Abstract
We study random perturbations of a Riemannian manifold \((\textsf{M},\textsf{g})\) by means of so-called Fractional Gaussian Fields, which are defined intrinsically by the given manifold. The fields \(h^\bullet : \omega \mapsto h^\omega \) will act on the manifold via the conformal transformation \(\textsf{g}\mapsto \textsf{g}^\omega := e^{2h^\omega }\,\textsf{g}\). Our focus will be on the regular case with Hurst parameter \(H>0\), the critical case \(H=0\) being the celebrated Liouville geometry in two dimensions. We want to understand how basic geometric and functional-analytic quantities like diameter, volume, heat kernel, Brownian motion, spectral bound, or spectral gap change under the influence of the noise. And if so, is it possible to quantify these dependencies in terms of key parameters of the noise? Another goal is to define and analyze in detail the Fractional Gaussian Fields on a general Riemannian manifold, a fascinating object of independent interest.
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Abramowitz, M., Stegun, I. A.: Handbook of mathematical functions. With Formulas, Graphs, and Mathematical Tables. Courier Corp. (1972)
Albeverio, S., Brasche, J., Röckner, M.: Dirichlet forms and generalized Schrödinger operators. In Holden, H., Jensen, A., (eds.) Schrödinger Operators – Proceedings of the Nordic Summer School in Mathematics – Sandbjerg Slot, Sønderborg, Denmark, August 1-12, 1988, volume 345 of Lecture Notes in Physics, pages 1–42. Springer-Verlag (1989)
Andres, S., Kajino, N.: Continuity and estimates of the Liouville heat kernel with applications to spectral dimensions. Probab. Theory Relat. Fields 166, 713–752 (2016)
Aubin, T.: Espaces de Sobolev sur les Variétés Riemanniennes. Bull. Sc. Math. 100, 149–173 (1976)
Barlow, M. T., Chen, Z.-Q., Murugan, M.: Stability of EHI and regularity of MMD spaces. (2022). arXiv:2008.05152v2
Baudoin, F., Lacaux, C.: Fractional Gaussian fields on the Sierpinski gasket and related fractals. (2020). arXiv:2003.04408
Berestycki, N.: Diffusion in planar Liouville quantum gravity. Ann. I. H. Poincaré B 51(3), 947–964 (2015)
Bogachev, V. I.: Gaussian measures, volume 62 of Mathematical Surveys and Monographs. Amer. Math. Soc. (1998)
Chavel, I.: Riemannian geometry, volume 98 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition. A modern introduction (2006)
Croke, C.: Some isoperimetric inequalities and eigenvalue estimates. Ann. Sci. Ecole Norm. Sup. 13(4), 419–435 (1980)
Dang, N. V.: Wick squares of the Gaussian Free Field and Riemannian rigidity (2019). arXiv:1902.07315
Dautray, R., Lions, J.-L.: Mathematical Analysis and Numerical Methods for Science and Technology - Voll. Springer-Verlag, I-V (1990)
David, F., Kupiainen, A., Rhodes, R., Vargas, V.: Liouville quantum gravity on the Riemann sphere. Commun. Math. Phys. 342(3), 869–907 (2016)
Davies, E. B.: Heat Kernels and Spectral Theory. Cambridge University Press (1989)
Dello Schiavo, L.: Ergodic decomposition of Dirichlet forms via direct integrals and applications. Potential Anal. 43 (2021)
Dello Schiavo, L., Herry, R., Kopfer, E., Sturm, K.-T.: Conformally Invariant Random Fields, Quantum Liouville Measures, and Random Paneitz Operators on Riemannian Manifolds of Even Dimension (2022). arXiv:2105.13925
Dixmier, J.: Von Neumann Algebras. North-Holland (1981)
Duplantier, B., Miller, J., Sheffield, S.: Liouville quantum gravity as a mating of trees. Astérisque, In press (2021+)
Eberle, A.: Girsanov-type transformations of local Dirichlet forms: An analytic approach. Osaka J. Math. 33(2), 497–531 (1996)
Fitzsimmons, P.J.: Absolute continuity of symmetric diffusions. Ann. Probab. 25(1), 230–258 (1997)
Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet forms and symmetric Markov processes, volume 19 of De Gruyter Studies in Mathematics. de Gruyter, extended edition, (2011)
Garban, C., Rhodes, R., Vargas, V.: On the heat kernel and the Dirichlet form of Liouville Brownian motion. Electron. J. Probab. 19(96), 1–25 (2014)
Garban, C., Rhodes, R., Vargas, V.: Liouville Brownian motion. Ann. Probab. 44(4), 3076–3110 (2016)
Gelbaum, Z.A.: Fractional Brownian fields over manifolds. Trans. Amer. Math. Soc. 366(9), 4781–4814 (2014)
Gilbarg D., Trudinger, N. S.: Elliptic partial differential equations of second order. reprint of the 1998 edition, Classics in Mathematics. Springer-Verlag, Berlin (2001)
Gradshteyn, I.S., Ryzhik, I.M.: Table of integrals, series, and products. Elsevier/Academic Press, Amsterdam, seventh edition (2007)
Grigor’yan, A., Noguchi, M.: The heat kernel on hyperbolic space. Bull. Lond. Math. Soc. 30, 643–650 (1998)
Große, N., Schneider, C.: Sobolev spaces on Riemannian manifolds with bounded geometry: General coordinates and traces. Math. Nachr. 286(16), 1586–1613 (2013)
Grosswald, E.: Bessel Polynomials, volume 698 of Lecture Notes in Mathematics. Springer-Verlag (1978)
Grothendieck, A.: Produits tensoriels topologiques et espaces nucléaires. Mem. Am. Math. Soc. 16 (1955)
Guillarmou, C., Rhodes, R., Vargas, V.: Polyakov’s formulation of \(2d\) bosonic string theory. Publ. Math. IHES 130, 111–185 (2019)
Han, B.-X., Sturm, K.-T.: Curvature-dimension conditions under time change. Ann. Matem. Pura Appl. 201(2), 801–822 (2021)
Hebey, E.: Sobolev spaces on Riemannian manifolds. Springer-Verlag (1996)
Kimura, Y., Okamoto, H.: Vortex motion on a sphere. J. Phys. Soc. Japan 56(12), 4203–4206 (1987)
Kuo, H.-H.: White Noise Distribution Theory. CRC Press, Probability and Stochastics Series (1996)
Ledoux, M., Talagrand, M.: Probability in banach spaces: isoperimetry and processes, volume 23 of ergebnisse der mathematik und ihrer grenzgebiete. 3. Folge – A Series of Modern Surveys in Mathematics. Springer (1991)
Li, J.: Gradient estimate for the heat kernel of a complete riemannian manifold and Its applications. J. Funct. Anal. 97, 293–310 (1991)
Li, P., Yau, S.-T.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156, 153–201 (1986)
Li, L., Zhang, Z.: On Li-Yau heat kernel estimate. Acta Math. Sinica 37(8), 1205–1218 (2021)
Lodhia, A., Sheffield, S., Sun, X., Watson, S.S.: Fractional Gaussian fields: a survey. Probab. Surv. 13, 1–56 (2016)
Lawson, H.B., Jr., Michelsohn, M.-L.: Spin Geometry. Princeton University Press, Princeton Mathematical Series (1989)
Le Gall, J.-F.: Brownian geometry. Jpn. J. Math. 14(2), 135–174 (2019)
Ma, Z.-M., Röckner, M.: Introduction to the theory of (Non-Symmetric) Dirichlet forms. Graduate Studies in Mathematics. Springer (1992)
Miller, J., Sheffield, S.: Liouville quantum gravity and the Brownian map I: the QLE(8/3,0) metric. Invent. Math. 219(1), 75–152 (2020)
Minakshisundaram, S., Pleijel, Å.: Some properties of the Eigenfunctions of the Laplace-Operator on Riemannian manifolds. Can. J. Math. 1(3), 242–256 (1949)
Müller, O., Nardmann, M.: Every conformal class contains a metric of bounded geometry. Math. Ann. 363(1–2), 143–174 (2015)
Norris, J.R.: Heat kernel asymptotics and the distance function in Lipschitz Riemannian manifolds. Acta Math. 179, 79–103 (1997)
Petersen, P.: Riemannian Geometry. Graduate Texts in Mathematics 171. Springer (2006)
Revuz, D., Yor, M.: Continuous Martingales and Brownian motion. Grundlehren der mathematischen Wissenschaften 293. Springer (1991)
Saloff-Coste, L.: A note on Poincaré, Sobolev, and Harnack inequalities. Int. Math. Res. Not. 2, 27–38 (1992)
Saloff-Coste, L.: Uniformly elliptic operators on Riemannian manifolds. J. Differ. Geom. 36, 417–450 (1992)
Schramm, O., Sheffield, S.: A contour line of the continuum Gaussian free field. Probab. Theory Relat. Fields 157, 47–80 (2013)
Schwartz, L.: Radon measures on arbitrary topological spaces and cylindrical measures. Tata Institute of Fundamental Research Studies in Mathematics. Oxford University Press (1973)
Sheffield, S.: Gaussian free fields for mathematicians. Probab. Theory Relat. Fields 139, 521–541 (2007)
Souplet, P., Zhang, Q.: Sharp gradient estimate and Yau’s Liouville theorem for the heat equation on noncompact manifolds. Bull. London Math. Soc. 38(6), 1045–1053 (2006)
Strichartz, R.S.: Analysis of the Laplacian on the complete Riemannian manifold. J. Funct. Anal. 52, 48–79 (1983)
Stroock, D.W., Turetsky, J.: Upper bounds on derivatives of the logarithm of the heat kernel. Comm. Anal. Geom. 6(4), 669–685 (1998)
Sturm, K.-T.: Analysis on local Dirichlet spaces III. The Parabolic Harnack Inequality. J. Math. Pures Appl. 75, 273–297 (1996)
Trèves, F.: Topological Vector Spaces, Distributions and Kernels, volume 25 of Pure and Applied Mathematics. Academic Press (1967)
Triebel, H.: Theory of Function Spaces – Volume II, volume 84 of Monographs in Mathematics. Birkhäuser (1992)
Vakhania, N. N., Tarieladze, V. I., Chobanyan, S. A.: Probability Distributions on Banach Spaces, volume 14 of Mathematics and its Applications (Soviet Series). D. Reidel Publishing Co., Dordrecht (1987)
Watson, G. A.: Treatise on the Theory of Bessel Functions. Cambridge University Press, 2nd edition (1944)
Acknowledgements
The authors would like to thank Matthias Erbar and Ronan Herry for valuable discussions on this project. They are also grateful to Nathanaël Berestycki, and Fabrice Baudoin for respectively pointing out the references [7], and [6, 24], and to Julien Fageot and Thomas Letendre for pointing out a mistake in a previous version of the proof of Proposition 3.10. The authors feel very much indebted to an anonymous reviewer for his/her careful reading and the many valuable suggestions that have significantly contributed to the improvement of the paper. L.D.S. gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft through CRC 1060 as well as through SPP 2265, and by the Austrian Science Fund (FWF) grant F65 at Institute of Science and Technology Austria. This research was funded in whole or in part by the Austrian Science Fund (FWF) ESPRIT 208. For the purpose of open access, the authors have applied a CC BY public copyright licence to any Author Accepted Manuscript version arising from this submission. E.K. and K.-T.S. gratefully acknowledge funding by the Deutsche Forschungsgemeinschaft through the Hausdorff Center for Mathematics and through CRC 1060 as well as through SPP 2265
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Dello Schiavo, L., Kopfer, E. & Sturm, KT. A Discovery Tour in Random Riemannian Geometry. Potential Anal (2024). https://doi.org/10.1007/s11118-023-10118-0
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DOI: https://doi.org/10.1007/s11118-023-10118-0
Keywords
- Fractional gaussian fields
- Gaussian free field
- Random geometry
- Liouville quantum gravity
- Liouville brownian motion
- Spectral gap estimates.