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Central Limit Theorems for Non-Symmetric Random Walks on Nilpotent Covering Graphs: Part II

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Abstract

In the present paper, as a continuation of our preceding paper (Ishiwata et al. 2018), we study another kind of central limit theorems (CLTs) for non-symmetric random walks on nilpotent covering graphs from a view point of discrete geometric analysis developed by Kotani and Sunada. We introduce a one-parameter family of random walks which interpolates between the original non-symmetric random walk and the symmetrized one. We first prove a semigroup CLT for the family of random walks by realizing the nilpotent covering graph into a nilpotent Lie group via discrete harmonic maps. The limiting diffusion semigroup is generated by the homogenized sub-Laplacian with a constant drift of the asymptotic direction on the nilpotent Lie group, which is equipped with the Albanese metric associated with the symmetrized random walk. We next prove a functional CLT (i.e., Donsker-type invariance principle) in a Hölder space over the nilpotent Lie group by combining the semigroup CLT, standard martingale techniques, and a novel pathwise argument inspired by rough path theory. Applying the corrector method, we finally extend these CLTs to the case where the realizations are not necessarily harmonic.

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References

  1. Alexopoulos, G.: Convolution powers on discrete groups of polynomial volume growth. Canad. Math. Soc. Conf. Proc. 21, 31–57 (1997)

    MathSciNet  MATH  Google Scholar 

  2. Alexopoulos, G.: Random walks on discrete groups of polynomial growth. Ann. Probab. 30, 723–801 (2002)

    Article  MathSciNet  Google Scholar 

  3. Alexopoulos, G.: Sub-Laplacians with drift on Lie groups of polynomial volume growth. Mem. Amer. Math. Soc. 155(739) (2002)

  4. Bensoussan, A., Lions, J.L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures. Studies in Mathematics and its Applications, vol. 5. North-Holland Publishing Co., Amsterdam (1978)

    MATH  Google Scholar 

  5. Biskup, M: Recent progress on the random conductance model. Probab. Surv. 8, 294–373 (2011)

    Article  MathSciNet  Google Scholar 

  6. Friz, P.K., Victoir, N.B.: Multidimensional Stochastic Processes as Rough Paths, Theory and Applications, Cambridge Studies in Advanced Mathematics, vol. 120. Cambridge Univ. Press, Cambridge (2010)

    MATH  Google Scholar 

  7. Gromov, M.: Groups of polynomial growth and expanding maps. IHES Publ. Math. 53, 53–73 (1981)

    Article  MathSciNet  Google Scholar 

  8. Ishiwata, S.: A central limit theorem on a covering graph with a transformation group of polynomial growth. J. Math. Soc. Japan 55, 837–853 (2003)

    Article  MathSciNet  Google Scholar 

  9. Ishiwata, S., Kawabi, H., Kotani, M.: Long time asymptotics of non-symmetric random walks on crystal lattices. J. Funct. Anal. 272, 1553–1624 (2017)

    Article  MathSciNet  Google Scholar 

  10. Ishiwata, S., Kawabi, H., Namba, R.: Central limit theorems for non-symmetric random walks on nilpotent covering graphs. Part I. Electron. J. Probab. 25, 46 pages (2020)

    MathSciNet  MATH  Google Scholar 

  11. Kipnis, C., Landim, C.: Scaling Limits of Interacting Particle Systems Grundlehren der mathematischen Wissenschaften, vol. 320. Springer, Berlin (1999)

    Book  Google Scholar 

  12. Kotani, M.: A central limit theorem for magnetic transition operators on a crystal lattice. J. London Math. Soc. 65, 464–482 (2002)

    Article  MathSciNet  Google Scholar 

  13. Kotani, M.: An asymptotic of the large deviation for random walks on a crystal lattice. Contemp. Math. 347, 141–152 (2004)

    Article  MathSciNet  Google Scholar 

  14. Kotani, M., Sunada, T.: Albanese maps and off diagonal long time asymptotics for the heat kernel. Comm. Math. Phys. 209, 633–670 (2000)

    Article  MathSciNet  Google Scholar 

  15. Kotani, M., Sunada, T.: Standard realizations of crystal lattices via harmonic maps. Trans. Amer. Math. Soc. 353, 1–20 (2000)

    Article  MathSciNet  Google Scholar 

  16. Kotani, M., Sunada, T.: Large deviation and the tangent cone at infinity of a crystal lattice. Math. Z. 254, 837–870 (2006)

    Article  MathSciNet  Google Scholar 

  17. Kotani, M., Shirai, T., Sunada, T.: Asymptotic behavior of the transition probability of a random walk on an infinite graph. J. Funct. Anal. 159, 664–689 (1998)

    Article  MathSciNet  Google Scholar 

  18. Kozlov, S.M.: The averaging method and walks in inhomogeneous environments. Russian Math. Surveys 40, 73–145 (1985)

    Article  MathSciNet  Google Scholar 

  19. Kumagai, T.: Random Walks on Disordered Media and their Scaling Limits, École d’Été de Probabilités de Saint-Flour XL-2010, LNM 2101. Springer, Cham (2014)

    MATH  Google Scholar 

  20. Lyons, T., Caruana, M., Lévy, T.: Differential Equations Driven by Rough Paths, École d’Été de Probabilités de Saint-Flour XXXIV-2004, LNM 1908. Springer, Berlin (2007)

    MATH  Google Scholar 

  21. Lyons, T: Differential equations driven by rough signals. Rev. Math. Iberoamericana 14, 215–310 (1998)

    Article  MathSciNet  Google Scholar 

  22. Lyons, T., Qian, Z.: System Control and Rough Paths, Oxford Mathematical Monographs. Oxford Univ. Press, Oxford (2002)

    Book  Google Scholar 

  23. Malćev, A.I.: On a class of homogeneous spaces. Amer. Math. Soc. Transl. 39, 276–307 (1951)

    MathSciNet  Google Scholar 

  24. Namba, R.: Central limit theorems for non-symmetric random walks on covering graphs, Ph.D. thesis at Okayama University, available at http://ousar.lib.okayama-u.ac.jp/files/public/5/56797/20190625084049750154/K0005956_fulltext.pdf (2019)

  25. Pap, G.: Central Limit Theorems on Stratified Lie Groups, Probability Theory and Mathematical Statistics (Vilnius, 1993), pp 613–627. TEV, Vilnius (1994)

    MATH  Google Scholar 

  26. Papanicolaou, G.C., Varadhan, S.R.S.: Boundary value problem with rapidly oscillating random coefficients. In: Random Fields, Vol. I, II (Esztergom, 1979), Colloq. Math. Soc. János Bolyai, 27, pp 835–873. North-Holland, Amsterdam-New York (1981)

  27. Raghunathan, M.S.: Discrete Subgroups of Lie Groups. Springer, Berlin (1972)

    Book  Google Scholar 

  28. Raugi, A: Thèoréme de la limite centrale sur les groupes nilpotents. Z. Wahrsch. Verw. Gebiete. 43, 149–172 (1978)

    Article  MathSciNet  Google Scholar 

  29. Robinson, D.W.: Elliptic Operators and Lie Groups, Oxford Mathematical Mono- graphs. Oxford Univ. Press, New York (1991)

    Google Scholar 

  30. Stroock, D.W.: Probability Theory: An Analytic View. Cambridge Univ. Press (1993)

  31. Sunada, T.: Discrete Geometric Analysis. In: Proc. Sympos. Pure Math. 77, Amer. Math. Soc. Analysis on Graphs and its Applications, pp 51–83, Providence (2008)

  32. Sunada, T.: Topological Crystallography with a View Towards Discrete Geometric Analysis, Surveys and Tutorials in the Applied Mathematical Sciences, vol. 6. Springer, Japan (2013)

    Google Scholar 

  33. Sunada, T.: Topics in mathematical crystallography. In: The Preceedings of the Symposium “Groups, Graphs and Random Walks”, London Math. Soc., Lecture Note Series, vol. 436, pp 473–513. Cambridge Univ. Press (2017)

  34. Tanaka, R: Hydrodynamic limit for weakly asymmetric simple exclusion processes in crystal lattices. Comm. Math. Phys. 315, 603–641 (2012)

    Article  MathSciNet  Google Scholar 

  35. Trotter, H.F.: Approximation of semi-groups of operators. Pac. J. Math. 8, 887–919 (1958)

    Article  MathSciNet  Google Scholar 

  36. Varopoulos, N.T., Saloff-Coste, L., Coulhon, T.: Analysis and Geometry on Groups, Cambridge Tracts in Mathematics, vol. 100. Cambridge Univ. Press, Cambridge (1992)

    MATH  Google Scholar 

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Acknowledgements

The authors are grateful to Professor Shoichi Fujimori for making pictures of the 3-dimensional Heisenberg hexagonal lattice and kindly allowing them to use these pictures in the present paper. They would like to thank Professor Seiichiro Kusuoka for reading of our manuscript carefully and for giving valuable comments. They also thank the anonymous referee who significantly helped to improve the quality of the present paper. A part of this work was done during the stay of R. N. at Hausdorff Center for Mathematics, Universität Bonn in March 2017 with the support of research fund of Research Institute for Interdisciplinary Science, Okayama University. He would like to thank Professor Massimiliano Gubinelli for warm hospitality and helpful discussions. S. I. was partially supported by JSPS Grant-in-Aid for Young Scientists (B) No. 25800034 and JSPS Grant-in-Aid for Scientific Research (C) No. 17K05215. H.K. was supported by JSPS Grant-in-Aid for Scientific Research (C) No. 26400134 and (C) No. 17K05300. R. N. was partially supported by JSPS Research Fellowships for Young Scientists No. 18J10225. The present paper is based on a part of R.N.’s Ph.D. thesis [24].

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Authors and Affiliations

  1. Department of Mathematical Sciences, Faculty of Science, Yamagata University, 1-4-12, Kojirakawa, Yamagata, 990-8560, Japan

    Satoshi Ishiwata

  2. Department of Mathematics, Keio University, Hiyoshi Campus, 4-1-1, Hiyoshi, Kohoku-ku, Yokohama, 223-8521, Japan

    Hiroshi Kawabi

  3. Department of Mathematical Sciences, College of Science and Engineering, Ritsumeikan University, 1-1-1, Noji-higashi, Kusatsu, 525-8577, Japan

    Ryuya Namba

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  1. Satoshi Ishiwata
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  2. Hiroshi Kawabi
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Correspondence to Ryuya Namba.

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Ishiwata, S., Kawabi, H. & Namba, R. Central Limit Theorems for Non-Symmetric Random Walks on Nilpotent Covering Graphs: Part II. Potential Anal 55, 127–166 (2021). https://doi.org/10.1007/s11118-020-09851-7

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