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Central limit theorem of Brownian motions in pinched negative curvature

Part of: Partial differential equations on manifolds; differential operators Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  22 April 2021

JAELIN KIM Open the ORCID record for JAELIN KIM [Opens in a new window]
JAELIN KIM*
Affiliation:
Department of Mathematical Sciences, Seoul National University, Seoul151-747, Republic of Korea
*
e-mail: kimjl@snu.ac.kr
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Abstract

We prove the central limit theorem of random variables induced by distances to Brownian paths and Green functions on the universal cover of Riemannian manifolds of finite volume with pinched negative curvature. We further provide some ergodic properties of Brownian motions and an application of the central limit theorem to the dynamics of geodesic flows in pinched negative curvature.

Keywords

foliated Brownian motioncentral limit theoremgeodesic flow in negative curvature

MSC classification

Primary: 58J65: Diffusion processes and stochastic analysis on manifolds
Secondary: 37D35: Thermodynamic formalism, variational principles, equilibrium states 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 7 , July 2022 , pp. 2352 - 2381
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Ancona, A.. Negatively curved manifolds, elliptic operators, and the Martin boundary. Ann. of Math. (2) 125 (1987), 495–536.CrossRefGoogle Scholar
Anderson, M. T. and Schoen, R.. Positive harmonic functions on complete manifolds of negative curvature. Ann. of Math. (2) 121 (1985), 429–461.CrossRefGoogle Scholar
Ballmann, W.. Lectures on Spaces of Nonpositive Curvature (Oberwolfach Seminars, 25). Birkhäuser and Springer, Basel, 1995.CrossRefGoogle Scholar
Benoist, Y., Foulon, P. and Labourie, F.. Flots d’Anosov á distributions stable et instable différentiables. J. Amer. Math. Soc. 5 (1992), 33–74.Google Scholar
Benoist, Y. and Hulin, D.. Harmonic measures on negatively curved manifolds . Ann. Inst. Fourier (Grenoble) 69 (2019), 2951–2971.CrossRefGoogle Scholar
Bridson, M. R. and Haeflinger, A.. Metric Spaces of Non-Positive Curvature (Grundlehren der mathematischen Wissenschaften, 319). Springer, Berlin, 2013.Google Scholar
Candel, A. and Conlon, L.. Foliations II (Graduate Studies in Mathematics, 23). American Mathematical Society, Providence, RI, 2003.CrossRefGoogle Scholar
Chavel, I. and Karp, L.. Large time behavior of the heat kernel: the parabolic $\lambda$ -potential alternative. Comment. Math. Helv. 66 (1991), 541–556.CrossRefGoogle Scholar
Dal’bo, F., Peiné, M., Picaud, J.-C. and Sambusetti, A.. On the growth of nonuniform lattices in pinched negatively curved manifolds. J. Reine Angew. Math. 627 (2009), 31–52.Google Scholar
Dal’bo, F., Peiné, M., Picaud, J.-C. and Sambusetti, A.. Convergence and counting in infinite measure. Ann. Inst. Fourier (Grenoble) 67 (2017), 483–520.CrossRefGoogle Scholar
Dodziuk, J.. A lower bound for the first eigenvalue of a finite-volume negatively curved manifold. Bull. Braz. Math. Soc. (N.S.) 18 (1987), 23–34.CrossRefGoogle Scholar
Donnelly, H.. Essential spectrum and heat kernel. J. Funct. Anal. 75 (1987), 362-381.CrossRefGoogle Scholar
Engel, K.-J. and Nagel, R.. One-Parameter Semigroups for Linear Evolution Equations (Graduate Texts in Mathematics, 194). Springer, New York, 1999.Google Scholar
Foulon, P. and Labourie, F.. Sur les variétés compactes asymptotiquement harmoniques. Invent. Math. 69 (1992), 375–392.Google Scholar
Franco, E.. Flows with unique equilibrium states. Amer. J. Math. 99 (1977), 486–514.CrossRefGoogle Scholar
Garnett, L.. Foliations, the ergodic theorem and Brownian motion. J. Funct. Anal. 51 (1983), 285–311.CrossRefGoogle Scholar
Grigor’yan, A.. Heat kernel upper bounds on a complete non-compact manifold. Rev. Mat. Iberoam. 10 (1994), 395–452.CrossRefGoogle Scholar
Guivarc’h, Y.. Mouvement brownien sur les revêtements d’une variété compacte. C. R. Math. Acad. Sci. Paris 292 (1981), 851–853.Google Scholar
Hamenstädt, U.. An explicit description of harmonic measure. Math. Z. 205 (1990), 287–299.CrossRefGoogle Scholar
Hamenstädt, U.. Harmonic measures for compact negatively curved manifolds. Acta Math. 178 (1997), 39–107.CrossRefGoogle Scholar
Hellund, I. S.. Central limit theorems for martingales with discrete or continuous time. Scand. J. Stat. 9 (1982), 79–94.Google Scholar
Hsu, E. P.. Stochasic Analysis on Manifolds (Graduate Studies in Mathematics, 38). American Mathematical Society, Providence, RI, 2002.Google Scholar
Kaimanovich, V. A.. Brownian motion and harmonic functions on covering manifolds. An entropic approach. Dokl. Akad. Nauk 288 (1986), 1045–1049.Google Scholar
Kifer, Y. and Ledrappier, F.. Hausdorff dimension of harmonic measures on negatively curved manifolds. Trans. Amer. Math. Soc. 318 (1990), 685–704.CrossRefGoogle Scholar
Kingman, J. F. C.. The ergodic theory and subadditive stochastic processes. J. R. Stat. Soc. Ser. B Stat. Methodol. 30 (1968), 499–510.Google Scholar
Ledrappier, F.. Propriété de Poisson et courbure négative. C. R. Math. Acad. Sci. Paris 305 (1987), 191–194.Google Scholar
Ledrappier, F.. Ergodic properties of Brownian motion on covers of compact negatively curved manifolds. Bull. Braz. Math. Soc. (N.S.) 19 (1988), 115–140.CrossRefGoogle Scholar
Ledrappier, F.. Profil d’entropie dans le cas continu (Astérisque, 236). Société Mathématique de France, Paris, 1988.Google Scholar
Ledrappier, F.. Harmonic measures and Bowen–Margulis measures. Israel J. Math. 71 (1990), 275–287.CrossRefGoogle Scholar
Ledrappier, F.. Central limit theorem in negative curvature. Ann. Probab. 23 (1995), 1219–1233.CrossRefGoogle Scholar
Ledrappier, F. and Shu, L.. Differentiating the stochastic entropy for compact negatively curved spaces under conformal changes. Ann. Inst. Fourier (Grenoble) 67 (2017), 1115–1183.CrossRefGoogle Scholar
Paulin, F., Pollicott, M. and Schapira, B.. Equilibrium States in Negative Curvature (Astérisque, 373). Société Mathématique de France, Paris, 2015.Google Scholar
Petersen, P.. Riemannian Geometry (Graduate Texts in Mathematics, 171). Springer, New York, 2016.CrossRefGoogle Scholar
Pinsky, M. A.. Stochastic Riemannian geometry. Probabilistic Analysis and Related Topics . Vol. 1. Ed. Bharucha-Reid, A. T.. Academic Press, New York, 1978, pp. 199–236.CrossRefGoogle Scholar
Pit, V. and Schapira, B.. Finiteness of Gibbs measures on noncompact manifolds with pinched negative curvature. Ann. Inst. Fourier (Grenoble) 68 (2018), 457–510.CrossRefGoogle Scholar
Prat, J.-J.. Étude asymptotique et convergence angulaire du mouvement Brownien sur une variété à courbure négative. C. R. Math. Acad. Sci. Paris 280 (1975), 1539–1542.Google Scholar
Riquelme, F.. Ruelle’s inequality in negative curvature. Discrete Contin. Dyn. Syst. 38 (2018), 2809–2825.CrossRefGoogle Scholar
Yue, C.. Integral formulas for the Laplacian along the unstable foliations to rigidity problems for manifolds of negative curvature. Ergod. Th. & Dynam.Sys. 11 (1991), 803–819.CrossRefGoogle Scholar