\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

A spectral theoretical approach for hypocoercivity applied to some degenerate hypoelliptic, and non-local operators

  • * Corresponding author: Aditya Vaidyanathan

    * Corresponding author: Aditya Vaidyanathan
This work was partially supported by NSF Grant DMS-1406599 and a CNRS grant
Abstract Full Text(HTML) Related Papers Cited by
  • The aim of this paper is to offer an original and comprehensive spectral theoretical approach to the study of convergence to equilibrium, and in particular of the hypocoercivity phenomenon, for contraction semigroups in Hilbert spaces. Our approach rests on a commutation relationship for linear operators known as intertwining, and we utilize this identity to transfer spectral information from a known, reference semigroup $ \widetilde{{P}} = (e^{-t\widetilde{{\mathbf{A}}}})_{t \geqslant 0} $ to a target semigroup $ P $ which is the object of study. This allows us to obtain conditions under which $ P $ satisfies a hypocoercive estimate with exponential decay rate given by the spectral gap of $ \widetilde{{\mathbf{A}}} $. Along the way we also develop a functional calculus involving the non-self-adjoint resolution of identity induced by the intertwining relations. We apply these results in a general Hilbert space setting to two cases: degenerate, hypoelliptic Ornstein-Uhlenbeck semigroups on $ \mathbb{R}^d $, and non-local Jacobi semigroups on $ [0,1]^d $, which have been introduced and studied for $ d = 1 $ in [12]. In both cases we obtain hypocoercive estimates and are able to explicitly identify the hypocoercive constants.

    Mathematics Subject Classification: Primary: 58J51, 47G20; Secondary: 60J75.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] F. AchleitnerA. Arnold and E. A. Carlen, On multi-dimensional hypocoercive BGK models, Kinet. Relat. Models, 11 (2018), 953-1009.  doi: 10.3934/krm.2018038.
    [2] J.-P. Antoine and C. Trapani, Some remarks on quasi-{H}ermitian operators, J. Math. Phys., 55 (2014), 013503, 17pp. doi: 10.1063/1.4853815.
    [3] A. ArnoldA. Einav and T. Wöhrer, On the rates of decay to equilibrium in degenerate and defective Fokker-Planck equations, J. Differential Equations, 264 (2018), 6843-6872.  doi: 10.1016/j.jde.2018.01.052.
    [4] A. Arnold and J. Erb, Sharp entropy decay for hypocoercive and non-symmetric Fokker-Planck equations with linear drift, arXiv: 1409.5425 [math.AP], Sep 2014.
    [5] D. Bakry, I. Gentil and M. Ledoux, Analysis and geometry of Markov diffusion operators, volume 348 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Cham, 2014. doi: 10.1007/978-3-319-00227-9.
    [6] F. Baudoin, Bakry-Émery meet Villani, J. Funct. Anal., 273 (2017), 2275-2291.  doi: 10.1016/j.jfa.2017.06.021.
    [7] A. Ben-Israel and T. N. E. Greville, Generalized Inverses, volume 15 of CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer-Verlag, New York, second edition, 2003. Theory and applications.
    [8] Y. M. Berezansky, Z. G. Sheftel and G. F. Us, Functional Analysis. Vol. II, volume 86 of Operator Theory: Advances and Applications, Birkhäuser Verlag, Basel, 1996. Translated from the 1990 Russian original by Peter V. Malyshev.
    [9] V. I. Bogachev, Ornstein-{U}hlenbeck operators and semigroups, Uspekhi Mat. Nauk, 73 (2018), 3-74.  doi: 10.4213/rm9812.
    [10] E. Bouin, J. Dolbeault, S. Mischler, C. Mouhot and C. Schmeiser, Hypocoercivity without confinement, 2017.
    [11] C. Burnap and P. F. Zweifel, A note on the spectral theorem, Integral Equations Operator Theory, 9 (1986), 305-324.  doi: 10.1007/BF01199348.
    [12] P. Cheridito, P. Patie, A. Srapionyan and A. Vaidyanathan, On non-local ergodic Jacobi semigroups: Spectral theory, convergence-to-equilibrium, and contractivity, arXiv: 1905.07832 [math.PR], 2019.
    [13] M. C. H. Choi and P. Patie, Analysis of non-reversible Markov chains via similarity orbit, Combinatorics, Probability and Computing, 2020. doi: 10.1017/S0963548320000024.
    [14] M. C. H. Choi and P. Patie, Skip-free Markov chains, Trans. Amer. Math. Soc., 371 (2019), 7301-7342.  doi: 10.1090/tran/7773.
    [15] E. B. DaviesOne-parameter Semigroups, volume 15 of London Mathematical Society Monographs, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980. 
    [16] J. DolbeaultC. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. Amer. Math. Soc., 367 (2015), 3807-3828.  doi: 10.1090/S0002-9947-2015-06012-7.
    [17] J. Dolbeault and X. Li, $\varphi$-entropies: Convexity, coercivity and hypocoercivity for Fokker-Planck and kinetic Fokker-Planck equations, Math. Models Methods Appl. Sci., 28 (2018), 2637-2666.  doi: 10.1142/S0218202518500574.
    [18] N. Dunford, Spectral operators, Pacific J. Math., 4 (1954), 321-354.  doi: 10.2140/pjm.1954.4.321.
    [19] N. Dunford, A survey of the theory of spectral operators, Bull. Amer. Math. Soc., 64 (1958), 217-274.  doi: 10.1090/S0002-9904-1958-10219-0.
    [20] N. Dunford and J. T. Schwartz, Linear Operators. Part Ⅲ, Wiley Classics Library. John Wiley & Sons, Inc., New York, 1988. Spectral operators, With the assistance of William G. Bade and Robert G. Bartle, Reprint of the 1971 original, A Wiley-Interscience Publication.
    [21] K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, volume 194 of Graduate Texts in Mathematics., Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt.
    [22] S. Gadat and L. Miclo, Spectral decompositions and $\Bbb L^2$-operator norms of toy hypocoercive semi-groups, Kinet. Relat. Models, 6 (2013), 317-372.  doi: 10.3934/krm.2013.6.317.
    [23] M. Grothaus and P. Stilgenbauer, Hilbert space hypocoercivity for the Langevin dynamics revisited, Methods Funct. Anal. Topology, 22 (2016), 152-168. 
    [24] M. Grothaus and F.-Y. Wang, Weak Poincaré inequalities for convergence rate of degenerate diffusion processes, Ann. Probab., 47 (2019), 2930-2952.  doi: 10.1214/18-AOP1328.
    [25] M. P. Gualdani, S. Mischler and C. Mouhot, Factorization of non-symmetric operators and exponential {$H$}-theorem, Mém. Soc. Math. Fr. (N.S.), 153 (2017), 137pp.
    [26] T. LelièvreF. Nier and G. A. Pavliotis, Optimal non-reversible linear drift for the convergence to equilibrium of a diffusion, J. Stat. Phys., 152 (2013), 237-274.  doi: 10.1007/s10955-013-0769-x.
    [27] L. Lorenzi and M. Bertoldi, Analytical Methods for Markov Semigroups, volume 283 of Pure and Applied Mathematics (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2007.
    [28] A. Lunardi, On the Ornstein-Uhlenbeck operator in $L^2$ spaces with respect to invariant measures, Trans. Amer. Math. Soc., 349 (1997), 155-169.  doi: 10.1090/S0002-9947-97-01802-3.
    [29] G. Metafune, $L^p$-spectrum of Ornstein-{U}hlenbeck operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 30 (2001), 97-124. 
    [30] G. MetafuneD. Pallara and E. Priola, Spectrum of Ornstein-Uhlenbeck operators in $L^p$ spaces with respect to invariant measures, J. Funct. Anal., 196 (2002), 40-60.  doi: 10.1006/jfan.2002.3978.
    [31] G. MetafuneJ. PrüssA. Rhandi and R. Schnaubelt, The domain of the Ornstein-Uhlenbeck operator on an $L^p$-space with invariant measure, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 1 (2002), 471-485. 
    [32] L. Miclo, On hyperboundedness and spectrum of {M}arkov operators, Invent. Math., 200 (2015), 311-343.  doi: 10.1007/s00222-014-0538-8.
    [33] L. Miclo and P. Patie, On a gateway between continuous and discrete Bessel and Laguerre processes, Annales Henri Lebesgue, 2 (2019), 59-98.  doi: 10.5802/ahl.13.
    [34] L. Miclo and P. Patie, On interweaving relations, arXiv: 1910.13709 [math.PR], 2019.
    [35] S. Mischler and C. Mouhot, Exponential stability of slowly decaying solutions to the kinetic-Fokker-Planck equation, Arch. Ration. Mech. Anal., 221 (2016), 677-723.  doi: 10.1007/s00205-016-0972-4.
    [36] P. Monmarché, Generalized $\Gamma$ calculus and application to interacting particles on a graph, Potential Anal., 50 (2019), 439-466.  doi: 10.1007/s11118-018-9689-3.
    [37] M. OttobreG. A. Pavliotis and K. Pravda-Starov, Some remarks on degenerate hypoelliptic Ornstein-Uhlenbeck operators, J. Math. Anal. Appl., 429 (2015), 676-712.  doi: 10.1016/j.jmaa.2015.04.019.
    [38] P. Patie and M. Savov, Spectral expansion of non-self-adjoint generalized Laguerre semigroups, Mem. Amer. Math. Soc., 2019,179pp.
    [39] P. PatieM. Savov and Y. Zhao, Intertwining, excursion theory and {K}rein theory of strings for non-self-adjoint {M}arkov semigroups, Ann. Probab., 47 (2019), 3231-3277.  doi: 10.1214/19-AOP1338.
    [40] W. Rudin, Functional Analysis, International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, second edition, 1991.
    [41] R. L. Schilling, R. Song and Z. Vondraček, Bernstein Functions, volume 37 of De Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, second edition, 2012. Theory and applications. doi: 10.1515/9783110269338.
    [42] B. Sz.-Nagy, C. Foias, H. Bercovici and L. Kérchy, Harmonic Analysis of Operators on Hilbert Space, Universitext. Springer, New York, enlarged edition, 2010. doi: 10.1007/978-1-4419-6094-8.
    [43] D. Talay, Stochastic Hamiltonian systems: Exponential convergence to the invariant measure, and discretization by the implicit Euler scheme, Markov Process. Related Fields, 8 (2002), 163–198. Inhomogeneous random systems (Cergy-Pontoise, 2001).
    [44] G. Teschl, Mathematical Methods in Quantum Mechanics, volume 157 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, second edition, 2014. With applications to Schrödinger operators.
    [45] C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), iv+141pp. doi: 10.1090/S0065-9266-09-00567-5.
  • 加载中
SHARE

Article Metrics

HTML views(1704) PDF downloads(258) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return