(1978), 162213.Google Scholar
[9] Varopoulos, N. Th., Potential theory in conical domains (III). To appear.Google Scholar
[10] Bensoussan, A., Lions, J. L. and Paparicolaou, G., Asymptotic analysis for periodic structures. North-Holland, 1978.Google Scholar
[11] Jikov, V. V., Kozlov, S. M. and Oleinik, O. A., Homogenization of Differential Operators and Integral Functionals. Springer-Verlag, Berlin, 1994.Google Scholar
[12] Kozlov, S. M., The method of averaging and walks in inhomogeneous environment. Uspekhi Math. Nauk. (2) 40(1985), 61120.Google Scholar
[13] Varopoulos, N. Th., Analysis on Lie groups. Rev. Mat. Iberoamericana (3) 12(1996), 791917.Google Scholar
[14] Mustapha, S. and Varopoulos, N. Th., Analysis on p-adic Lie groups. To appear.Google Scholar
[15] Carleson, L., On the existence of boundary values for harmonic functions in several variables. Ark. Mat. 4(1962), 339393.Google Scholar
[16] Kenig, C. E., Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems. CBMS Regional Conferences Series in Math. 83, Amer. Math. Soc., 1994.Google Scholar
[17] Dahlberg, B., Estimates of harmonic measure. Arch. Rational Mech. Anal. 65(1977), 275288.Google Scholar
[18] Salsa, S., Some properties of nonnegative solutions of parabolic differential operators. Ann. Mat. Pura. Appl. 128(1981), 193206.Google Scholar
[19] Doob, J. L., Stochastic Processes. John Wiley, New York, 1953.Google Scholar
[20] Hebisch, W. and Saloff-Coste, L., Gaussian estimates for Markov chains and random walks on groups. Ann. Probab. (2) 21(1993), 673709.Google Scholar
[21] Stampacchia, G., Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier Grenoble 15(1965), 189258.Google Scholar
[22] Stroock, D. W. and Varadhan, S. R. S., Multidimensional Diffusion Processes. Springer-Verlag, Berlin-New York, 1979.Google Scholar
[23] Kaufman, R. and Wu, J. M., Singular parabolic measures. Compositio Math. (2) 40(1980), 243250.Google Scholar
[24] Ito, K. and McKean, H. P. Jr., Diffusion Processes and their sample paths. Springer-Verlag, 1974.Google Scholar
[25] Heurteaux, Y., Solutions positives et mesure harmonique pour des opérateurs paraboliques dans des ouverts“lipschitzien”. Ann. Inst. Fourier (3) 41(1991), 601649.Google Scholar
[26] Lawler, G. F. and Polaski, T. W., Harnack inequalities and difference estimates for random walks with infinite range. J. Theoret. Probab. 6(1993), 781802.Google Scholar
[27] Friedman, A., Partial Differential Equations of Parabolic Type. Prentice Hall, Englewood Cliffs, NJ, 1964.Google Scholar
[28] Feller, W., An introduction to Probability Theory, Vols. I & II. John Wiley & Sons.Google Scholar
[29] Lawler, G. F., Estimates for differences and Harnack inequality for difference operators coming from random walks with symmetric, spatially inhomogeneous, increments. Proc. London Math. Soc. (3) 63(1991), 552568.Google Scholar
[30] Dunford, N. and Schwartz, J. T., Linear operators. Interscience.Google Scholar
[31] Hörmander, L., The Analysis of Linear Partial Differential Operators, Vols. I & II. Google Scholar
[32] Varopoulos, N. Th., Marches aléatoires et diffusions dans les domaines lipschitziens. C. R. Acad. Sci. Paris Sér. I Math. 330(2000), 317320.Google Scholar
[33] Kozlov, S. M., Math. U.S.S.R. Sbornik (4) 35(1979), 481498.Google Scholar
[34] Avellaneda, M. and Lin, F. H., Compactness methods in the theory of homogenization. II. Equations in nondivergence form. Comm. Pure Appl. Math. 42(1989), 139172.Google Scholar
[35] Blumenthal, R. M. and Getoor, R. K., Markov Processes and Potential Theory. Academic Press, New York-London, 1968.Google Scholar
[36] Aronson, D. G., Bounds for the fundamental solution of a parabolic equation. Bull. Amer.Math. Soc. 73(1967), 890896.Google Scholar
[37] Kao, H. G., Tridinger N.S. Duke Math. J. 91(1998), 587607.Google Scholar
[38] Krylov, N. and Safonov, M., Math. USSR Izv. 16(1981), 151164.Google Scholar