Escape rate of brownian motion on riemanian manifolds

References

• Albeverio , S. and Kolokoltsov , V.N. in press . The rate of escape for some Gaussian processes and the scattering theory for their small perturbations .
   Google Scholar
• Ancona , A. 1990 . “ Théorie du potentiel sur des graphes et des variétés ” . In Cours de l'Ecole d'été de probabilités de Saint-Flour, 1988 , Vol. 1427 , 4 – 112 . Springer Lecture Notes .
   Google Scholar
• Bakry , D. , Coulhon , T. , Leodoux , M. and Saloff-coste , L. 1995 . Sobolev inequalities in disguise . Indiana Univ. Math. J , 124 ( 4 ) : 1033 – 1074 .
   Google Scholar
• Barlow , M. and Perkins , A. 1989 . Symmetric markov chains in Zd: how fast can they move? . Probab. Th. Rel. Fields , 82 ( 4 ) : 95 – 108 .
   Web of Science ®Google Scholar
• Bingham , N.H. 1986 . Variants on the law of the iterated logarithm . Bull. London Math. Soc , 18 ( 4 ) : 433 – 467 .
   Google Scholar
• Carron G. Inégalités isopérimétriques de Faber-Krahn et conséquences, in: “Actes de la table ronde de géométrie différentielle en l'honneur de Marcel Berger” 1994 Collection SMF Séminairés et congrès, No. 1
   Google Scholar
• Chavel , I. and Feldman , E.A. 1991 . Modified isoperimetric constants, and large time heat diffusion in Riemannian manifolds . Duke. Math. J , 64 ( 3 ) : 473 – 499 .
   Web of Science ®Google Scholar
• Davies , E.B. 1992 . Heat kernel bounds, conservation of probability and the Feller property . J. rl'Analyse Math , 58 ( 3 ) : 99 – 119 .
   Web of Science ®Google Scholar
• Dodziuk , J. 1983 . Maximum principle for parabolic inequalities and the heat flow on open manifolds . Indiana Univ, Math. J , 32 ( 5 ) : 703 – 716 .
   Web of Science ®Google Scholar
• Dvoretzky , A. and Erdös , P. 1951 . “ Some problems on random walk in space ” . In Proc. Second Berkeley Symposium on Math. Stat. and Probability , University of California Press .
   Google Scholar
• Grigor'yan , A. 1985 . On the existence of positive fundamental solution of the Laplace equation on Riemannian manifolds, (in Russian) . Mat. Sbornik , 128 ( 3 ) : 354 – 363 . Engl. transl. Math. USSR Sb.
   Google Scholar
• Grigor'yan , A. 1991 . The heat equation on non-compact Riemannian manifolds, (in Russian) . Matem. Sbornik , 182 ( 1 ) : 58 – 87 . Engl. transl. Math. USSR Sb., 72 (1992) No. 1, 47–77.
   Google Scholar
• Grigor'yan , A. 1994 . Heat kernel upper bounds on a complete non-compact manifold . Revista Alathemàtica Iberoamericana , 10 ( 2 ) : 395 – 452 .
   Google Scholar
• Grigor'yan , A. 1994 . Integral maximum principle and its applications . Proc. Roy. Soc. Edinburgh , 124A ( 2 ) : 353 – 362 .
   Google Scholar
• Grigor'yan A. Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds preprint
   Google Scholar
• Grigor'yan A. Kelbert M. Range of fluctuation of Brownian motion on a complete Riemannian manifold 1998 Ann. Prob.
   Google Scholar
• Grigor'yan A. Kelbert M. On Hardy-Littlewood inequality for Brownian motion on Riemannian manifolds 1998 J. London Math. Soc.,to appear.
   Google Scholar
• Grigor'yan A. Saloff-Coste L. Heat kernel on manifolds with ends in preparation
   Google Scholar
• Hebish , W. and Saloff-coste , L. 1993 . Gaussian estimates for Markov chains and random walks on groups . Ann. Prob , 21 ( 2 ) : 673 – 709 .
   Web of Science ®Google Scholar
• Hoffman , D. and Spruck , J. 1974 . Sobolev and isoperimetric inequalities for Riemannian submanifolds . Comm. Pure Appl. Math , 27 ( 2 ) : 715 – 725 .
   Web of Science ®Google Scholar
• Itô , K. and Mckean , H. 1965 . Diffusion process and their sample paths , Berlin : Springer .
   Google Scholar
• Kakutani , S. 1961 . Random walk and the type problem of Riemann surfaces , 95 – 101 . Princeton Univ. Press .
   Google Scholar
• Karpl , L. 1982 . “ Subharmonic functions, harmonic mappings and isometric immersions ” . In Ann. Math. Stud , Edited by: Yau , S.T. Vol. 102 , Pinceton . Seminar on Differential Geometry
   Google Scholar
• Li , P. and schoen , R. 1984 . LP and mean value properties of subharmonic functions on Riemannian manifolds . Acta Math , 153 : 279 – 301 .
   Web of Science ®Google Scholar
• Lyons T. Random thoughts on reversible potential theory, in: “Summer School in Potential Theory, Joensuu 1990” University of Joensuu. Publications in Sciences 26 71 114 IIpo Laine,ISBN 951–696–837–6.
   Google Scholar
• Maz'ya , V.G. 1985 . “Sobolev spaces”, (in Russian) ” . In Izdat , Leningrad : Leningrad Gos. Univ . Engl. transl. Springer-Verlag
   Google Scholar
• Maz'ya , V.G. 1991 . “ Classes of domains, measures and capacities in the theory of differentiable functions ” . In Encyclopaedia Math. Sci , Vol. 26 , Berlin : Springer .
   Google Scholar
• Moser , J. 1964 . A Harnack inequality for parabolic differential equations . Comm. Pure Appl. Math , 17 : 101 – 134 .
   Web of Science ®Google Scholar
• Sturm , K-th. 1995 . Sharp estimates for capacities and applications to symmetrical diffusions . Probability theo1:y and related fields , 103 ( 1 ) : 73 – 89 .
   Web of Science ®Google Scholar
• Takeda , M. 1989 . On a martingale method for symmetric diffusion process and its applications . Osalca J. Math , 26 ( 1 ) : 605 – 623 .
   Google Scholar
• Tsuji M. Potential theory in modern function theory Maruzen Tokyo 1959 hepth 9706099
   Google Scholar
• Varopoulos N.Th. Potential theory and diffusion of Riemannian manifolds, in: Conference on harmonic Analysis in honor of Antoni Zygmund. Belmont Calif 1983 I, II 821 837 Wadsworth Math. Ser.
   Google Scholar