On non-local ergodic Jacobi semigroups: spectral theory, convergence-to-equilibrium and contractivity
[Sur les semi-groupes de Jacobi ergodiques et non locaux : théorie spectrale, convergence vers l’équilibre et contractivité]
Journal de l’École polytechnique — Mathématiques, Tome 8 (2021) , pp. 331-378.

Dans cet article, nous introduisons et étudions des opérateurs de Jacobi non locaux, qui généralisent les opérateurs de Jacobi classiques (locaux). Nous montrons que ces opérateurs s’étendent aux générateurs de semi-groupes de Markov ergodiques avec des mesures de probabilité invariantes uniques et étudions leurs propriétés spectrales et de convergence. En particulier, nous dérivons un développement en série du semi-groupe en termes de polynômes explicitement définis, qui généralisent les polynômes orthogonaux de Jacobi classiques. De plus, nous donnons une caractérisation complète du spectre du générateur et du semi-groupe non auto-adjoint. Nous montrons que la convergence de la variance du semi-groupe est hypocoercive avec des constantes explicites, ce qui fournit une généralisation naturelle de l’estimation donnée par le trou spectral. Après un temps de préchauffage aléatoire, le semi-groupe décroît également de manière exponentielle en entropie et est à la fois hypercontractif et ultracontractif. Nos preuves s’articulent autour du développement d’identités de commutation, appelées relations d’entrelacement, entre opérateurs et semi-groupes de Jacobi locaux et non locaux, les objets locaux servant de points de référence pour le transfert de propriétés du cas local au cas non local.

In this paper, we introduce and study non-local Jacobi operators, which generalize the classical (local) Jacobi operators. We show that these operators extend to generators of ergodic Markov semigroups with unique invariant probability measures and study their spectral and convergence properties. In particular, we derive a series expansion of the semigroup in terms of explicitly defined polynomials, which generalize the classical Jacobi orthogonal polynomials. In addition, we give a complete characterization of the spectrum of the non-self-adjoint generator and semigroup. We show that the variance decay of the semigroup is hypocoercive with explicit constants, which provides a natural generalization of the spectral gap estimate. After a random warm-up time, the semigroup also decays exponentially in entropy and is both hypercontractive and ultracontractive. Our proofs hinge on the development of commutation identities, known as intertwining relations, between local and non-local Jacobi operators and semigroups, with the local objects serving as reference points for transferring properties from the local to the non-local case.

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/jep.148
Classification : 37A30,  47D06,  47G20,  60J75
Mots clés : Semi-groupes de Markov, théorie spectrale, opérateurs non auto-adjoints, convergence vers l’équilibre, hypercontractivité, ultracontractivité, estimations du noyau de la chaleur
@article{JEP_2021__8__331_0,
     author = {Patrick Cheridito and Pierre Patie and Anna Srapionyan and Aditya Vaidyanathan},
     title = {On non-local ergodic {Jacobi} semigroups: spectral theory, convergence-to-equilibrium and contractivity},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {331--378},
     publisher = {\'Ecole polytechnique},
     volume = {8},
     year = {2021},
     doi = {10.5802/jep.148},
     mrnumber = {4218161},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.148/}
}
Patrick Cheridito; Pierre Patie; Anna Srapionyan; Aditya Vaidyanathan. On non-local ergodic Jacobi semigroups: spectral theory, convergence-to-equilibrium and contractivity. Journal de l’École polytechnique — Mathématiques, Tome 8 (2021) , pp. 331-378. doi : 10.5802/jep.148. https://jep.centre-mersenne.org/articles/10.5802/jep.148/

[1] F. Achleitner, A. Arnold & E. A. Carlen - “On multi-dimensional hypocoercive BGK models”, Kinet. and Relat. Mod. 11 (2018) no. 4, p. 953-1009 | Article | MR 3810853 | Zbl 1405.82023

[2] C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto & G. Scheffer - Sur les inégalités de Sobolev logarithmiques, Panoramas & Synthèses, vol. 10, Société Mathématique de France, Paris, 2000 | Zbl 0982.46026

[3] D. Bakry - “Remarques sur les semigroupes de Jacobi”, in Hommage à P. A. Meyer et J. Neveu, Astérisque, vol. 236, Société Mathématique de France, Paris, 1996, p. 23-39 | Numdam | Zbl 0859.47026

[4] D. Bakry - “Étude des transformations de Riesz dans les variétés riemanniennes à courbure de Ricci minorée”, in Séminaire de Probabilités, XXI, Lect. Notes in Math., vol. 1247, Springer, Berlin, 1987, p. 137-172 | Article | Numdam | Zbl 0629.58018

[5] D. Bakry, I. Gentil & M. Ledoux - Analysis and geometry of Markov diffusion operators, Grundlehren Math. Wiss., vol. 348, Springer, Cham, 2014 | Article | MR 3155209 | Zbl 1376.60002

[6] F. Baudoin - “Bakry-Émery meet Villani”, J. Funct. Anal. 273 (2017) no. 7, p. 2275-2291 | Article | MR 3677826 | Zbl 1373.35061

[7] C. Berg & A. J. Durán - “A transformation from Hausdorff to Stieltjes moment sequences”, Ark. Mat. 42 (2004) no. 2, p. 239-257 | Article | MR 2101386 | Zbl 1057.44002

[8] J. Bertoin - “Subordinators: examples and applications”, in Lectures on probability theory and statistics (Saint-Flour, 1997), Lect. Notes in Math., vol. 1717, Springer, Berlin, 1999, p. 1-91 | Article | MR 1746300 | Zbl 0955.60046

[9] K. Bogdan, T. Byczkowski, T. Kulczycki, M. Ryznar, R. Song & Z. Vondraček - Potential analysis of stable processes and its extensions, Lect. Notes in Math., vol. 1980, Springer-Verlag, Berlin, 2009 | Article | MR 2569321

[10] A. Borodin & G. Olshanski - “Markov dynamics on the Thoma cone: a model of time-dependent determinantal processes with infinitely many particles”, Electron. J. Probab. 18 (2013), article ID 75, 43 pages | Article | MR 3091721 | Zbl 1293.60074

[11] A. N. Borodin & P. Salminen - Handbook of Brownian motion—facts and formulae, Probability and its Applications, Birkhäuser Verlag, Basel, 2002 | Article | Zbl 1012.60003

[12] B. Böttcher, R. Schilling & J. Wang - Lévy. III, Lect. Notes in Math., vol. 2099, Springer, Cham, 2013 | Article | Zbl 1384.60004

[13] B. L. J. Braaksma - “Asymptotic expansions and analytic continuations for a class of Barnes-integrals”, Compositio Math. 15 (1964), p. 239-341 (1964) | MR 167651 | Zbl 0129.28604

[14] D. Chafaï - “Entropies, convexity, and functional inequalities: on Φ-entropies and Φ-Sobolev inequalities”, J. Math. Kyoto Univ. 44 (2004) no. 2, p. 325-363 | Article | MR 2081075 | Zbl 1079.26009

[15] M. Chazal, A. Kyprianou & P. Patie - “A transformation for Lévy processes with one-sided jumps with applications”, 2010 | arXiv:1010.3819

[16] O. Christensen - An introduction to frames and Riesz bases, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, Cham, 2016 | Article | Zbl 1348.42033

[17] C. Cuchiero, M. Larsson & S. Svaluto-Ferro - “Polynomial jump-diffusions on the unit simplex”, Ann. Appl. Probab. 28 (2018) no. 4, p. 2451-2500 | Article | MR 3843834 | Zbl 1402.60105

[18] G. Da Prato - An introduction to infinite-dimensional analysis, Universitext, Springer-Verlag, Berlin, 2006 | Article | Zbl 1109.46001

[19] F. Delbaen & H. Shirakawa - “An Interest Rate Model with Upper and Lower Bounds”, Asia-Pacific Financial Markets 9 (2002) no. 3, p. 191-209 | Article | Zbl 1071.91020

[20] N. Demni & M. Zani - “Large deviations for statistics of the Jacobi process”, Stochastic Process. Appl. 119 (2009) no. 2, p. 518-533 | Article | MR 2494002 | Zbl 1158.60008

[21] J. Dolbeault, C. Mouhot & C. Schmeiser - “Hypocoercivity for linear kinetic equations conserving mass”, Trans. Amer. Math. Soc. 367 (2015) no. 6, p. 3807-3828 | Article | MR 3324910 | Zbl 1342.82115

[22] B. K. Driver - “Analysis tools with applications” (2003), http://www.math.ucsd.edu/~bdriver/231-02-03/Lecture_Notes/PDE-Anal-Book/analpde1.pdf

[23] E. B. Dynkin - Markov processes. Vols. I, II, Grundlehren Math. Wiss., vol. 121; 122, Academic Press Inc., New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1965 | Zbl 0132.37901

[24] K.-J. Engel & R. Nagel - One-parameter semigroups for linear evolution equations, Graduate Texts in Math., vol. 194, Springer-Verlag, New York, 2000 | MR 1721989 | Zbl 0952.47036

[25] S. N. Ethier & T. G. Kurtz - Markov processes, Wiley Series in Proba. and Math. Stat., John Wiley & Sons, Inc., New York, 1986 | Article

[26] É. Fontenas - “Sur les minorations des constantes de Sobolev et de Sobolev logarithmiques pour les opérateurs de Jacobi et de Laguerre”, in Séminaire de Probabilités, XXXII, Lect. Notes in Math., vol. 1686, Springer, Berlin, 1998, p. 14-29 | Article | Numdam | MR 1651226 | Zbl 0914.60047

[27] C. Gourieroux & J. Jasiak - “Multivariate Jacobi process with application to smooth transitions”, J. Econometrics 131 (2006) no. 1-2, p. 475-505 | Article | MR 2276008 | Zbl 1337.62365

[28] R. C. Griffiths, P. A. Jenkins & D. Spanó - “Wright-Fisher diffusion bridges”, Theoret. Population Biol. 122 (2018), p. 67-77 | Article | Zbl 1405.92173

[29] R. C. Griffiths & D. Spanó - “Diffusion processes and coalescent trees”, in Probability and mathematical genetics, London Math. Soc. Lecture Note Ser., vol. 378, Cambridge Univ. Press, Cambridge, 2010, p. 358-379 | Article | MR 2744247 | Zbl 1402.92297

[30] L. Gross - “Logarithmic Sobolev inequalities”, Amer. J. Math. 97 (1975) no. 4, p. 1061-1083 | Article | MR 420249

[31] T. Huillet - “On Wright-Fisher diffusion and its relatives”, J. Stat. Mech. Theory Exp. (2007) no. 11, article ID P11006, 30 pages | Zbl 07120350

[32] M. E. H. Ismail - Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Math. and its Appl., vol. 98, Cambridge University Press, Cambridge, 2005 | Article | MR 2191786 | Zbl 1082.42016

[33] A. E. Kyprianou - Fluctuations of Lévy processes with applications, Universitext, Springer, Heidelberg, 2014 | Article | Zbl 1384.60003

[34] L. Miclo & P. Patie - “On a gateway between continuous and discrete Bessel and Laguerre processes”, Ann. H. Lebesgue 2 (2019), p. 59-98 | Article | MR 3974488 | Zbl 07099975

[35] L. Miclo & P. Patie - “On interweaving relations”, J. Funct. Anal. 280 (2021) no. 3, p. 108816, 53 | Article | MR 4170787 | Zbl 07285374

[36] S. Mischler & C. Mouhot - “Exponential stability of slowly decaying solutions to the kinetic-Fokker-Planck equation”, Arch. Rational Mech. Anal. 221 (2016) no. 2, p. 677-723 | Article | MR 3488535 | Zbl 1338.35430

[37] O. P. Misra & J. L. Lavoine - Transform analysis of generalized functions, North-Holland Math. Studies, vol. 119, North-Holland Publishing Co., Amsterdam, 1986 | MR 832188 | Zbl 0583.46032

[38] S. Pal - “Wright-Fisher diffusion with negative mutation rates”, Ann. Probab. 41 (2013) no. 2, p. 503-526 | Article | MR 3077518 | Zbl 1268.60077

[39] P. Patie - “A refined factorization of the exponential law”, Bernoulli 17 (2011) no. 2, p. 814-826 | Article | MR 2787616 | Zbl 1253.60020

[40] P. Patie & M. Savov - “Bernstein-gamma functions and exponential functionals of Lévy processes”, Electron. J. Probab. 23 (2018), article ID 75, 101 pages | Article | Zbl 1396.30001

[41] P. Patie & M. Savov - Spectral expansion of non-self-adjoint generalized Laguerre semigroups, Mem. Amer. Math. Soc., American Mathematical Society, Providence, RI, 2019, to appear

[42] P. Patie, M. Savov & Y. Zhao - “Intertwining, excursion theory and Krein theory of strings for non-self-adjoint Markov semigroups”, Ann. Probab. 47 (2019) no. 5, p. 3231-3277 | Article | MR 4021250 | Zbl 1451.60088

[43] P. Patie & A. Vaidyanathan - “A spectral theoretical approach for hypocoercivity applied to some degenerate hypoelliptic, and non-local operators”, Kinet. and Relat. Mod. 13 (2020) no. 3, p. 479-506 | Article | MR 4097722 | Zbl 1447.58032

[44] A. Pazy - Semigroups of linear operators and applications to partial differential equations, Applied Math. Sciences, vol. 44, Springer-Verlag, New York, 1983 | Article | MR 710486 | Zbl 0516.47023

[45] J. M. Pearson - “Best constants in Sobolev inequalities for ultraspherical polynomials”, Arch. Rational Mech. Anal. 116 (1992) no. 4, p. 361-374 | Article | MR 1132767 | Zbl 0793.46014

[46] L. C. G. Rogers & J. W. Pitman - “Markov functions”, Ann. Probab. 9 (1981) no. 4, p. 573-582 | Article | MR 624684 | Zbl 0466.60070

[47] W. Rudin - Functional analysis, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973 | Zbl 0253.46001

[48] K.-i. Sato - Lévy processes and infinitely divisible distributions, Cambridge Studies in Advanced Math., vol. 68, Cambridge University Press, Cambridge, 1999 | Zbl 0973.60001

[49] R. L. Schilling, R. Song & Z. Vondraček - Bernstein functions, De Gruyter Studies in Math., vol. 37, Walter de Gruyter & Co., Berlin, 2012 | Article | MR 2978140 | Zbl 1257.33001

[50] G. Szegő - Orthogonal polynomials, American Math. Soc., Colloquium Publ., vol. XXIII, American Mathematical Society, Providence, RI, 1975 | Zbl 0305.42011

[51] C. Villani - Hypocoercivity, Mem. Amer. Math. Soc., vol. 202, no. 950, American Mathematical Society, Providence, RI, 2009 | Article