Nous donnons une condition nécessaire et suffisante pour la convergence uniforme de séries aléatoires de fonctions propres sur une variété riemannienne compacte sans bord. Comme conséquence, nous généralisons une estimation de Burq et Lebeau concernant les bornes d’une fonction propre aléatoire. Finalement, nous prouvons que nos résultats sont universels par rapport aux variables aléatoires utilisées (il s’agit d’un analogue riemannien d’un résultat de Marcus et Pisier), par rapport aux sous-variétés compactes et à la structure riemannienne de la variété compacte. Nos preuves reposent sur plusieurs outils dont le théorème de Dudley-Fernique, le théorème de comparaison de Slepian et un calcul fonctionnel semi-classique pour les opérateurs elliptiques d’une variété compacte.
We give a necessary and sufficient condition for the uniform convergence of random series of eigenfunctions on a boundaryless compact Riemannian manifold. As a consequence, we generalize an estimate of Burq and Lebeau about the supremum of a random eigenfunction. Finally, we prove that our results are universal with respect to the random variables (this is a Riemannian analogue of a result of Marcus and Pisier), with respect to compact submanifolds and with respect to the Riemannian structure of the manifold. Our proofs rely on several tools like the Dudley-Fernique theorem, the Slepian comparison theorem and the semi-classical functional calculus for elliptic operators on compact manifolds.
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@article{JEP_2022__9__747_0,
author = {Rafik Imekraz},
title = {A necessary and sufficient condition for probabilistic continuity on a boundaryless compact {Riemannian} manifold},
journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
pages = {747--805},
publisher = {\'Ecole polytechnique},
volume = {9},
year = {2022},
doi = {10.5802/jep.194},
language = {en},
url = {https://jep.centre-mersenne.org/articles/10.5802/jep.194/}
}
TY - JOUR AU - Rafik Imekraz TI - A necessary and sufficient condition for probabilistic continuity on a boundaryless compact Riemannian manifold JO - Journal de l’École polytechnique — Mathématiques PY - 2022 SP - 747 EP - 805 VL - 9 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.194/ DO - 10.5802/jep.194 LA - en ID - JEP_2022__9__747_0 ER -
%0 Journal Article %A Rafik Imekraz %T A necessary and sufficient condition for probabilistic continuity on a boundaryless compact Riemannian manifold %J Journal de l’École polytechnique — Mathématiques %D 2022 %P 747-805 %V 9 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.194/ %R 10.5802/jep.194 %G en %F JEP_2022__9__747_0
Rafik Imekraz. A necessary and sufficient condition for probabilistic continuity on a boundaryless compact Riemannian manifold. Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 747-805. doi : 10.5802/jep.194. https://jep.centre-mersenne.org/articles/10.5802/jep.194/
[AT08] - “ properties for Gaussian random series”, Trans. Amer. Math. Soc. 360 (2008) no. 8, p. 4425-4439 | DOI | MR | Zbl
[BGT04] - “Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds”, Amer. J. Math. 126 (2004) no. 3, p. 569-605 | DOI | Zbl
[Bin04] - “Derivatives of the spectral function and Sobolev norms of eigenfunctions on a closed Riemannian manifold”, Ann. Global Anal. Geom. 26 (2004) no. 3, p. 231-252 | DOI | MR | Zbl
[BL13] - “Injections de Sobolev probabilistes et applications”, Ann. Sci. École Norm. Sup. (4) 46 (2013) no. 6, p. 917-962 | DOI | Numdam | MR | Zbl
[BL14] - “Probabilistic Sobolev embeddings, applications to eigenfunctions estimates”, in Geometric and spectral analysis, Contemp. Math., vol. 630, American Mathematical Society, Providence, RI, 2014, p. 307-318 | DOI | MR | Zbl
[CE08] - Comparison theorems in Riemannian geometry, AMS Chelsea Publishing, Providence, RI, 2008, Revised reprint of the 1975 original | DOI
[CH15a] - “Fixed frequency eigenfunction immersions and supremum norms of random waves”, Electron. Res. Announc. Math. Sci. 22 (2015), p. 76-86 | DOI | MR | Zbl
[CH15b] - “Scaling limit for the kernel of the spectral projector and remainder estimates in the pointwise Weyl law”, Anal. PDE 8 (2015) no. 7, p. 1707-1731 | DOI | MR | Zbl
[CH18] - “ scaling asymptotics for the spectral projector of the Laplacian”, J. Geom. Anal. 28 (2018) no. 1, p. 111-122 | DOI | MR | Zbl
[Die72] - Treatise on analysis. Vol. III, Pure and Applied Math., vol. 10-III, Academic Press, New York-London, 1972
[Dud67] - “The sizes of compact subsets of Hilbert space and continuity of Gaussian processes”, J. Functional Analysis 1 (1967), p. 290-330 | DOI | MR | Zbl
[Dud14] - Uniform central limit theorems, Cambridge Studies in Advanced Mathematics, vol. 142, Cambridge University Press, New York, 2014
[Far08] - Analysis on Lie groups. An introduction, Cambridge Studies in Advanced Math., vol. 110, Cambridge University Press, Cambridge, 2008 | DOI
[FTR67] - “A theorem on random Fourier series on noncommutative groups”, Pacific J. Math. 21 (1967) no. 3, p. 487-492 | DOI | MR | Zbl
[Gri10] - “Almost sure convergence of some random series”, C. R. Acad. Sci. Paris 348 (2010) no. 3-4, p. 155-159 | DOI | MR | Zbl
[Hör68] - “The spectral function of an elliptic operator”, Acta Math. 121 (1968), p. 193-218 | DOI | MR | Zbl
[Ime18] - “Concentration et randomisation universelle de sous-espaces propres”, Anal. PDE 11 (2018) no. 2, p. 263-350 | DOI | MR | Zbl
[Ime19] - “Multidimensional Paley-Zygmund theorems and sharp estimates for some elliptic operators”, Ann. Inst. Fourier (Grenoble) 69 (2019) no. 6, p. 2723-2809 | DOI | Numdam | MR | Zbl
[IRT16] - “On random Hermite series”, Trans. Amer. Math. Soc. 368 (2016) no. 4, p. 2763-2792 | DOI | MR | Zbl
[Kah85] - Some random series of functions, Cambridge Studies in Advanced Math., vol. 5, Cambridge University Press, Cambridge, 1985
[Ler10] - Metrics on the phase space and non-selfadjoint pseudo-differential operators, Pseudo-Differential Operators. Theory and Appl., vol. 3, Birkhäuser Verlag, Basel, 2010 | DOI | MR
[LQ18a] - Introduction to Banach spaces: analysis and probability. Vol. 1, Cambridge Studies in Advanced Math., vol. 166, Cambridge University Press, Cambridge, 2018
[LQ18b] - Introduction to Banach spaces: analysis and probability. Vol. 2, Cambridge Studies in Advanced Math., vol. 167, Cambridge University Press, Cambridge, 2018
[LT79] - Classical Banach spaces. II, Ergeb. Math. Grenzgeb. (3), vol. 97, Springer-Verlag, Berlin-New York, 1979, Function spaces
[LT91] - Probability in Banach spaces. Isoperimetry and processes, Ergeb. Math. Grenzgeb. (3), vol. 23, Springer-Verlag, Berlin, 1991 | DOI
[Mar73] - “Continuity of Gaussian processes and random Fourier series”, Ann. Probab. 1 (1973) no. 6, p. 968-981 | DOI | MR | Zbl
[Mar75] - “Uniform convergence of random Fourier series”, Ark. Mat. 13 (1975) no. 1, p. 107-122 | DOI | MR | Zbl
[Mau74] - “Type et cotype dans les espaces munis de structures locales inconditionnelles”, in Espaces , applications radonifiantes et géométrie des espaces de Banach, Séminaire Maurey-Schwartz, Ecole Polytechnique, 1974, Exp. nos. 24 & 25 | Numdam | Zbl
[MP81] - Random Fourier series with applications to harmonic analysis, Annals of Math. Studies, vol. 101, Princeton University Press, Princeton, NJ, 1981
[MS70] - “Continuity of Gaussian processes”, Trans. Amer. Math. Soc. 151 (1970), p. 377-391 | DOI | MR | Zbl
[Pis78] - “Sur l’espace de Banach des séries de Fourier aléatoires presque sûrement continues”, in Séminaire sur la géométrie des espaces de Banach (1977–1978), École Polytechnique, Palaiseau, 1978, Exp. no. 17–18, 33p. | Zbl
[Pis89] - The volume of convex bodies and Banach space geometry, Cambridge Tracts in Math., vol. 94, Cambridge University Press, Cambridge, 1989 | DOI
[PZ30] - “On some series of functions. I, II”, Proc. Camb. Phil. Soc. 26 (1930), p. 337-357, 458–474 | DOI
[Saf88] - “Asymptotic of the spectral function of a positive elliptic operator without the nontrap condition”, Funct. Anal. Appl. 22 (1988) no. 3, p. 213-223 | DOI | Zbl
[Sch42] - “Positive definite functions on spheres”, Duke Math. J. 9 (1942), p. 96-108 | DOI | MR | Zbl
[Shu01] - Pseudodifferential operators and spectral theory, Springer-Verlag, Berlin, 2001 | DOI
[Sog17] - Fourier integrals in classical analysis, Cambridge Tracts in Math., vol. 210, Cambridge University Press, Cambridge, 2017 | DOI
[SW71] - Introduction to Fourier analysis on Euclidean spaces, Princeton Math. Series, vol. 32, Princeton University Press, Princeton, NJ, 1971
[SZ54] - “Some properties of trigonometric series whose terms have random signs”, Acta Math. 91 (1954) no. 1, p. 245-301 | DOI | MR | Zbl
[Sze75] - Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. XXIII, American Mathematical Society, Providence, RI, 1975
[Tzv10] - “Riemannian analogue of a Paley-Zygmund theorem”, in Séminaire Équations aux Dérivées Partielles (2008–2009), École Polytechnique, Palaiseau, 2010, Exp. no. XV, 14p. | MR | Zbl
[Zel09] - “Real and complex zeros of Riemannian random waves”, in Spectral analysis in geometry and number theory, Contemp. Math., vol. 484, American Mathematical Society, Providence, RI, 2009, p. 321-342 | DOI | MR | Zbl
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