Volume growth and escape rate of symmetric diffusion processes

References

• M. Biroli and U. Mosco, Formes de Dirichlet et estimations structurelles dans les milieux discontinus [Dirichlet forms and structural estimates on discontinuous media]. C. R. Acad. Sci. Paris S\’{e}r. I Math. 313(9) (1991), pp. 593–598.
   Google Scholar
• M. Biroli and U. Mosco, A Saint-Venant type principle for Dirichlet forms on discontinuous media, Ann. Mat. Pura Appl. 4(169) (1995), pp. 125–181.
   Web of Science ®Google Scholar
• N. Bouleau, and F. Hirsch, Dirichlet Forms and Analysis on Wiener space, de Gruyter Studies in Mathematics Vol. 14, Walter de Gruyter & Co., Berlin, 1991.
   Google Scholar
• Z.-Q. Chen, and M. Fukushima, Symmetric Markov Processes, Time Change, and Boundary Theory, London Mathematical Society Monographs Series Vol. 35, Princeton University Press, Princeton, NJ, 2012.
   Google Scholar
• E.B. Davies, L1properties of second order elliptic operators, Bull. London Math. Soc. 17(5) (1985), pp. 417–436.
   Web of Science ®Google Scholar
• A. Friedman, Stochastic Differential Equations and Applications, Dover Publications, Mineola, NY, 2006. Two volumes bound as one, Reprint of the 1975 and 1976 original published in two volumes.
   Google Scholar
• M. Fukushima, Y. Oshima, and M. Takeda, Dirichlet Forms and Symmetric Markov processes, de Gruyter Studies in Mathematics, extended ed., Vol. 9, Walter de Gruyter & Co., Berlin, 2011.
   Google Scholar
• {\u{I}}.{\=I}. G{\={\i}}hman and A.V. Skorohod, Stochastic Differential Equations, Springer-Verlag, New York, 1972. Translated from the Russian by Kenneth Wickwire, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 72.
   Google Scholar
• A.A. Grigor’yan, Stochastically complete manifolds, Dokl. Akad. Nauk SSSR 290(3) (1986), pp. 534–537.
   Google Scholar
• A. Grigor’yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. (N.S.). 36(2) (1999), pp. 135–249.
   Google Scholar
• A. Grigor’yan, Escape rate of Brownian motion on Riemannian manifolds, Appl. Anal. 71(1–4) (1999), pp. 63–89.
   Google Scholar
• A. Grigor’yan and E. Hsu, Volume growth and escape rate of Brownian motion on a Cartan-Hadamard manifold, in Sobolev Spaces in Mathematics. II, International Mathematics Series (New York) Vol. 9, Springer, New York, 2009, pp. 209–225.
   Google Scholar
• A. Grigor’yan, and M. Kelbert, Range of fluctuation of Brownian motion on a complete Riemannian manifold, Ann. Probab. 26(1) (1998), pp. 78–111.
   Web of Science ®Google Scholar
• A. Grigor’yan and M. Kelbert, On Hardy--Littlewood inequality for Brownian motion on Riemannian manifolds, J. London Math. Soc. (2) 62(2) (2000), pp. 625–639.
   Google Scholar
• P.E. Hsu, and G. Qin, Volume growth and escape rate of Brownian motion on a complete Riemannian manifold, Ann. Probab. 38(4) (2010), pp. 1570–1582.
   Web of Science ®Google Scholar
• K. Itô, and H.P. McKean, Diffusion Processes and their Sample Paths, Springer, Berlin, 1974.
   Google Scholar
• T.J. Lyons and W.A. Zheng, A crossing estimate for the canonical process on a Dirichlet space and a tightness result, Ast\’{e}risque 157--158 (1988), pp. 249-271. Colloque Paul L\’{e}vy sur les Processus Stochastiques (Palaiseau, 1987).
   Google Scholar
• X. Mao, Stochastic Differential Equations and Applications, 2nd ed., Horwood Publishing Limited, Chichester, 2008.
   Google Scholar
• M. Motoo, Proof of the law of iterated logarithm through diffusion equation, Ann. Inst. Statist. Math. 10 (1959), pp. 21–28.
   Web of Science ®Google Scholar
• T. Shiga, and S. Watanabe, Bessel diffusions as a one-parameter family of diffusion processes, Z. Wahr. verw. Geb. 27 (1973), pp. 37–46.
   Web of Science ®Google Scholar
• Y. Shiozawa, Escape rate of symmetric jump-diffusion processes, Trans. Amer. Math. Soc., preprint (2015).
   Google Scholar
• K.-T. Sturm, Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and Lp-Liouville properties, J. Reine Angew. Math. 456 (1994), pp. 173–196.
   Web of Science ®Google Scholar
• K.-T. Sturm, On the geometry defined by Dirichlet forms, in Seminar on Stochastic Analysis, Random Fields and Applications (Ascona, 1993), Progress in Probability Vol. 36, Birkh\"{a}user, Basel, 2015, pp. 231–242.
   Google Scholar
• K.-T. Sturm, The geometric aspect of Dirichlet forms, in New Directions in Dirichlet Forms, AMS/IP Studies in Advanced Mathematics Vol. 8, Amercan Mathematical Society, Providence, RI, 1998, pp. 233–277.
   Google Scholar
• M. Takeda, On a martingale method for symmetric diffusion processes and its applications, Osaka J. Math. 26(3) (1989), pp. 605–623.
   Web of Science ®Google Scholar
• M. Takeda, On the conservativeness of the Brownian motion on a Riemannian manifold, Bull. London Math. Soc. 23(1) (1991), pp. 86–88.
   Web of Science ®Google Scholar
• M. Takeda, and G. Trutnau, Conservativeness of non-symmetric diffusion processes generated by perturbed divergence forms, Forum Math. 24(2) (2012), pp. 419–444.
   Web of Science ®Google Scholar
• G. Trutnau, Skorokhod decomposition of reflected diffusions on bounded Lipschitz domains with singular non-reflection part, Probab. Theory Related Fields 127(4) (2003), pp. 455–495.
   Web of Science ®Google Scholar
• G. Trutnau, A short note on Lyons--Zheng decomposition in the non-sectorial case, Proceedings of RIMS Workshop on Stochastic Analysis and Applications, RIMS K\^{o}ky\^{u}roku Bessatsu, B6, Res. Inst. Math. Sci. (RIMS) Kyoto, 2008, pp. 237–245.
   Google Scholar