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 Uspekhi Mat. Nauk, 2014, Volume 69, Issue 4(418), Pages 3–102 (Mi umn9602)

Isotropic Markov semigroups on ultra-metric spaces

A. D. Bendikova, A. A. Grigor'yanb, Ch. Pittetc, W. Woessd

a Institute of Mathematics, Wroclaw University, Wroclaw, Poland
b Bielefeld University, Germany
c LATP, Université d'Aix-Marseille, Marseille, France
d Institut für Mathematische Strukturtheorie, Technische Universität Graz, Graz, Austria

Abstract: Let $(X,d)$ be a separable ultra-metric space with compact balls. Given a reference measure $\mu$ on $X$ and a distance distribution function $\sigma$ on $[0,\infty)$, a symmetric Markov semigroup $\{P^{t}\}_{t\geqslant 0}$ acting in $L^{2}(X,\mu )$ is constructed. Let $\{\mathcal{X}_{t}\}$ be the corresponding Markov process. The authors obtain upper and lower bounds for its transition density and its Green function, give a transience criterion, estimate its moments, and describe the Markov generator $\mathcal{L}$ and its spectrum, which is pure point. In the particular case when $X=\mathbb{Q}_{p}^{n}$, where $\mathbb{Q}_{p}$ is the field of $p$-adic numbers, the construction recovers the Taibleson Laplacian (spectral multiplier), and one can also apply the theory to the study of the Vladimirov Laplacian. Even in this well-established setting, several of the results are new. The paper also describes the relation between the processes involved and Kigami's jump processes on the boundary of a tree which are induced by a random walk. In conclusion, examples illustrating the interplay between the fractional derivatives and random walks are provided.
Bibliography: 66 titles.

Keywords: ultra-metric measure space, metric trees, isotropic Markov semigroups, Markov generators, heat kernels, transition density, $p$-number field, Vladimirov–Taibleson operator, nearest neighbour random walk on a tree, Dirichlet form, harmonic functions with finite energy, traces of harmonic functions with finite energy.

 Funding Agency Grant Number Deutsche Forschungsgemeinschaft SFB 701 National Science Centre (Narodowe Centrum Nauki) 2012/05/B/ST 1/00613 Centre National de la Recherche Scientifique Austrian Science Fund W1230-N13P24028-N18 This work was begun and finished at Bielefeld University under the support of SFB 701 of the German Research Council. The first author was supported by Polish Government Scientific Research Fund (grant no. 2012/05/B/ST 1/00613). The second author was supported by the German Research Council (SFB 701). The third author was supported by the French National Centre for Scientific Research (CNRS). The fourth author was supported by the Austrian Science Fund (projects FWF W1230-N13 and FWF P24028-N18).

DOI: https://doi.org/10.4213/rm9602

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English version:
Russian Mathematical Surveys, 2014, 69:4, 589–680

Bibliographic databases:

Document Type: Article
UDC: 519.217.5+519.217.13+517.518.14
MSC: Primary 46S10, 60J25; Secondary 05C05, 11S80, 35S05

Citation: A. D. Bendikov, A. A. Grigor'yan, Ch. Pittet, W. Woess, “Isotropic Markov semigroups on ultra-metric spaces”, Uspekhi Mat. Nauk, 69:4(418) (2014), 3–102; Russian Math. Surveys, 69:4 (2014), 589–680

Citation in format AMSBIB
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This publication is cited in the following articles:
1. A. Bendikov, P. Krupski, “On the spectrum of the hierarchical Laplacian”, Potential Anal., 41:4 (2014), 1247–1266
2. A. D. Bendikov, A. A. Grigor'yan, S. A. Molchanov, G. P. Samorodnitsky, “On a class of random perturbations of the hierarchical Laplacian”, Izv. Math., 79:5 (2015), 859–893
3. J. Angulo, S. Velasco-Forero, “Morphological semigroups and scale-spaces on ultrametric spaces”, Mathematical morphology and its applications to signal and image processing, Lecture Notes in Comput. Sci., 10225, Springer, Cham, 2017, 28–39
4. Sh.-L. Kong, K.-S. Lau, T.-K. L. Wong, “Random walks and induced Dirichlet forms on self-similar sets”, Adv. Math., 320 (2017), 1099–1134
5. Theory Probab. Appl., 63:1 (2018), 94–116
6. A. Bendikov, “Heat kernels for isotropic-like Markov generators on ultrametric spaces: a survey”, P-Adic Numbers Ultrametric Anal. Appl., 10:1 (2018), 1–11
7. A. Torresblanca-Badillo, W. A. Zuniga-Galindo, “Ultrametric diffusion, exponential landscapes, and the first passage time problem”, Acta Appl. Math., 157:1 (2018), 93–116
8. M. L. Lapidus, H. Lu, M. van Frankenhuijsen, “Minkowski dimension and explicit tube formulas for $p$-adic fractal strings”, Fractal Pract., 2:4 (2018), 26
9. Bendikov A., Cygan W., Woess W., “Oscillating Heat Kernels on Ultrametric Spaces”, J. Spectr. Theory, 9:1 (2019), 195–226
10. Bendikov A., Cygan W., “On the Rate of Convergence in the Central Limit Theorem For Hierarchical Laplacians”, ESAIM-Prob. Stat., 23 (2019), 68–81
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