Maximum principles for subharmonic functions via local semi-Dirichlet forms

Canadian Journal of Mathematics

Volumn 60, Issue 4, 2008, Pages 822-874

  Kuwae, Kazuhiro ^a  

^a KUMAMOTO UNIVERSITY   (Japan)

Author keywords

Absolute continuity condition; Dirichlet form; Girsanov transformation; Harmonic functions; Irreducibility; Positivity preserving form; Semi Dirichlet form; Strong maximum principle; Subharmonic functions; Superharmonic functions; Weak maximum principle

Indexed keywords

EID: 49149118183     PISSN: 0008414X     EISSN: None     Source Type: Journal    
DOI: 10.4153/CJM-2008-036-8     Document Type: Article

References (47)

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