Journal of Differential Geometry

Instability of the Liouville property for quasi-isometric Riemannian manifolds and reversible Markov chains

Terry Lyons

Full-text: Open access

Article information

Source
J. Differential Geom., Volume 26, Number 1 (1987), 33-66.

Dates
First available in Project Euclid: 26 June 2008

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1214441175

Digital Object Identifier
doi:10.4310/jdg/1214441175

Mathematical Reviews number (MathSciNet)
MR892030

Zentralblatt MATH identifier
0599.60011

Subjects
Primary: 31C12: Potential theory on Riemannian manifolds [See also 53C20; for Hodge theory, see 58A14]
Secondary: 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx] 30F15: Harmonic functions on Riemann surfaces 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20] 58G05 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Citation

Lyons, Terry. Instability of the Liouville property for quasi-isometric Riemannian manifolds and reversible Markov chains. J. Differential Geom. 26 (1987), no. 1, 33--66. doi:10.4310/jdg/1214441175. https://projecteuclid.org/euclid.jdg/1214441175


Export citation

References

  • [1] A. Ancona, Principe de Harnack a la frontiere et theoreme de Fatou pour operateur elliptique dans un domaine Lipschitzien, Ann. Inst. Fourier (Grenoble) 28 (1978) 169-213.
  • [2] P. Billingsley, Convergence of probability measures, Wiley, New York, 1968.
  • [3] G. Choquet and J. Deny, Sur equation de convolution [math], C. R. Acad. Sci. Paris Ser. I. Math. 250 (1960) 799-801.
  • [4] C. Constantinescu and A. Cornea, Potential theory on harmonic spaces, Springer, New York, 1972.
  • [5] P. Doyle, Random walk on the Speiser graph of a Riemann Surface, Bull. Amer. Math. Soc. (N. S.) 11 (1984) 371-377.
  • [6] M. Kanai, Rough isometries and the parabolicity of Riemannian manifolds, J. Math. Soc. Japan 38 (1986) 227-238.
  • [7] J. C. F. Kingman, The ergodic theory of subadditive stochastic processes, J. Roy. Statist. Soc. Ser. B 30 (1968) 499-510.
  • [8] T. J. Lyons, A simple criterion for transience of a reversible Markov chain, Ann. Probability 11 (1983) 393-402.
  • [9] T. J. Lyons and D. Sullivan, Function theory, random paths and covering spaces, J. Differential Geometry 19 (1984) 299-323.
  • [10] J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961) 577-591.
  • [11] H. L. Royden, Harmonic functions on open Riemann surfaces, Trans. Amer. Math. Soc. 73 (1952) 191-201.
  • [12] H. L. Royden, Open Riemann surfaces, Ann. Acad. Sci. Fenn. Ser. A I Math. 249 (1958).
  • [13] S. Segawa and T. Tada, Martin compactifications and quasi-conformal mappings, preprint.
  • [14] N. Varopoulos, Brownian motion and random walks on manifolds, Ann. Inst. Fourier (Grenoble) 34 (1984) 243-269.
  • [15] H. Watanabe, Spectral dimension of a wire network, J. Phys. A (1985) 2807-2823.
  • [16] S.-T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975) 201-228.