Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds

Grigor’yan, Alexander

doi:10.1090/S0273-0979-99-00776-4

Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds
HTML articles powered by AMS MathViewer

by Alexander Grigor’yan PDF

Bull. Amer. Math. Soc. 36 (1999), 135-249 Request permission

Abstract:

We provide an overview of such properties of the Brownian motion on complete non-compact Riemannian manifolds as recurrence and non-explosion. It is shown that both properties have various analytic characterizations, in terms of the heat kernel, Green function, Liouville properties, etc. On the other hand, we consider a number of geometric conditions such as the volume growth, isoperimetric inequalities, curvature bounds, etc., which are related to recurrence and non-explosion.

References

Ahlfors L.V. Sur le type d’une surface de Riemann, C.R. Acad. Sci. Paris 201 (1935), pp. 30-32.
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Lars V. Ahlfors and Leo Sario, Riemann surfaces, Princeton Mathematical Series, No. 26, Princeton University Press, Princeton, N.J., 1960. MR 0114911, DOI 10.1515/9781400874538
Sergio Albeverio and Vassily N. Kolokoltsov, The rate of escape for some Gaussian processes and the scattering theory for their small perturbations, Stochastic Process. Appl. 67 (1997), no. 2, 139–159. MR 1449828, DOI 10.1016/S0304-4149(97)00013-6
A. Ancona, Théorie du potentiel sur les graphes et les variétés, École d’été de Probabilités de Saint-Flour XVIII—1988, Lecture Notes in Math., vol. 1427, Springer, Berlin, 1990, pp. 1–112 (French). MR 1100282, DOI 10.1007/BFb0103041
Robert Azencott, Behavior of diffusion semi-groups at infinity, Bull. Soc. Math. France 102 (1974), 193–240. MR 356254, DOI 10.24033/bsmf.1778
Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951
Martin T. Barlow and Edwin A. Perkins, Symmetric Markov chains in $\mathbf Z^d$: how fast can they move?, Probab. Theory Related Fields 82 (1989), no. 1, 95–108. MR 997432, DOI 10.1007/BF00340013
H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, Proc. London Math. Soc. (3) 25 (1972), 603–614. MR 379672, DOI 10.1112/plms/s3-25.4.603
C. J. K. Batty, Asymptotic stability of Schrödinger semigroups: path integral methods, Math. Ann. 292 (1992), no. 3, 457–492. MR 1152946, DOI 10.1007/BF01444631
Itai Benjamini and Jianguo Cao, Examples of simply-connected Liouville manifolds with positive spectrum, Differential Geom. Appl. 6 (1996), no. 1, 31–50. MR 1384877, DOI 10.1016/0926-2245(96)00005-8
Bernstein S. Sur un théorème de géométrie et ses applications aux équations aux dérivées partielles du type elliptique, Comm. Inst. Sci. Math. Mech. Univ. Kharkov 15 (1915-17), pp. 38-45.
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
N. H. Bingham, Variants on the law of the iterated logarithm, Bull. London Math. Soc. 18 (1986), no. 5, 433–467. MR 847984, DOI 10.1112/blms/18.5.433
R. L. Bishop and B. O’Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1–49. MR 251664, DOI 10.1090/S0002-9947-1969-0251664-4
J. Bliedtner and W. Hansen, Potential theory, Universitext, Springer-Verlag, Berlin, 1986. An analytic and probabilistic approach to balayage. MR 850715, DOI 10.1007/978-3-642-71131-2
E. Bombieri and E. Giusti, Harnack’s inequality for elliptic differential equations on minimal surfaces, Invent. Math. 15 (1972), 24–46. MR 308945, DOI 10.1007/BF01418640
Robert Brooks, A relation between growth and the spectrum of the Laplacian, Math. Z. 178 (1981), no. 4, 501–508. MR 638814, DOI 10.1007/BF01174771
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 (English, with French summary). MR 898496
Gilles Carron, Inégalités isopérimétriques de Faber-Krahn et conséquences, Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992) Sémin. Congr., vol. 1, Soc. Math. France, Paris, 1996, pp. 205–232 (French, with English and French summaries). MR 1427759
Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
Isaac Chavel, Riemannian geometry—a modern introduction, Cambridge Tracts in Mathematics, vol. 108, Cambridge University Press, Cambridge, 1993. MR 1271141
Isaac Chavel and Edgar A. Feldman, Isoperimetric constants, the geometry of ends, and large time heat diffusion in Riemannian manifolds, Proc. London Math. Soc. (3) 62 (1991), no. 2, 427–448. MR 1085648, DOI 10.1112/plms/s3-62.2.427
Isaac Chavel and Edgar A. Feldman, Modified isoperimetric constants, and large time heat diffusion in Riemannian manifolds, Duke Math. J. 64 (1991), no. 3, 473–499. MR 1141283, DOI 10.1215/S0012-7094-91-06425-2
Isaac Chavel and Leon Karp, Large time behavior of the heat kernel: the parabolic $\lambda$-potential alternative, Comment. Math. Helv. 66 (1991), no. 4, 541–556. MR 1129796, DOI 10.1007/BF02566664
Jeff Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, Problems in analysis (Papers dedicated to Salomon Bochner, 1969) Princeton Univ. Press, Princeton, N. J., 1970, pp. 195–199. MR 0402831
Jeff Cheeger and Shing Tung Yau, A lower bound for the heat kernel, Comm. Pure Appl. Math. 34 (1981), no. 4, 465–480. MR 615626, DOI 10.1002/cpa.3160340404
S. Y. Cheng and S. T. Yau, Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28 (1975), no. 3, 333–354. MR 385749, DOI 10.1002/cpa.3160280303
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
Kai Lai Chung and Zhong Xin Zhao, From Brownian motion to Schrödinger’s equation, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 312, Springer-Verlag, Berlin, 1995. MR 1329992, DOI 10.1007/978-3-642-57856-4
L. O. Chung, Existence of harmonic $L^{1}$ functions in complete Riemannian manifolds, Proc. Amer. Math. Soc. 88 (1983), no. 3, 531–532. MR 699427, DOI 10.1090/S0002-9939-1983-0699427-2
Corneliu Constantinescu and Aurel Cornea, Potential theory on harmonic spaces, Die Grundlehren der mathematischen Wissenschaften, Band 158, Springer-Verlag, New York-Heidelberg, 1972. With a preface by H. Bauer. MR 0419799, DOI 10.1007/978-3-642-65432-9
Thierry Coulhon, Noyau de la chaleur et discrétisation d’une variété riemannienne, Israel J. Math. 80 (1992), no. 3, 289–300 (French, with English summary). MR 1202573, DOI 10.1007/BF02808072
Thierry Coulhon, Ultracontractivity and Nash type inequalities, J. Funct. Anal. 141 (1996), no. 2, 510–539. MR 1418518, DOI 10.1006/jfan.1996.0140
Thierry Coulhon and Alexander Grigor′yan, Heat kernels, volume growth and anti-isoperimetric inequalities, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 11, 1027–1032 (English, with English and French summaries). MR 1396634
Thierry Coulhon and Alexander Grigor’yan, On-diagonal lower bounds for heat kernels and Markov chains, Duke Math. J. 89 (1997), no. 1, 133–199. MR 1458975, DOI 10.1215/S0012-7094-97-08908-0
Coulhon T., Grigor’yan A. Random walks on graphs with regular volume growth, Geom. and Funct. Analysis 8 (1998), pp. 656-701.
Thierry Coulhon and Laurent Saloff-Coste, Isopérimétrie pour les groupes et les variétés, Rev. Mat. Iberoamericana 9 (1993), no. 2, 293–314 (French). MR 1232845, DOI 10.4171/RMI/138
Coulhon T., Saloff-Coste L. Harnack inequality and hyperbolicity for $p$-Laplacian with applications to Picard type theorems, preprint. . (1997).
M. Cranston, A probabilistic approach to Martin boundaries for manifolds with ends, Probab. Theory Related Fields 96 (1993), no. 3, 319–334. MR 1231927, DOI 10.1007/BF01292675
E. B. Davies, $L^1$ properties of second order elliptic operators, Bull. London Math. Soc. 17 (1985), no. 5, 417–436. MR 806008, DOI 10.1112/blms/17.5.417
E. B. Davies, Gaussian upper bounds for the heat kernels of some second-order operators on Riemannian manifolds, J. Funct. Anal. 80 (1988), no. 1, 16–32. MR 960220, DOI 10.1016/0022-1236(88)90062-6
E. B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1989. MR 990239, DOI 10.1017/CBO9780511566158
E. B. Davies, Heat kernel bounds, conservation of probability and the Feller property, J. Anal. Math. 58 (1992), 99–119. Festschrift on the occasion of the 70th birthday of Shmuel Agmon. MR 1226938, DOI 10.1007/BF02790359
E. B. Davies, Non-Gaussian aspects of heat kernel behaviour, J. London Math. Soc. (2) 55 (1997), no. 1, 105–125. MR 1423289, DOI 10.1112/S0024610796004607
E. B. Davies and M. M. H. Pang, Sharp heat kernel bounds for some Laplace operators, Quart. J. Math. Oxford Ser. (2) 40 (1989), no. 159, 281–290. MR 1010819, DOI 10.1093/qmath/40.3.281
J. Deny, Méthodes hilbertiennes en théorie du potentiel, Potential Theory (C.I.M.E., I Ciclo, Stresa, 1969) Edizioni Cremonese, Rome, 1970, pp. 121–201 (French). MR 0284609
Jozef Dodziuk, Maximum principle for parabolic inequalities and the heat flow on open manifolds, Indiana Univ. Math. J. 32 (1983), no. 5, 703–716. MR 711862, DOI 10.1512/iumj.1983.32.32046
Jozef Dodziuk, Difference equations, isoperimetric inequality and transience of certain random walks, Trans. Amer. Math. Soc. 284 (1984), no. 2, 787–794. MR 743744, DOI 10.1090/S0002-9947-1984-0743744-X
Harold Donnelly, Bounded harmonic functions and positive Ricci curvature, Math. Z. 191 (1986), no. 4, 559–565. MR 832813, DOI 10.1007/BF01162345
J. L. Doob, Classical potential theory and its probabilistic counterpart, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 262, Springer-Verlag, New York, 1984. MR 731258, DOI 10.1007/978-1-4612-5208-5
Peter G. Doyle, On deciding whether a surface is parabolic or hyperbolic, Geometry of random motion (Ithaca, N.Y., 1987) Contemp. Math., vol. 73, Amer. Math. Soc., Providence, RI, 1988, pp. 41–48. MR 954627, DOI 10.1090/conm/073/954627
Peter G. Doyle and J. Laurie Snell, Random walks and electric networks, Carus Mathematical Monographs, vol. 22, Mathematical Association of America, Washington, DC, 1984. MR 920811, DOI 10.5948/UPO9781614440222
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
E. B. Dynkin, Markov processes. Vols. I, II, Die Grundlehren der mathematischen Wissenschaften, Band 121, vol. 122, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1965. Translated with the authorization and assistance of the author by J. Fabius, V. Greenberg, A. Maitra, G. Majone. MR 0193671, DOI 10.1007/978-3-662-00031-1
Einstein A. On the motion of small particles suspended in liquids at rest required by the molecular-kinetic theory of heat, Annalen der Physik 1905. See also Einstein’s miraculous year, , ed. John Stachel, Princeton University Press 1998. pp. 71-85.
K. D. Elworthy, Stochastic differential equations on manifolds, London Mathematical Society Lecture Note Series, vol. 70, Cambridge University Press, Cambridge-New York, 1982. MR 675100, DOI 10.1017/CBO9781107325609
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
William Feller, An introduction to probability theory and its applications. Vol. I, 3rd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0228020
José L. Fernández, On the existence of Green’s function in Riemannian manifolds, Proc. Amer. Math. Soc. 96 (1986), no. 2, 284–286. MR 818459, DOI 10.1090/S0002-9939-1986-0818459-7
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
Masatoshi Fukushima, Dirichlet forms and Markov processes, North-Holland Mathematical Library, vol. 23, North-Holland Publishing Co., Amsterdam-New York; Kodansha, Ltd., Tokyo, 1980. MR 569058
Masatoshi Fukushima, Y\B{o}ichi Ōshima, and Masayoshi Takeda, Dirichlet forms and symmetric Markov processes, De Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin, 1994. MR 1303354, DOI 10.1515/9783110889741
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
Matthew P. Gaffney, The conservation property of the heat equation on Riemannian manifolds, Comm. Pure Appl. Math. 12 (1959), 1–11. MR 102097, DOI 10.1002/cpa.3160120102
Peter Gerl, Random walks on graphs, Probability measures on groups, VIII (Oberwolfach, 1985) Lecture Notes in Math., vol. 1210, Springer, Berlin, 1986, pp. 285–303. MR 879011, DOI 10.1007/BFb0077189
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977. MR 0473443, DOI 10.1007/978-3-642-96379-7
R. E. Greene and H. Wu, Function theory on manifolds which possess a pole, Lecture Notes in Mathematics, vol. 699, Springer, Berlin, 1979. MR 521983, DOI 10.1007/BFb0063413
A. A. Grigor′yan, Existence of the Green function on a manifold, Uspekhi Mat. Nauk 38 (1983), no. 1(229), 161–162 (Russian). MR 693728
A. A. Grigor′yan, The existence of positive fundamental solutions of the Laplace equation on Riemannian manifolds, Mat. Sb. (N.S.) 128(170) (1985), no. 3, 354–363, 446 (Russian). MR 815269
A. A. Grigor′yan, Stochastically complete manifolds, Dokl. Akad. Nauk SSSR 290 (1986), no. 3, 534–537 (Russian). MR 860324
A. A. Grigor′yan, On Liouville theorems for harmonic functions with finite Dirichlet integral, Mat. Sb. (N.S.) 132(174) (1987), no. 4, 496–516, 592 (Russian); English transl., Math. USSR-Sb. 60 (1988), no. 2, 485–504. MR 886642
A. A. Grigor′yan, On the set of positive solutions of the Laplace-Beltrami equation on Riemannian manifolds of a special form, Izv. Vyssh. Uchebn. Zaved. Mat. 2 (1987), 30–37, 84 (Russian). MR 889193
A. A. Grigor′yan, Stochastically complete manifolds and summable harmonic functions, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 5, 1102–1108, 1120 (Russian); English transl., Math. USSR-Izv. 33 (1989), no. 2, 425–432. MR 972099
A. A. Grigor′yan, Bounded solutions of the Schrödinger equation on noncompact Riemannian manifolds, Trudy Sem. Petrovsk. 14 (1989), 66–77, 265–266 (Russian, with English summary); English transl., J. Soviet Math. 51 (1990), no. 3, 2340–2349. MR 1001354, DOI 10.1007/BF01094993
A. A. Grigor′yan, Dimensions of spaces of harmonic functions, Mat. Zametki 48 (1990), no. 5, 55–61, 159 (Russian); English transl., Math. Notes 48 (1990), no. 5-6, 1114–1118 (1991). MR 1092153, DOI 10.1007/BF01236296
A. A. Grigor′yan, The heat equation on noncompact Riemannian manifolds, Mat. Sb. 182 (1991), no. 1, 55–87 (Russian); English transl., Math. USSR-Sb. 72 (1992), no. 1, 47–77. MR 1098839
Alexander Grigor′yan, Heat kernel of a noncompact Riemannian manifold, Stochastic analysis (Ithaca, NY, 1993) Proc. Sympos. Pure Math., vol. 57, Amer. Math. Soc., Providence, RI, 1995, pp. 239–263. MR 1335475, DOI 10.1090/pspum/057/1335475
Alexander Grigor′yan, Heat kernel upper bounds on a complete non-compact manifold, Rev. Mat. Iberoamericana 10 (1994), no. 2, 395–452. MR 1286481, DOI 10.4171/RMI/157
Alexander Grigor′yan, Integral maximum principle and its applications, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), no. 2, 353–362. MR 1273753, DOI 10.1017/S0308210500028511
Alexander Grigor′yan, Heat kernel on a manifold with a local Harnack inequality, Comm. Anal. Geom. 2 (1994), no. 1, 111–138. MR 1312681, DOI 10.4310/CAG.1994.v2.n1.a7
Grigor’yan A. Escape rate of Brownian motion on weighted manifolds, to appear in Applicable Analysis.
Grigor’yan A. On non-parabolicity of Riemannian manifolds, preprint. .
Grigor’yan A., Hansen W. A Liouville property for Schrödinger operators, Math. Ann. 312 (1998), pp. 659-716.
Grigor’yan A., Kelbert M. Range of fluctuation of Brownian motion on a complete Riemannian manifold, Ann. Prob. 26 (1998), pp. 78-111.
Grigor’yan A., Kelbert M. On Hardy-Littlewood inequality for Brownian motion on Riemannian manifolds, to appear in J. London Math. Soc. (1998).
A. A. Grigor′yan and N. S. Nadirashvili, The Liouville theorems and exterior boundary value problems, Izv. Vyssh. Uchebn. Zaved. Mat. 5 (1987), 25–33, 88 (Russian). MR 904371
Mikhael Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53–73. MR 623534, DOI 10.1007/BF02698687
Y. Guivarc’h, Sur la loi des grands nombres et le rayon spectral d’une marche aléatoire, Conference on Random Walks (Kleebach, 1979) Astérisque, vol. 74, Soc. Math. France, Paris, 1980, pp. 47–98, 3 (French, with English summary). MR 588157
A. K. Gushchin, A criterion for uniform stabilization of solutions of the second mixed problem for a parabolic equation, Dokl. Akad. Nauk SSSR 264 (1982), no. 5, 1041–1045 (Russian). MR 672010
Emmanuel Hebey, Sobolev spaces on Riemannian manifolds, Lecture Notes in Mathematics, vol. 1635, Springer-Verlag, Berlin, 1996. MR 1481970, DOI 10.1007/BFb0092907
W. Hebisch and L. Saloff-Coste, Gaussian estimates for Markov chains and random walks on groups, Ann. Probab. 21 (1993), no. 2, 673–709. MR 1217561, DOI 10.1214/aop/1176989263
Ilkka Holopainen, Rough isometries and $p$-harmonic functions with finite Dirichlet integral, Rev. Mat. Iberoamericana 10 (1994), no. 1, 143–176. MR 1271760, DOI 10.4171/RMI/148
Ilkka Holopainen, Solutions of elliptic equations on manifolds with roughly Euclidean ends, Math. Z. 217 (1994), no. 3, 459–477. MR 1306672, DOI 10.1007/BF02571955
Holopainen I. Volume growth, Green’s functions and parabolicity of ends, to appear in Duke Math. J.
Pei Hsu, Heat semigroup on a complete Riemannian manifold, Ann. Probab. 17 (1989), no. 3, 1248–1254. MR 1009455
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
Kanji Ichihara, Curvature, geodesics and the Brownian motion on a Riemannian manifold. I. Recurrence properties, Nagoya Math. J. 87 (1982), 101–114. MR 676589, DOI 10.1017/S0027763000019978
Kanji Ichihara, Curvature, geodesics and the Brownian motion on a Riemannian manifold. I. Recurrence properties, Nagoya Math. J. 87 (1982), 101–114. MR 676589, DOI 10.1017/S0027763000019978
Ishige K., Murata M. Parabolic equations whose nonnegative solutions are determined only by their initial values, preprint.
Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951
Kiyoshi Itô and Henry P. McKean Jr., Diffusion processes and their sample paths, Die Grundlehren der mathematischen Wissenschaften, Band 125, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-New York, 1965. MR 0199891
V. A. Kaĭmanovich and A. M. Vershik, Random walks on discrete groups: boundary and entropy, Ann. Probab. 11 (1983), no. 3, 457–490. MR 704539, DOI 10.1214/aop/1176993497
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
Masahiko Kanai, Rough isometries, and combinatorial approximations of geometries of noncompact Riemannian manifolds, J. Math. Soc. Japan 37 (1985), no. 3, 391–413. MR 792983, DOI 10.2969/jmsj/03730391
Masahiko Kanai, Rough isometries and the parabolicity of Riemannian manifolds, J. Math. Soc. Japan 38 (1986), no. 2, 227–238. MR 833199, DOI 10.2969/jmsj/03820227
Karp L. Subharmonic functions, harmonic mappings and isometric immersions, in Seminar on Differential Geometry , ed. S.T.Yau, Ann. Math. Stud. 102, Princeton 1982.
Karp L., Li P. The heat equation on complete Riemannian manifolds, unpublished. (1983).
V. M. Kesel′man, Riemannian manifolds of $\alpha$-parabolic type, Izv. Vyssh. Uchebn. Zaved. Mat. 4 (1985), 81–83, 88 (Russian). MR 796595
R. Z. Has′minskiĭ, Ergodic properties of recurrent diffusion processes and stabilization of the solution of the Cauchy problem for parabolic equations, Teor. Verojatnost. i Primenen. 5 (1960), 196–214 (Russian, with English summary). MR 0133871
Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol. I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1996. Reprint of the 1963 original; A Wiley-Interscience Publication. MR 1393940
V. A. Kondrat′ev and E. M. Landis, Qualitative theory of second-order linear partial differential equations, Partial differential equations, 3 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1988, pp. 99–215, 220 (Russian). MR 1133457, DOI 10.1134/S0081543814050101
A. Korányi and J. C. Taylor, Minimal solutions of the heat equation and uniqueness of the positive Cauchy problem on homogeneous spaces, Proc. Amer. Math. Soc. 94 (1985), no. 2, 273–278. MR 784178, DOI 10.1090/S0002-9939-1985-0784178-8
Ju. T. Kuz′menko and S. A. Molčanov, Counterexamples to theorems of Liouville type, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 6 (1979), 39–43, 107 (Russian, with English summary). MR 561406
E. M. Landis, Second order equations of elliptic and parabolic type, Translations of Mathematical Monographs, vol. 171, American Mathematical Society, Providence, RI, 1998. Translated from the 1971 Russian original by Tamara Rozhkovskaya; With a preface by Nina Ural′tseva. MR 1487894, DOI 10.1090/mmono/171
Peter Li, Uniqueness of $L^1$ solutions for the Laplace equation and the heat equation on Riemannian manifolds, J. Differential Geom. 20 (1984), no. 2, 447–457. MR 788288
Peter Li, Large time behavior of the heat equation on complete manifolds with nonnegative Ricci curvature, Ann. of Math. (2) 124 (1986), no. 1, 1–21. MR 847950, DOI 10.2307/1971385
Li P. Curvature and function theory on Riemannian manifolds, preprint. . (1996).
Peter Li and Richard Schoen, $L^p$ and mean value properties of subharmonic functions on Riemannian manifolds, Acta Math. 153 (1984), no. 3-4, 279–301. MR 766266, DOI 10.1007/BF02392380
Peter Li and Luen-Fai Tam, Positive harmonic functions on complete manifolds with nonnegative curvature outside a compact set, Ann. of Math. (2) 125 (1987), no. 1, 171–207. MR 873381, DOI 10.2307/1971292
Peter Li and Luen-Fai Tam, Symmetric Green’s functions on complete manifolds, Amer. J. Math. 109 (1987), no. 6, 1129–1154. MR 919006, DOI 10.2307/2374588
Peter Li and Luen-Fai Tam, Harmonic functions and the structure of complete manifolds, J. Differential Geom. 35 (1992), no. 2, 359–383. MR 1158340
Peter Li and Luen-Fai Tam, Green’s functions, harmonic functions, and volume comparison, J. Differential Geom. 41 (1995), no. 2, 277–318. MR 1331970
Li P., Wang J. Convex hull properties of harmonic maps, J. Diff. Geom., 48 (1998) 497–530.
Peter Li and Shing-Tung Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), no. 3-4, 153–201. MR 834612, DOI 10.1007/BF02399203
W. Littman, G. Stampacchia, and H. F. Weinberger, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 17 (1963), 43–77. MR 161019
A. G. Losev, Some Liouville theorems on Riemannian manifolds of a special type, Izv. Vyssh. Uchebn. Zaved. Mat. 12 (1991), 15–24 (Russian); English transl., Soviet Math. (Iz. VUZ) 35 (1991), no. 12, 15–23. MR 1205018
Terry Lyons, Instability of the Liouville property for quasi-isometric Riemannian manifolds and reversible Markov chains, J. Differential Geom. 26 (1987), no. 1, 33–66. MR 892030
Terry Lyons, Random thoughts on reversible potential theory, Summer School in Potential Theory (Joensuu, 1990) Joensuun Yliop. Luonnont. Julk., vol. 26, Univ. Joensuu, Joensuu, 1992, pp. 71–114. MR 1175217
Terry Lyons, Instability of the conservative property under quasi-isometries, J. Differential Geom. 34 (1991), no. 2, 483–489. MR 1131440
Terry Lyons and Dennis Sullivan, Function theory, random paths and covering spaces, J. Differential Geom. 19 (1984), no. 2, 299–323. MR 755228
Paul Malliavin, Stochastic analysis, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 313, Springer-Verlag, Berlin, 1997. MR 1450093, DOI 10.1007/978-3-642-15074-6
Vladimir G. Maz’ja, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova. MR 817985, DOI 10.1007/978-3-662-09922-3
Ivor McGillivray, A recurrence condition for some subordinated strongly local Dirichlet forms, Forum Math. 9 (1997), no. 2, 229–246. MR 1431122, DOI 10.1515/form.1997.9.229
H. P. McKean Jr., Stochastic integrals, Probability and Mathematical Statistics, No. 5, Academic Press, New York-London, 1969. MR 0247684
H. P. McKean, An upper bound to the spectrum of $\Delta$ on a manifold of negative curvature, J. Differential Geometry 4 (1970), 359–366. MR 266100, DOI 10.4310/jdg/1214429509
V. M. Mīkljukov, A new approach to the Bernšteĭn theorem and to related questions of equations of minimal surface type, Mat. Sb. (N.S.) 108(150) (1979), no. 2, 268–289, 304 (Russian). MR 525842
John Milnor, On deciding whether a surface is parabolic or hyperbolic, Amer. Math. Monthly 84 (1977), no. 1, 43–46. MR 428232, DOI 10.2307/2318308
S. A. Molčanov, Diffusion processes, and Riemannian geometry, Uspehi Mat. Nauk 30 (1975), no. 1(181), 3–59 (Russian). MR 0413289
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
Minoru Murata, Positive harmonic functions on rotationary symmetric Riemannian manifolds, Potential theory (Nagoya, 1990) de Gruyter, Berlin, 1992, pp. 251–259. MR 1167241
Minoru Murata, Uniqueness and nonuniqueness of the positive Cauchy problem for the heat equation on Riemannian manifolds, Proc. Amer. Math. Soc. 123 (1995), no. 6, 1923–1932. MR 1242097, DOI 10.1090/S0002-9939-1995-1242097-3
Myrberg P.J. Über die Existenz der Greenschen Funktionen auf einer gegebenen Riemannschen Fläche, Acta Math. 61 (1933).
N. S. Nadirashvili, A theorem of Liouville type on a Riemannian manifold, Uspekhi Mat. Nauk 40 (1985), no. 5(245), 259–260 (Russian). MR 810821
Mitsuru Nakai, On Evans potential, Proc. Japan Acad. 38 (1962), 624–629. MR 150296
J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958), 931–954. MR 100158, DOI 10.2307/2372841
Nevanlinna R. Über die Lösbarkeit des Dirichletschen Problems für eine Riemannsche Fläche, Göttinger Nachr. 1 no. 14, (1939).
Leonard Eugene Dickson, New First Course in the Theory of Equations, John Wiley & Sons, Inc., New York, 1939. MR 0000002
P. Erdös and T. Grünwald, On polynomials with only real roots, Ann. of Math. (2) 40 (1939), 537–548. MR 7, DOI 10.2307/1968938
Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
Hiroyuki Ôkura, Capacitary inequalities and global properties of symmetric Dirichlet forms, Dirichlet forms and stochastic processes (Beijing, 1993) de Gruyter, Berlin, 1995, pp. 291–303. MR 1366444
O. A. Oleĭnik and E. V. Radkevič, The method of introducing a parameter for the investigation of evolution equations, Uspekhi Mat. Nauk 33 (1978), no. 5(203), 7–76, 237 (Russian). MR 511881
Robert Osserman, Remarks on minimal surfaces, Comm. Pure Appl. Math. 12 (1959), 233–239. MR 107868, DOI 10.1002/cpa.3160120204
Robert Osserman, Hyperbolic surfaces of the form $z=f(x,\,y)$, Math. Ann. 144 (1961), 77–79. MR 176062, DOI 10.1007/BF01396545
Robert Osserman, A survey of minimal surfaces, 2nd ed., Dover Publications, Inc., New York, 1986. MR 852409
Peter Petersen, Riemannian geometry, Graduate Texts in Mathematics, vol. 171, Springer-Verlag, New York, 1998. MR 1480173, DOI 10.1007/978-1-4757-6434-5
Pólya G. Über eine Aufgabe der Wahrscheinlichkeitstheorie betreffend die Irrfahrt im Straßennetz, Math. Ann. 84 (1921), pp. 149-160.
Yehuda Pinchover, On nonexistence of any $\lambda _0$-invariant positive harmonic function, a counterexample to Stroock’s conjecture, Comm. Partial Differential Equations 20 (1995), no. 9-10, 1831–1846. MR 1349233, DOI 10.1080/03605309508821153
Derek W. Robinson, Elliptic operators and Lie groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1991. Oxford Science Publications. MR 1144020
Steven Rosenberg, The Laplacian on a Riemannian manifold, London Mathematical Society Student Texts, vol. 31, Cambridge University Press, Cambridge, 1997. An introduction to analysis on manifolds. MR 1462892, DOI 10.1017/CBO9780511623783
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
L. Saloff-Coste, A note on Poincaré, Sobolev, and Harnack inequalities, Internat. Math. Res. Notices 2 (1992), 27–38. MR 1150597, DOI 10.1155/S1073792892000047
L. Saloff-Coste, Parabolic Harnack inequality for divergence-form second-order differential operators, Potential Anal. 4 (1995), no. 4, 429–467. Potential theory and degenerate partial differential operators (Parma). MR 1354894, DOI 10.1007/BF01053457
Laurent Saloff-Coste, Lectures on finite Markov chains, Lectures on probability theory and statistics (Saint-Flour, 1996) Lecture Notes in Math., vol. 1665, Springer, Berlin, 1997, pp. 301–413. MR 1490046, DOI 10.1007/BFb0092621
Leo Sario, Mitsuru Nakai, Cecilia Wang, and Lung Ock Chung, Classification theory of Riemannian manifolds, Lecture Notes in Mathematics, Vol. 605, Springer-Verlag, Berlin-New York, 1977. Harmonic, quasiharmonic and biharmonic functions. MR 0508005, DOI 10.1007/BFb0064417
R. Schoen and L. Simon, Regularity of simply connected surfaces with quasiconformal Gauss map, Seminar on minimal submanifolds, Ann. of Math. Stud., vol. 103, Princeton Univ. Press, Princeton, NJ, 1983, pp. 127–145. MR 795232
R. Schoen and S.-T. Yau, Lectures on differential geometry, Conference Proceedings and Lecture Notes in Geometry and Topology, I, International Press, Cambridge, MA, 1994. Lecture notes prepared by Wei Yue Ding, Kung Ching Chang [Gong Qing Zhang], Jia Qing Zhong and Yi Chao Xu; Translated from the Chinese by Ding and S. Y. Cheng; With a preface translated from the Chinese by Kaising Tso. MR 1333601
R. Schoen and S. T. Yau, Lectures on harmonic maps, Conference Proceedings and Lecture Notes in Geometry and Topology, II, International Press, Cambridge, MA, 1997. MR 1474501
Robert S. Strichartz, Analysis of the Laplacian on the complete Riemannian manifold, J. Functional Analysis 52 (1983), no. 1, 48–79. MR 705991, DOI 10.1016/0022-1236(83)90090-3
Daniel W. Stroock, Probability theory, an analytic view, Cambridge University Press, Cambridge, 1993. MR 1267569
Karl-Theodor Sturm, Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and $L^p$-Liouville properties, J. Reine Angew. Math. 456 (1994), 173–196. MR 1301456, DOI 10.1515/crll.1994.456.173
K.-T. Sturm, Sharp estimates for capacities and applications to symmetric diffusions, Probab. Theory Related Fields 103 (1995), no. 1, 73–89. MR 1347171, DOI 10.1007/BF01199032
Sung C.-J., Tam L.-F., Wang J. Spaces of harmonic functions, preprint.
Täcklind S. Sur les classes quasianalytiques des solutions des équations aux dérivées partielles du type parabolique, Nova Acta Regalis Societatis Scientiarum Uppsaliensis, (4) 10 no. 3, (1936), pp. 3-55.
Masayoshi Takeda, On a martingale method for symmetric diffusion processes and its applications, Osaka J. Math. 26 (1989), no. 3, 605–623. MR 1021434
Tichonov A.N. Uniqueness theorems for the equation of heat conduction, Matem. Sbornik 42 (1935), pp. 199-215.
M. Tsuji, Potential theory in modern function theory, Maruzen Co. Ltd., Tokyo, 1959. MR 0114894
Hajime Urakawa, Geometry of Laplace-Beltrami operator on a complete Riemannian manifold, Progress in differential geometry, Adv. Stud. Pure Math., vol. 22, Math. Soc. Japan, Tokyo, 1993, pp. 347–406. MR 1274959, DOI 10.2969/aspm/02210347
Nicholas Th. Varopoulos, The Poisson kernel on positively curved manifolds, J. Functional Analysis 44 (1981), no. 3, 359–380. MR 643040, DOI 10.1016/0022-1236(81)90015-X
Nicholas Th. Varopoulos, The Poisson kernel on positively curved manifolds, J. Functional Analysis 44 (1981), no. 3, 359–380. MR 643040, DOI 10.1016/0022-1236(81)90015-X
Nicholas Th. Varopoulos, Random walks on soluble groups, Bull. Sci. Math. (2) 107 (1983), no. 4, 337–344 (English, with French summary). MR 732356
N. T. Varopoulos, Potential theory and diffusion on Riemannian manifolds, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981) Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 821–837. MR 730112
Nicolas Th. Varopoulos, Brownian motion and random walks on manifolds, Ann. Inst. Fourier (Grenoble) 34 (1984), no. 2, 243–269. MR 746500, DOI 10.5802/aif.972
N. Th. Varopoulos, Hardy-Littlewood theory for semigroups, J. Funct. Anal. 63 (1985), no. 2, 240–260. MR 803094, DOI 10.1016/0022-1236(85)90087-4
Nicholas Th. Varopoulos, Théorie du potentiel sur des groupes et des variétés, C. R. Acad. Sci. Paris Sér. I Math. 302 (1986), no. 6, 203–205 (French, with English summary). MR 832044
N. Th. Varopoulos, L. Saloff-Coste, and T. Coulhon, Analysis and geometry on groups, Cambridge Tracts in Mathematics, vol. 100, Cambridge University Press, Cambridge, 1992. MR 1218884
Wiener N. Differential space, J. Math. Phys. Mass. Techn. 2 (1923), pp. 131-174.
Wolfgang Woess, Random walks on infinite graphs and groups—a survey on selected topics, Bull. London Math. Soc. 26 (1994), no. 1, 1–60. MR 1246471, DOI 10.1112/blms/26.1.1
Woess W. Random walks on infinite graphs and groups, in preparation.
Shing Tung Yau, Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold, Ann. Sci. École Norm. Sup. (4) 8 (1975), no. 4, 487–507. MR 397619, DOI 10.24033/asens.1299
Shing Tung Yau, Some function-theoretic properties of complete Riemannian manifold and their applications to geometry, Indiana Univ. Math. J. 25 (1976), no. 7, 659–670. MR 417452, DOI 10.1512/iumj.1976.25.25051
Shing Tung Yau, On the heat kernel of a complete Riemannian manifold, J. Math. Pures Appl. (9) 57 (1978), no. 2, 191–201. MR 505904
Kôsaku Yosida, Functional analysis, 6th ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 123, Springer-Verlag, Berlin-New York, 1980. MR 617913