Abstract
This paper gives complementary results of Folz (Trans Am Math Soc, 2013). We first generalize the weak Omori–Yau maximum principle to the setting of strongly local Dirichlet forms. As an application, we obtain an analytic approach to compare the stochastic completeness of a weighted graph with that of an associated metric graph. This comparison result played an essential role in the volume growth criterion of Folz (Trans Am Math Soc, 2013), who first proved it via a probabilistic approach. We also give an alternative analytic proof based on a criterion in Fukushima et al. (1994).
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Bär, C., Bessa, G.P.: Stochastic completeness and volume growth. Proc. Am. Math. Soc. 138(7), 2629–2640 (2010)
Biroli, M., Mosco, U.: Formes de Dirichlet et estimations structurelles dans les milieux discontinus. C. R. Acad. Sci. Paris Sér. I Math. 313(9), 593–598 (1991)
Biroli, M., Mosco, U.: A Saint–Venant type principle for Dirichlet forms on discontinuous media. Ann. Mat. Pura Appl. 169(4), 125–181 (1995)
Brezis, H.: Functional analysis, Sobolev spaces and partial differential equations. In: Universitext. Springer, New York (2011)
Chung, K.L.: Markov chains with stationary transition probabilities, 2nd edn. In: Die Grundlehren der Mathematischen Wissenschaften, Band 104. Springer, New York, Inc., New York (1967)
Courant, R., Friedrichs, K., Lewy, H.: Über die partiellen differenzengleichungen der mathematischen physik. Math. Ann. 100(1), 32–74 (1928)
Davies, E.B.: Heat kernels and spectral theory. Cambridge University Press, Cambridge (1989)
Davies, E.B.: Heat kernel bounds, conservation of probability and the Feller property. J. Anal. Math. 58, 99–119 (1992)
Colin de Verdière, Y.: Spectres de graphes. In: Cours Spécialisés [Specialized Courses], vol. 4. Société Mathématique de France, Paris (1998)
Dodziuk, J.: Elliptic operators on infinite graphs. In: Analysis, Geometry and Topology of Elliptic Operators, pp. 353–368. World Sci. Publ., Hackensack, NJ (2006)
Dodziuk, J., Mathai, V.: Kato’s inequality and asymptotic spectral properties for discrete magnetic Laplacians. In: The Ubiquitous Heat Kernel, Contemp. Math., vol. 398, pp. 69–81. Amer. Math. Soc., Providence, RI (2006)
Feller, W.: Boundaries induced by non-negative matrices. Trans. Am. Math. Soc. 83, 19–54 (1956)
Feller, W.: On boundaries and lateral conditions for the Kolmogorov differential equations. Ann. Math. 65(2), 527–570 (1957). MR 0090928 (19,892b)
Feller, W.: An Introduction to Probability Theory and its Applications. Wiley, New York (1966)
Folz, M.: Volume growth and stochastic completeness of graphs. Trans. Am. Math. Soc. arXiv:1201.5908 (2013, to appear). Accessed 21 April 2012
Folz, M.: Gaussian upper bounds for heat kernels of continuous time simple random walks. Electr. J. Probab. 16, 1693–1722 (2011)
Frank, R., Lenz, D., Wingert, D.: Intrinsic metrics for (non-local) symmetric Dirichlet forms and applications to spectral theory. arXiv:1012.5050v1 (2010). Accessed 22 Dec 2010
Freedman, D.: Markov Chains. Holden-Day, San Francisco, California (1971)
Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter, Berlin (1994)
Grigor’yan, A.: On stochastically complete manifolds. DAN SSSR 290, 534–537 (1986, in Russian). Engl. transl.: Sov. Math. Dokl. 34(2), 310–313 (1987)
Grigor’yan, A.: Stochastically complete manifolds and summable harmonic functions. Izv. Akad. Nauk SSSR Ser. Mat. 52(5), 1102–1108, 1120 (1988)
Grigor’yan, A.: Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Am. Math. Soc. 36, 135–249 (1999)
Grigor’yan, A.: Heat kernel and analysis on manifolds. In: AMS-IP Studies in Advanced Mathematics, vol. 47 (2009)
Grigor’yan, A., Hu, J.: Off-diagonal upper estimates for the heat kernel of the Dirichlet forms on metric spaces. Invent. Math. 174(1), 81–126 (2008)
Grigor’yan, A., Huang, X., Masamune, J.: On stochastic completeness of jump processes. Math. Z. 271(3–4), 1211–1239 (2012)
Haeseler, S.: Heat kernel estimates and related inequalities on metric graphs. arXiv:1101.3010 [math-ph] (2011). Accessed 15 Jan 2011
Hsu, E.P.: Heat semigroup on a complete Riemannian manifold. Ann. Probab. 17, 1248–1254 (1989)
Huang, X.: On uniqueness class for a heat equation on graphs. J. Math. Anal. Appl. 393(2), 377–388 (2011)
Huang, X.: Stochastic incompleteness for graphs and weak Omori–Yau maximum principle. J. Math. Anal. Appl. 379(2), 764–782 (2011)
Huang, X.: Escape rate of markov chains on infinite graphs. J. Theory Probab. (2012). doi:10.1007/s10959-012-0456-x
Huang, X., Keller, M., Masamune, J., Wojciechowski, R.K.: A note on self-adjoint extensions of the laplacian on weighted graphs. arXiv:1208.6358v1 [math.FA] (2012). Accessed 31 Aug 2012
Le Jan, Y.: Mesures associées à une forme de Dirichlet. Appl. Bull. Soc. Math. France 106(1), 61–112 (1978)
Karp, L., Li, P.: The heat equation on complete Riemannian manifolds. http://math.uci.edu/~pli/heat.pdf (1983). Accessed 19 April 2013
Keller, M., Lenz, D.: Dirichlet forms and stochastic completeness of graphs and subgraphs. J. Reine Angew. Math. 666, 189–223 (2012)
Khas’minskii, R.Z.: Ergodic properties of recurrent diffusion processes and stabilization of solutions to the Cauchy problem for parabolic equations. Theory Probab. Appl. 5, 179–195 (1960)
Kuchment, P.: Quantum graphs. I. Some basic structures. Waves Random Media 14(1), S107–S128 (2004). Special section on quantum graphs
Kuchment, P.: Quantum graphs. II. Some spectral properties of quantum and combinatorial graphs. J. Phys. A 38(22), 4887–4900 (2005)
Masamune, J., Uemura, T.: Conservation property of symmetric jump processes. Ann. Inst. Henri. Poincare Probab. Stat. 47(3), 650–662 (2011)
Masamune, J., Uemura, T., Wang, J.: On the conservativeness and recurrence of symmetric jump-diffusions. J. Funct. Anal. 263(12), 3984–4008 (2012)
Omori, H.: Isometric immersions of Riemannian manifolds. J. Math. Soc. Jpn. 19, 205–214 (1967)
Pigola, S., Rigoli, M., Setti, A.G.: A remark on the maximum principle and stochastic completeness. Proc. Am. Math. Soc. 131(4), 1283–1288 (2003)
Pigola, S., Rigoli, M., Setti, A.G.: Volume growth, “a priori” estimates, and geometric applications. Geom. Funct. Anal. 13(6), 1302–1328 (2003)
Pigola, S., Rigoli, M., Setti, A.G.: Maximum principles on Riemannian manifolds and applications. Mem. Am. Math. Soc. 174(822), x+99 (2005)
Reuter, G.E.H.: Denumerable Markov processes and the associated contraction semigroups on l. Acta Math. 97, 1–46 (1957)
Richardson, W.: Steepest descent and the least C for Sobolev’s inequality. Bull. Lond. Math. Soc. 18(5), 478–484 (1986)
Shiozawa, Y.: Conservation property of symmetric jump-diffusion processes. Forum Math. (2012). doi:10.1515/forum-2012-0043
Sturm, K.T.: Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and L p-Liouville properties. J. Reine Angew. Math. 456, 173–196 (1994)
Takeda, T.: On a martingale method for symmetric diffusion processes and its applications. Osaka J. Math. 26(3), 605–623 (1989)
Watanabe, K., Kametaka, Y., Nagai, A., Takemura, K., Yamagishi, H.: The best constant of Sobolev inequality on a bounded interval. J. Math. Anal. Appl. 340(1), 699–706 (2008)
Weber, A.: Analysis of the laplacian and the heat flow on a locally finite graph. J. Math. Anal. Appl. 370, 146–158 (2010)
Wojciechowski, R.K.: Heat kernel and essential spectrum of infinite graphs. Indiana Univ. Math. J. 58(3), 1419–1441 (2009)
Wojciechowski, R.K.: Stochastically incomplete manifolds and graphs. Progr. Probab. 64, 163–179 (2011)
Yau, S.T.: Harmonic functions on complete Riemannian manifolds. Commun. Pure Appl. Math. 28, 201–228 (1975)
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Research supported by Project CRC701.
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Huang, X. A Note on the Volume Growth Criterion for Stochastic Completeness of Weighted Graphs. Potential Anal 40, 117–142 (2014). https://doi.org/10.1007/s11118-013-9342-0
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DOI: https://doi.org/10.1007/s11118-013-9342-0
Keywords
- Stochastic completeness
- Weighted graphs
- Metric graphs
- Weak Omori–Yau maximum principle
- Strongly local Dirichlet spaces