Hostname: page-component-7b7567c8c-fvjsw Total loading time: 0 Render date: 2026-11-15T02:05:19.936Z Has data issue: false hasContentIssue false
  • English
  • Français

Maximum Principles for Subharmonic Functions Via Local Semi-Dirichlet Forms

Published online by Cambridge University Press:  20 November 2018

Kazuhiro Kuwae
Kazuhiro Kuwae*
Affiliation:
Department of Mathematics and Engineering, Faculty of Engineering, Kumamoto University, Kumamoto, 860-8555, Japan e-mail: kuwae@gpo.kumamoto-u.ac.jp
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

Maximum principles for subharmonic functions in the framework of quasi-regular local semi-Dirichlet forms admitting lower bounds are presented. As applications, we give weak and strong maximum principles for (local) subsolutions of a second order elliptic differential operator on the domain of Euclidean space under conditions on coefficients, which partially generalize the results by Stampacchia.

Keywords

31C2535B5060J4535J53C58positivity preserving formsemi-Dirichlet formDirichlet formsubharmonic functionssuperharmonic functionsharmonic functionsweak maximum principlestrong maximum principleirreducibilityabsolute continuity conditionGirsanov transformationKato class functionDynkin class functionHardy class function

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 60 , Issue 4 , 01 August 2008, pp. 822 - 874
Copyright
Copyright © Canadian Mathematical Society 2008

References

[1] Aizenman, M. and B. Simon, Brownian motion and Harnack inequality for Schrödinger operators. Comm. Pure Appl. Math. 35(1982), no. 2, 209–273.Google Scholar
[2] Albeverio, S., Ma, Z. M., and Röckner, M., Characterization of (non-symmetric) semi-Dirichlet formsassociated with Hunt processes. Random Oper. Stochastic Equations 3(1995), no. 2, 161–179.Google Scholar
[3] T. Barlow, M., Bass, R.F. and Kumagai, T., Note on the equivalence of parabolic Harnack inequalities and heat kernel estimates, http://www.math.kyoto-u.ac.jp/˜ kumagai/kumpre.html. Google Scholar
[4] Blumenthal, R. M. and Getoor, R. K., Markov Processes and Potential Theory. Pure and Applied Mathematics 29, Academic Press, New York, 1968.Google Scholar
[5] Carlen, E. A., Kusuoka, S., and Stroock, D.W., Upper bounds for symmetric Markov transition functions. Ann. Inst. H. Poincaré Probab. Statist. 23(1987), no. 2, suppl., 245–287.Google Scholar
[6] Carrillo-Menendez, S., Processus de Markov associé à une forme de Dirichlet non symétrique. (German) Z.Wahrsch. Verw. Gebiete 33(1975/76), no. 2, 139–154.Google Scholar
[7] Chen, Y.-Z. and Wu, L.-C., Second order elliptic equations and elliptic systems. Translations of Mathematical Monographs 174, American Mathematical Society, Providence, RI, 1998.Google Scholar
[8] Chung, K. L., Doubly-Feller process with multiplicative functional. In: Seminar on Stochastic Processes. Prog. Probab. Statist. 12 Birkhäuser Boston, Boston,MA, 1986, pp. 63–78.Google Scholar
[9] Chung, K. L. and Zhao, Z. X., From Brownian motion to Schrödinger's equation. Grundlehren der MathematischenWissenschaften 312, Springer-Verlag, Berlin, 1995.Google Scholar
[10] Davies, E. B., A review of Hardy inequalities . Oper. Theory Adv. Appl. 110(1998), 55–67.Google Scholar
[11] Edmunds, D. E. andEvans, W. D., Spectral Theory and Differential Operators. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1987.Google Scholar
[12] Fitzsimmons, P. J., Hardy's inequality for Dirichlet forms. J. Math. Anal. Appl. 250(2000), no. 2, 548–560.Google Scholar
[13] Fitzsimmons, P. J., On the quasi regularity of semi-Dirichlet forms. Potential Anal. 15(2001), no. 3, 151–185.Google Scholar
[14] Fitzsimmons, P. J. and Kuwae, K., Non-symmetric perturbations of symmetric Dirichlet forms. J. Funct. Anal. 208(2004), no. 1, 140–162.Google Scholar
[15] Fuglede, B., The quasi topology associated with a countably subadditive set function. Ann. Inst. Fourier, Grenoble 21(1971), 123–169.Google Scholar
[16] Fukushima, M., A note on irreducibility and ergodicity of symmetric Markov processes. In: Stochastic Processes in Quantum Theory and Statistical Physics. Lecture Notes in Phys. 173, Springer, Berlin, 1982, pp. 200–207.Google Scholar
[17] Fukushima, M., On a decomposition of additive functionals in the strict sense for a symmetric Markov process. In: Dirichlet Forms and Stochastic Processes. de Gruyter, Berlin, 1995, pp. 155–169.Google Scholar
[18] Fukushima, M., Oshima, Y., and Takeda, M., Dirichlet Forms and Symmetric Markov Processes, de Gruyter Studies in Mathematics 19,Walter de Gruyter, Berlin, 1994.Google Scholar
[19] Getoor, R. K., Markov processes: Ray processes and right processes. Lecture Notes in Mathematics 440, Springer-Verlag, Berlin, 1975.Google Scholar
[20] Getoor, R. K., Transience and recurrence of Markov processes. In: Seminar on Probability. Lecture Notes in Mathematics 784, Springer, Berlin, 1980, pp. 397–409.Google Scholar
[21] Getoor, R. K. and Sharpe, M. J., Naturality, standardness, and weak duality for Markov processes. Z.Wahrsch. Verw. Gebiete 67(1984), no. 1, 1–62.Google Scholar
[22] Gilbarg, D. and Trudinger, N. S., Elliptic partial differential equations of second order. Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001.Google Scholar
[23] Hervé, R.-M. and Hervé, M., Les fonctions surharmoniques associées à une opérateur elliptique du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 19(1969), no. 1, 305–359.Google Scholar
[24] Kunita, H., Sub-Markov semi-groups in Banach lattices. In: Proc. Internat. Conf. on Functional Analysis and Related Topics. Univ. of Tokyo Press, Tokyo, 1970, pp. 332–343.Google Scholar
[25] Kurata, K., Continuity and Harnack's inequality for solutions of elliptic partial differential equations of second order. Indiana. Univ.Math. J. 43(1994), no. 2, 411–440.Google Scholar
[26] Kuwae, K., Functional calculus for Dirichlet forms. Osaka J. Math. 35(1998), no. 3, 683–715.Google Scholar
[27] Kuwae, K., On a strong maximum principle for Dirichlet forms. In: Stochastic Processes, Physics and Geometry: New Interplays, II. CMS Conf. Proc. 29, American Mathematical Society, Providence, RI, 2000, pp. 423–429.Google Scholar
[28] Kuwae, K., Invariant sets and ergodic decomposition of local semi-Dirichlet forms. Preprint, 2005. http://www.srik.kumamoto-u.ac.jp/˜ kuwaeGoogle Scholar
[29] Kuwae, K., On Calabi's strong maximum principle via local semi-Dirichlet forms. Preprint, 2005. http://www.srik.kumamoto-u.ac.jp/˜ kuwae Google Scholar
[30] Kuwae, K., Machigashira, Y., and T. Shioya, Sobolev spaces, Laplacian, and heat kernel on Alexandrov spaces. Math. Z. 238(2001), no. 2, 269–316.Google Scholar
[31] Kuwae, K., Beginning of analysis on Alexandrov spaces. In: Geometry and Topology. Contemp.Math. 228, American Mathematical Society, Providence, RI, 2000, pp. 275–284.Google Scholar
[32] Ma, Z. M., Overbeck, L., and Röckner, M.,Markov processes associated with semi-Dirichlet forms. Osaka J. Math. 32(1995), 97–119.Google Scholar
[33] Ma, Z. M. and Röckner, M., Introduction to the Theory of (Non-Symmetric) Dirichlet Forms, Springer-Verlag, Berlin, 1992.Google Scholar
[34] Ma, Z. M. and Röckner, M., Markov processes associated with positivity preserving coercive forms. Canad. J. Math. 47(1995), no. 4, 817–840.Google Scholar
[35] Maz, V. G.’ya, Sobolev Spaces. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1985.Google Scholar
[36] Moser, J., On a pointwise estimate for parabolic differential equations. Comm. Pure Appl. Math. 24(1971), 727–740.Google Scholar
[37] Oshima, Y., Lectures on Dirichlet spaces. Lecture Notes Universität Erlangen Nürnberg 1988. http://www.srik.kumamoto-u.ac.jp Google Scholar
[38] Saloff-Coste, L., Aspects of Sobolev-Type Inequalities. LondonMathematical Society Lecture Note Series 289, Cambridge University Press, Cambridge, 2002.Google Scholar
[39] Schmuland, B., On the local property for positivity preserving coercive forms. In: Dirichlet Forms and Stochastic Processes. de Gruyter, Berlin, 1995, pp. 345–354.Google Scholar
[40] Sharpe, M., General Theory of Markov Processes. Pure and Applied Mathematics 133, Academic Press, Boston,MA, 1988.Google Scholar
[41] Shigekawa, I., A non-symmetric diffusion process on theWiener space. Preprint. http://www.math.kyoto-u.ac.jp/˜ ichiroGoogle Scholar
[42] Stampacchia, G., Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15(1965), no. 1, 189–258.Google Scholar
[43] W. Stroock, D., Diffusion semigroups corresponding to uniformly elliptic divergence form operators. In: Séminaire de Probabilités XXI. Lecture Notes in Mathematics 1321, Springer, Berlin, 1988, p. 316–347.Google Scholar
[44] K.-T. Sturm, Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and Lp-Liouville properties. J. Reine Angew.Math. 456(1994), 173–196.Google Scholar
[45] K.-T. Sturm, Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality. J.Math. Pures Appl. 75(1996), no. 3, 273–297.Google Scholar
[46] K.-T. Sturm, Diffusion processes and heat kernels on metric spaces. Ann. Probab. 26(1998), no. 1, 1–55.Google Scholar
[47] Takeda, M., On exit times of symmetric Levy processes from connected open sets. In: Probability Theory and Mathematical Statistics. World Sci. Publishing, River Edge, NJ, 1996, pp. 478–484.Google Scholar