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On two properties of the Fisher information
Université Grenoble-Alpes & CNRS, LPMMC (UMR 5493), B.P. 166, F-38042 Grenoble, France |
Alternative proofs for the superadditivity and the affinity (in the large system limit) of the usual and some fractional Fisher informations of a probability density of many variables are provided. They are consequences of the fact that such informations can be interpreted as quantum kinetic energies.
References:
[1] |
J. Bourgain, H. Brézis and P. Mironescu, Another look at Sobolev spaces, in Optimal control and Partial Differential equations, IOS Press, (2001), 439–455. |
[2] |
J. Bourgain, H. Brezis and P. Mironescu,
Limiting embedding theorems for $W^{s,p}$ when $s\uparrow 1$ and applications, J. Anal. Math., 87 (2002), 77-101.
doi: 10.1007/BF02868470. |
[3] |
E. A. Carlen,
Superadditivity of Fisher's information and logarithmic Sobolev inequalities, J. Funct. Anal., 101 (1991), 194-211.
doi: 10.1016/0022-1236(91)90155-X. |
[4] |
G. F. dell'Antonio,
On the limits of sequences of normal states, Comm. Pure Appl. Math., 20 (1967), 413-429.
doi: 10.1002/cpa.3160200209. |
[5] |
N. Fournier, M. Hauray and S. Mischler,
Propagation of chaos for the 2d viscous vortex model, J. Eur. Math. Soc., 16 (2014), 1423-1466.
doi: 10.4171/JEMS/465. |
[6] |
F. Golse, On the dynamics of large particle systems in the mean field limit, arXiv: 1301.5494, (2013)., Lecture notes for a course at the NDNS+ Applied Dynamical Systems Summer School "Macroscopic and large scale phenomena", Universiteit Twente, Enschede (The Netherlands). Google Scholar |
[7] |
M. Hauray, Limite de Champ Moyen et Propagation du Chaos Pour des Systèmes de Particules, Limites Gyro-cinétique et Quasi-neutre Pour Les Plasmas., Habilitation thesis, 2014. Google Scholar |
[8] |
M. Hauray and S. Mischler,
On Kac's chaos and related problems, J. Func. Anal., 266 (2014), 6055-6157.
doi: 10.1016/j.jfa.2014.02.030. |
[9] |
E. Hewitt and L. J. Savage,
Symmetric measures on Cartesian products, Trans. Amer. Math. Soc., 80 (1955), 470-501.
doi: 10.1090/S0002-9947-1955-0076206-8. |
[10] |
M. Hoffmann-Ostenhof and T. Hoffmann-Ostenhof,
Schrödinger inequalities and asymptotic behavior of the electron density of atoms and molecules, Phys. Rev. A, 16 (1977), 1782-1785.
doi: 10.1103/PhysRevA.16.1782. |
[11] |
R. L. Hudson and G. R. Moody,
Locally normal symmetric states and an analogue of de Finetti's theorem, Z. Wahrscheinlichkeitstheor. und Verw. Gebiete, 33 (1975/76), 343-351.
doi: 10.1007/BF00534784. |
[12] |
M. K.-H. Kiessling, The Hartree limit of Born's ensemble for the ground state of a bosonic atom or ion, J. Math. Phys., 53 (2012), 095223, 21 pp.
doi: 10.1063/1.4752475. |
[13] |
M. Lewin, Mean-Field limit of Bose systems: Rigorous results, arXiv: 1510.04407, Proceedings of the International Congress of Mathematical Physics, 2015 Google Scholar |
[14] |
M. Lewin, P. T. Nam and N. Rougerie,
Derivation of Hartree's theory for generic mean-field Bose systems, Adv. Math., 254 (2014), 570-621.
doi: 10.1016/j.aim.2013.12.010. |
[15] |
E. H. Lieb and M. Loss, Analysis, vol. 14 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2nd ed., 2001.
doi: 10.1090/gsm/014. |
[16] |
W. Masja and J. Nagel,
Über äquivalente normierung der anisotropen Funktionalraüme $H ^{\mu} ( { {\mathbb R} } ^n)$, Beiträge zur Analysis, 12 (1978), 7-17.
|
[17] |
V. Maz'ya and T. Shaposhnikova,
On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Func. Anal., 195 (2002), 230-238.
doi: 10.1006/jfan.2002.3955. |
[18] |
D. W. Robinson and D. Ruelle,
Mean entropy of states in classical statistical mechanics, Commun. Math. Phys., 5 (1967), 288-300.
doi: 10.1007/BF01646480. |
[19] |
N. Rougerie, De Finetti theorems, mean-field limits and Bose-Einstein condensation, arXiv: 1506.05263, 2014. LMU lecture notes. Google Scholar |
[20] |
——, Théorèmes de De Finetti, Limites de Champ Moyen et Condensation de Bose-Einstein, Les cours Peccot, Spartacus IDH, Paris, 2016., Cours Peccot, Collège de France : février-mars 2014. Google Scholar |
[21] |
S. Salem, Propagation of chaos for fractional Keller Segel equations in diffusion dominated and fair competition cases, Journal de Mathématiques Pures et Appliquées, 132 (2019), 79-132.
doi: 10.1016/j.matpur.2019.04.011. |
[22] |
S. Salem, Propagation of chaos for the Boltzmann equation with moderately soft potentials, arXiv: 1910.01883, 2019. Google Scholar |
[23] |
R. Schatten, Norm Ideals of Completely Continuous Operators, vol. 2 of Ergebnisse der Mathematik und ihrer Grenzgebiete, Folge, 1960. |
[24] |
B. Simon, Trace Ideals and Their Applications, vol. 35 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1979.
![]() |
[25] |
G. Toscani,
The fractional Fisher information and the central limit theorem for stable laws, Ric. Mat., 65 (2016), 71-91.
doi: 10.1007/s11587-015-0253-9. |
[26] |
G. Toscani,
The information-theoretic meaning of Gagliardo-Nirenberg type inequalities, Rend. Lincei Mat. Appl., 30 (2019), 237-253.
doi: 10.4171/RLM/845. |
[27] |
G. Toscani, Score functions, generalized relative Fisher information and applications, Ricerche mat., 66 (2017) 15–26.
doi: 10.1007/s11587-016-0281-0. |
show all references
References:
[1] |
J. Bourgain, H. Brézis and P. Mironescu, Another look at Sobolev spaces, in Optimal control and Partial Differential equations, IOS Press, (2001), 439–455. |
[2] |
J. Bourgain, H. Brezis and P. Mironescu,
Limiting embedding theorems for $W^{s,p}$ when $s\uparrow 1$ and applications, J. Anal. Math., 87 (2002), 77-101.
doi: 10.1007/BF02868470. |
[3] |
E. A. Carlen,
Superadditivity of Fisher's information and logarithmic Sobolev inequalities, J. Funct. Anal., 101 (1991), 194-211.
doi: 10.1016/0022-1236(91)90155-X. |
[4] |
G. F. dell'Antonio,
On the limits of sequences of normal states, Comm. Pure Appl. Math., 20 (1967), 413-429.
doi: 10.1002/cpa.3160200209. |
[5] |
N. Fournier, M. Hauray and S. Mischler,
Propagation of chaos for the 2d viscous vortex model, J. Eur. Math. Soc., 16 (2014), 1423-1466.
doi: 10.4171/JEMS/465. |
[6] |
F. Golse, On the dynamics of large particle systems in the mean field limit, arXiv: 1301.5494, (2013)., Lecture notes for a course at the NDNS+ Applied Dynamical Systems Summer School "Macroscopic and large scale phenomena", Universiteit Twente, Enschede (The Netherlands). Google Scholar |
[7] |
M. Hauray, Limite de Champ Moyen et Propagation du Chaos Pour des Systèmes de Particules, Limites Gyro-cinétique et Quasi-neutre Pour Les Plasmas., Habilitation thesis, 2014. Google Scholar |
[8] |
M. Hauray and S. Mischler,
On Kac's chaos and related problems, J. Func. Anal., 266 (2014), 6055-6157.
doi: 10.1016/j.jfa.2014.02.030. |
[9] |
E. Hewitt and L. J. Savage,
Symmetric measures on Cartesian products, Trans. Amer. Math. Soc., 80 (1955), 470-501.
doi: 10.1090/S0002-9947-1955-0076206-8. |
[10] |
M. Hoffmann-Ostenhof and T. Hoffmann-Ostenhof,
Schrödinger inequalities and asymptotic behavior of the electron density of atoms and molecules, Phys. Rev. A, 16 (1977), 1782-1785.
doi: 10.1103/PhysRevA.16.1782. |
[11] |
R. L. Hudson and G. R. Moody,
Locally normal symmetric states and an analogue of de Finetti's theorem, Z. Wahrscheinlichkeitstheor. und Verw. Gebiete, 33 (1975/76), 343-351.
doi: 10.1007/BF00534784. |
[12] |
M. K.-H. Kiessling, The Hartree limit of Born's ensemble for the ground state of a bosonic atom or ion, J. Math. Phys., 53 (2012), 095223, 21 pp.
doi: 10.1063/1.4752475. |
[13] |
M. Lewin, Mean-Field limit of Bose systems: Rigorous results, arXiv: 1510.04407, Proceedings of the International Congress of Mathematical Physics, 2015 Google Scholar |
[14] |
M. Lewin, P. T. Nam and N. Rougerie,
Derivation of Hartree's theory for generic mean-field Bose systems, Adv. Math., 254 (2014), 570-621.
doi: 10.1016/j.aim.2013.12.010. |
[15] |
E. H. Lieb and M. Loss, Analysis, vol. 14 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2nd ed., 2001.
doi: 10.1090/gsm/014. |
[16] |
W. Masja and J. Nagel,
Über äquivalente normierung der anisotropen Funktionalraüme $H ^{\mu} ( { {\mathbb R} } ^n)$, Beiträge zur Analysis, 12 (1978), 7-17.
|
[17] |
V. Maz'ya and T. Shaposhnikova,
On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Func. Anal., 195 (2002), 230-238.
doi: 10.1006/jfan.2002.3955. |
[18] |
D. W. Robinson and D. Ruelle,
Mean entropy of states in classical statistical mechanics, Commun. Math. Phys., 5 (1967), 288-300.
doi: 10.1007/BF01646480. |
[19] |
N. Rougerie, De Finetti theorems, mean-field limits and Bose-Einstein condensation, arXiv: 1506.05263, 2014. LMU lecture notes. Google Scholar |
[20] |
——, Théorèmes de De Finetti, Limites de Champ Moyen et Condensation de Bose-Einstein, Les cours Peccot, Spartacus IDH, Paris, 2016., Cours Peccot, Collège de France : février-mars 2014. Google Scholar |
[21] |
S. Salem, Propagation of chaos for fractional Keller Segel equations in diffusion dominated and fair competition cases, Journal de Mathématiques Pures et Appliquées, 132 (2019), 79-132.
doi: 10.1016/j.matpur.2019.04.011. |
[22] |
S. Salem, Propagation of chaos for the Boltzmann equation with moderately soft potentials, arXiv: 1910.01883, 2019. Google Scholar |
[23] |
R. Schatten, Norm Ideals of Completely Continuous Operators, vol. 2 of Ergebnisse der Mathematik und ihrer Grenzgebiete, Folge, 1960. |
[24] |
B. Simon, Trace Ideals and Their Applications, vol. 35 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1979.
![]() |
[25] |
G. Toscani,
The fractional Fisher information and the central limit theorem for stable laws, Ric. Mat., 65 (2016), 71-91.
doi: 10.1007/s11587-015-0253-9. |
[26] |
G. Toscani,
The information-theoretic meaning of Gagliardo-Nirenberg type inequalities, Rend. Lincei Mat. Appl., 30 (2019), 237-253.
doi: 10.4171/RLM/845. |
[27] |
G. Toscani, Score functions, generalized relative Fisher information and applications, Ricerche mat., 66 (2017) 15–26.
doi: 10.1007/s11587-016-0281-0. |
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