Abstract
We show that the LiouvilleD p -property is invariant under rough isometries between a Riemannian manifold of bounded geometry and a graph of bounded degree.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Benjamini, I.: Instability of the Liouville property for quasi-isometric graphs and manifolds of polynomial volume growth. J. Theoret. Probab.4 631–637 (1991)
Benjamini, I., Chavel, I., E. Feldman: Heat kernel lower bounds on Riemannian manifolds using the old ideas of Nash. Proc. London Math. Soc.72 215–240 (1996)
Coulhon, T.: Noyau de la chaleur et discrétisation d’une variété riemannienne. Israel J. Math.80 289–300 (1992)
Coulhon, T., Saloff-Coste, L.: Variétés riemanniennes isométriques à l’infini. Rev. Mat. Iberoamericana11 687–726 (1995)
Dodziuk, J.: Difference equations, isoperimetric inequality and transience of certain random walks. Trans. Amer. Math. Soc.284 787–794 (1984)
Grigor’yan, A.: On Liouville theorems for harmonic functions with finite Dirichlet integral. Math. Sb.132(174) 496–516 (1987) (Russian) [English transl.: Math. USSR Sb.69 485–504 (1988)]
Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. (Oxford Mathematical Monographs) Oxford, New York, Tokyo: Clarendon Press 1993
Holopainen, I.: Nonlinear potential theory and quasiregular mappings on Riemannian manifolds. Ann. Acad. Sci. Fenn. Ser. AI Math. Diss.74 1–45 (1990)
Holopainen, I.: Positive solutions of quasilinear elliptic equations on Riemannian manifolds. Proc. London Math. Soc. (3)65 651–672 (1992)
Holopainen, I.: Rough isometries andp-harmonic functions with finite Dirichlet integral. Rev. Mat. Iberoamericana10 143–176 (1994)
Holopainen, I.: Solutions of elliptic equations on manifolds with roughly Euclidean ends. Math. Z.217 459–477 (1994)
Kanai, M.: Rough isometries and combinatorial approximations of geometries of non-compact Riemannian manifolds. J. Math. Soc. Japan37 391–413 (1985)
Kanai, M.: Rough isometries and the parabolicity of Riemannian manifolds. J. Math. Soc. Japan38 227–238 (1986)
Kilpeläinen, T., Malý, J.: The Wiener test and potential estimates for quasilinear elliptic equations. Acta Math.172 137–161 (1994)
Lyons, T.: Instability of the Liouville property for quasi-isometric Riemannian manifolds and reversible Markov chains. J. Differential Geometry26 33–66 (1987)
Lyons, T., Sullivan, D.: Function theory, random paths and covering spaces. J. Differential Geometry19 299–323 (1984)
Markvorsen, S., McGuinness, S., Thomassen, C.: Transient random walks on graphs and metric spaces with applications to hyperbolic surfaces. Proc. London Math. Soc. (3)64 1–20 (1992)
Murakami, A., Yamasaki, M.: Nonlinear potentials on an infinite network. Mem. Fac. Sci. Shimane Univ.26 15–28 (1992)
Soardi, P.M.: Rough isometries and Dirichlet finite harmonic functions on graphs. Proc. Amer. Math. Soc.119 1239–1248 (1993)
Soardi, P.M.: Potential Theory on Infinite Networks (Lecture Notes in Math., vol. 1590) Springer, Berlin Heidelberg New York, 1994
Soardi, P.M., Yamasaki, M.: Parabolic index and rough isometries. Hiroshima Math. J.23 333–342 (1993)
Soardi, P.M., Yamasaki, M.: Classification of infinite networks and its application. Circuit Systems Signal Process12 133–149 (1993)
Varopoulos, N.: Brownian motion and random walks on manifolds. Ann. Inst. Fourier (Grenoble)34 243–269 (1984)
Yamasaki, M.: Parabolic and hyperbolic infinite networks. Hiroshima Math. J.7 135–146 (1977)
Yamasaki, M.: Discrete potentials on an infinite network. Mem. Fac. Sci. Shimane Univ.13 31–44 (1979)
Author information
Authors and Affiliations
Additional information
The first author was supported partly by the EU HCM contract No. CHRX-CT92-0071.
This article was processed by the author using the Springer-Verlag TEX P Jourlg macro package 1991.