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June  2020, 13(3): 479-506. doi: 10.3934/krm.2020016

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

1. 

School of Operations Research and Information Engineering, Cornell University, Ithaca, NY, 14853, USA

2. 

Center for Applied Mathematics, Cornell University, Ithaca, NY, 14853, USA

* Corresponding author: Aditya Vaidyanathan

Received  May 2019 Revised  November 2019 Published  March 2020

Fund Project: This work was partially supported by NSF Grant DMS-1406599 and a CNRS grant

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.

Citation: Pierre Patie, Aditya Vaidyanathan. A spectral theoretical approach for hypocoercivity applied to some degenerate hypoelliptic, and non-local operators. Kinetic & Related Models, 2020, 13 (3) : 479-506. doi: 10.3934/krm.2020016
References:
[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[6]

F. Baudoin, Bakry-Émery meet Villani, J. Funct. Anal., 273 (2017), 2275-2291.  doi: 10.1016/j.jfa.2017.06.021.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

V. I. Bogachev, Ornstein-{U}hlenbeck operators and semigroups, Uspekhi Mat. Nauk, 73 (2018), 3-74.  doi: 10.4213/rm9812.  Google Scholar

[10]

E. Bouin, J. Dolbeault, S. Mischler, C. Mouhot and C. Schmeiser, Hypocoercivity without confinement, 2017. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15] E. B. Davies, One-parameter Semigroups, volume 15 of London Mathematical Society Monographs, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[18]

N. Dunford, Spectral operators, Pacific J. Math., 4 (1954), 321-354.  doi: 10.2140/pjm.1954.4.321.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

M. Grothaus and P. Stilgenbauer, Hilbert space hypocoercivity for the Langevin dynamics revisited, Methods Funct. Anal. Topology, 22 (2016), 152-168.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

G. Metafune, $L^p$-spectrum of Ornstein-{U}hlenbeck operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 30 (2001), 97-124.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[32]

L. Miclo, On hyperboundedness and spectrum of {M}arkov operators, Invent. Math., 200 (2015), 311-343.  doi: 10.1007/s00222-014-0538-8.  Google Scholar

[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.  Google Scholar

[34]

L. Miclo and P. Patie, On interweaving relations, arXiv: 1910.13709 [math.PR], 2019. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

P. Patie and M. Savov, Spectral expansion of non-self-adjoint generalized Laguerre semigroups, Mem. Amer. Math. Soc., 2019,179pp. Google Scholar

[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.  Google Scholar

[40]

W. Rudin, Functional Analysis, International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, second edition, 1991.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[45]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), iv+141pp. doi: 10.1090/S0065-9266-09-00567-5.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[6]

F. Baudoin, Bakry-Émery meet Villani, J. Funct. Anal., 273 (2017), 2275-2291.  doi: 10.1016/j.jfa.2017.06.021.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

V. I. Bogachev, Ornstein-{U}hlenbeck operators and semigroups, Uspekhi Mat. Nauk, 73 (2018), 3-74.  doi: 10.4213/rm9812.  Google Scholar

[10]

E. Bouin, J. Dolbeault, S. Mischler, C. Mouhot and C. Schmeiser, Hypocoercivity without confinement, 2017. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15] E. B. Davies, One-parameter Semigroups, volume 15 of London Mathematical Society Monographs, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[18]

N. Dunford, Spectral operators, Pacific J. Math., 4 (1954), 321-354.  doi: 10.2140/pjm.1954.4.321.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

M. Grothaus and P. Stilgenbauer, Hilbert space hypocoercivity for the Langevin dynamics revisited, Methods Funct. Anal. Topology, 22 (2016), 152-168.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

G. Metafune, $L^p$-spectrum of Ornstein-{U}hlenbeck operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 30 (2001), 97-124.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[32]

L. Miclo, On hyperboundedness and spectrum of {M}arkov operators, Invent. Math., 200 (2015), 311-343.  doi: 10.1007/s00222-014-0538-8.  Google Scholar

[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.  Google Scholar

[34]

L. Miclo and P. Patie, On interweaving relations, arXiv: 1910.13709 [math.PR], 2019. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

P. Patie and M. Savov, Spectral expansion of non-self-adjoint generalized Laguerre semigroups, Mem. Amer. Math. Soc., 2019,179pp. Google Scholar

[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.  Google Scholar

[40]

W. Rudin, Functional Analysis, International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, second edition, 1991.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[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.  Google Scholar

[45]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), iv+141pp. doi: 10.1090/S0065-9266-09-00567-5.  Google Scholar

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