Abstract
This note is concerned with two families of operators related to the fractional Laplacian, the first arising from the Caffarelli-Silvestre extension problem and the second from the fractional heat equation. They both include the Poisson semigroup. We show that on a complete, connected, and non-compact Riemannian manifold of non-negative Ricci curvature, in both cases, the solution with \(L^1\) initial data behaves asymptotically as the mass times the fundamental solution. Similar long-time convergence results remain valid on more general manifolds satisfying the Li-Yau two-sided estimate of the heat kernel. The situation changes drastically on hyperbolic space, and more generally on rank one non-compact symmetric spaces: we show that for the Poisson semigroup, the convergence to the Poisson kernel fails -but remains true under the additional assumption of radial initial data.
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References
Abadias, L., Alvarez, E.: Asymptotic behavior for the discrete in time heat equation. Mathematics 10, 3128 (2022)
Abadias, L., González-Camus, J., Miana, P., Pozo, J.: Large time behaviour for the heat equation on \(\mathbb{Z}\), moments and decay rates. J. Math. Anal. Appl. 500, (2021)
Anker, J.-Ph., Ji, L.: Heat kernel and Green function estimates on noncompact symmetric spaces. Geom. Funct. Anal. 9, 1035–1091 (1999)
Anker J.-Ph., Ostellari P.: The heat kernel on noncompact symmetric spaces. Lie Groups and Symmetric Spaces: In Memory of F.I. Karpelevich. Amer. Math. Soc. 210(2) (2003)
Anker, J.-Ph., Papageorgiou, E., Zhang, H.-W.: Asymptotic behavior of solutions to the heat equation on noncompact symmetric spaces. J. Funct. Anal. 284(6), (2023)
Barlow, M.: Diffusions on fractals. In: Lectures On Probability Theory And Statistics (Saint-Flour, 1995), pp. 1–121. Springer, Berlin (1998)
Banica, V., González, M.M., Sáez, M.: Some constructions for the fractional Laplacian on noncompact manifolds. Rev. Mat. Iberoam. 31(2), 681–712 (2007)
Bonforte, M., Sire, Y., Vázquez, J.L.: Optimal existence and uniqueness theory for the fractional heat equation. Nonlinear Anal. 153, 142–168 (2017)
Bhowmik, M., Pusti, S.: An extension problem and Hardy’s inequality for the fractional Laplace-Beltrami operator on Riemannian symmetric spaces of noncompact type. J. Funct. Anal. 282(9), (2022)
Blumenthal, R.M., Getoor, R.K.: Some Theorems on Stable Processes. Trans. Amer. Math. Soc. 95, 263–273 (1960)
Bogdan, K., Jakubowski, T.: Estimates of the heat kernel of fractional Laplacian perturbed by gradient operators. Commun. Math. Phys. 271(1), 179–198 (2007)
Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32(7–9), 1245–1260 (2007)
Chen, X., Hassell, A.: The heat kernel on asymptotically hyperbolic manifolds. Commun. Partial Differ. Equ. 45, 1031–1071 (2020)
Chen, Z.-Q., Kumagai, T.: Heat kernel estimates for stable-like processes on \(d\)-sets. Stoch. Proc. Their Appl. 108(1), 27–62 (2003)
Chen, Z.-Q., Kumagai, T.: Heat kernel estimates for jump processes of mixed types on metric measure spaces. Probab. Theory Related Fields 140, 277–317 (2008)
Cowling, M.G., Giulini, S., Meda, S.: \(L^p-L^q\) estimates for functions of the Laplace-Beltrami operator on noncompact symmetric spaces II. J. Lie Theory 5, 1–14 (1995)
Davies, E., Mandouvalos, N.: Heat kernel bounds on hyperbolic space and Kleinian groups. Proc. London Math. Soc. 3(57), 182–208 (1988)
Dziubański, J., Preisner, M.: Hardy spaces for semigroups with Gaussian bounds. Ann. Mat. Pura Appl. 4(197), 965–987 (2018)
Fabes, E., Stroock, D.: A new proof of Moser’s parabolic Harnack inequality using the old ideas of Nash. Arch. Ration. Mech. Anal. 96, 327–338 (1986)
Getoor, R.K.: Infinitely divisible probabilities on the hyperbolic plane. Pacific J. Math. 11, 1287–1308 (1961)
Graczyk, P., Stos, A.: Transition density estimates for stable processes on symmetric spaces. Pacific J. Math. 217, 87–100 (2004)
Grigor’yan, A.: The heat equation on noncompact Riemannian manifolds. Mat. Sb. 182, 55–87 (1991)
Grigor’yan, A.: Heat kernel and analysis on manifolds. AMS International Press (2009)
Grigor’yan, A.: Upper bounds of derivatives of the heat kernel on an arbitrary complete manifold. J. Funct. Anal. 127, 363–389 (1995)
Grigor’yan, A., Papageorgiou, E., Zhang, H.-W.: Asymptotic behavior of the heat semigroup on certain Riemannian manifolds. To appear in "From Classical Analysis to Analysis on Fractals - The Robert Strichartz Memorial Volume", Springer, (2023). arXiv:2205.06105
Helgason, S.: Geometric Analysis on Symmetric Spaces, Mathematical Surveys and Monographs. Second Edition. Amer. Math. Soc. 637, (2008)
Li, P., Yau, S.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156, 153–201 (1986)
Meerschaert M., Sikorskii A.: Stochastic Models for Fractional Calculus. De Gruyter (2011)
Papageorgiou E.: Asymptotics for the infinite Brownian loop on noncompact symmetric spaces. Preprint (2023). arXiv:2301.09924
Saloff-Coste, L.: A note on Poincaré, Sobolev, and Harnack inequalities. Internat. Math. Res. Notices, 27–38 (1992)
Saloff-Coste, L.: Aspects of Sobolev-type inequalities. Cambridge University Press (2002)
Saloff-Coste, L.: The heat kernel and its estimates. In: Probabilistic Approach To Geometry, pp. 405–436. Adv. Stud. Pure Math. 57, Math. Soc. Japan, Tokyo (2010)
Shiozawa, Y.: Bottom crossing probability for symmetric jump processes. Math. Z. 287, 1355–1376 (2017)
Stinga, P.R.: User’s guide to the fractional Laplacian and the method of semigroups. In: Kochubei, Anatoly, Luchko, Yuri (eds.) Volume 2 Fractional Differential Equations, pp. 235–266. De Gruyter, Berlin, Boston (2019)
Stinga, P.R., Torrea, J.L.: Extension problem and Harnack’s inequality for some fractional operators. Commun. Partial Differ. 35(10–12), 2092–2122 (2012)
Strichartz, R.: Analysis of the Laplacian on the complete Riemannian manifold. J. Funct. Anal. 52, 48–79 (1983)
Vázquez J.L.: Asymptotic behaviour methods for the heat equation. Convergence to the Gaussian. Course Notes (2017). arXiv:1706.10034
Vázquez J.L.: Asymptotic behaviour for the fractional Heat equation in the Euclidean space. Complex Var. Elliptic Equ., Special volume in honor of Vladimir I. Smirnov’s 130th anniversary, 63 (7-8),1216–1231 (2018)
Vázquez, J.L.: Asymptotic behaviour for the heat equation in hyperbolic space. Comm. Anal. Geom. 30(9), 2123–2156 (2022)
Yosida, K.: Functional Analysis. Springer, Berlin (1980)
Acknowledgements
The author would like to thank the referee for useful comments which improved the presentation, and for pointing out the references [14] and [33]. This work was supported by the Hellenic Foundation for Research and Innovation, Project HFRI-FM17-1733. Currently the author is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)–SFB-Geschäftszeichen –Projektnummer SFB-TRR 358/1 2023 –491392403.
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Open Access funding enabled and organized by Projekt DEAL. Hellenic Foundation for Research and Innovation, Project HFRI-FM17-1733
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Papageorgiou, E. Large-Time Behavior of Two Families of Operators Related to the Fractional Laplacian on Certain Riemannian Manifolds. Potential Anal (2023). https://doi.org/10.1007/s11118-023-10109-1
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DOI: https://doi.org/10.1007/s11118-023-10109-1
Keywords
- Fractional laplacian
- Extension problem
- Fractional heat equation
- Asymptotic behavior
- Long-time convergence
- Noncompact symmetric spaces