Boundary theory on the Hata tree. (English) Zbl 1282.31004
Summary: We prove that for a certain Markov chain on the symbolic space of the Hata tree \(K\), the Martin boundary \(\mathcal M\) is homeomorphic to the trunk of the Hata tree, and the minimal Martin boundary is the post-critical set \(\{1\dot 2,\dot 1, \dot 2\}\), which corresponds to the three vertices of the trunk. Moreover, the class of \(P\)-harmonic functions on \(\mathcal M\) coincides with Kigami’s class of harmonic functions on \(K\).
MSC:
| 31C20 | Discrete potential theory |
| 31C35 | Martin boundary theory |
| 60J50 | Boundary theory for Markov processes |
| 28A80 | Fractals |
| 60J10 | Markov chains (discrete-time Markov processes on discrete state spaces) |
References:
| [1] | Denker, M.; Sato, H., Sierpiński gasket as a Martin boundary I: Martin kernels, Potential Anal., 14, 211-232 (2001) · Zbl 0980.60094 |
| [2] | Denker, M.; Sato, H., Sierpiński gasket as a Martin boundary II: the intrinsic metric, Publ. Res. Inst. Math. Sci., 35, 769-794 (1999) · Zbl 0980.60095 |
| [3] | Denker, M.; Sato, H., Reflections on harmonic analysis of the Sierpiński gasket, Math. Nachr., 241, 32-55 (2002) · Zbl 1020.60066 |
| [4] | Kaimanovich, V., Random walks on Sierpiński graphs: hyperbolicity and stochastic homogenization, (Fractals in Graz 2001. Fractals in Graz 2001, Trends Math. (2003), Birhäuser), 145-183 · Zbl 1031.60033 |
| [5] | Ju, H.; Lau, K.-S.; Wang, X.-Y., Post-critically finite fractal and Martin boundary, Trans. Amer. Math. Soc., 364, 103-118 (2012) · Zbl 1244.28004 |
| [6] | Kigami, J., Dirichlet forms and associated heat kernels on the Cantor set induced by random walks on trees, Adv. Math., 225, 2674-2730 (2010) · Zbl 1234.60077 |
| [7] | Lau, K.-S.; Wang, X.-Y., Self-similar sets as hyperbolic boundaries, Indiana Univ. Math. J., 58, 1777-1795 (2009) · Zbl 1188.28005 |
| [9] | Lau, K.-S.; Ngai, S.-M., Martin boundary and exit space on the Sierpinski gasket, Sci. China Math., 55, 475-494 (2012) · Zbl 1241.31014 |
| [10] | Jorgensen, P. E.T.; Pearse, E. P.J., Gel’fand triples and boundaries of infinite networks, New York J. Math., 17, 745-781 (2011) · Zbl 1241.05080 |
| [12] | Kigami, J., (Analysis on Fractals. Analysis on Fractals, Cambridge Tracts in Mathematics, vol. 143 (2001), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0998.28004 |
| [14] | Hajnal, J., Weak ergodicity in non-homogeneous Markov chains, Proc. Cambridge Philos. Soc., 54, 233-246 (1958) · Zbl 0082.34501 |
| [15] | Doob, J. L., Discrete potential theory and boundaries, J. Math. Mech., 8, 433-458 (1959) · Zbl 0101.11503 |
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