January  2021, 14(1): 77-88. doi: 10.3934/krm.2020049

On two properties of the Fisher information

Université Grenoble-Alpes & CNRS, LPMMC (UMR 5493), B.P. 166, F-38042 Grenoble, France

Received  January 2020 Revised  August 2020 Published  November 2020

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.

Citation: Nicolas Rougerie. On two properties of the Fisher information. Kinetic & Related Models, 2021, 14 (1) : 77-88. doi: 10.3934/krm.2020049
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.  Google Scholar

[2]

J. BourgainH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

N. FournierM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. LewinP. 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.  Google Scholar

[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.  Google Scholar

[16]

W. Masja and J. Nagel, Über äquivalente normierung der anisotropen Funktionalraüme $H ^{\mu} ( { {\mathbb R} } ^n)$, Beiträge zur Analysis, 12 (1978), 7-17.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24] B. Simon, Trace Ideals and Their Applications, vol. 35 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1979.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[27]

G. Toscani, Score functions, generalized relative Fisher information and applications, Ricerche mat., 66 (2017) 15–26. doi: 10.1007/s11587-016-0281-0.  Google Scholar

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.  Google Scholar

[2]

J. BourgainH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

N. FournierM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. LewinP. 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.  Google Scholar

[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.  Google Scholar

[16]

W. Masja and J. Nagel, Über äquivalente normierung der anisotropen Funktionalraüme $H ^{\mu} ( { {\mathbb R} } ^n)$, Beiträge zur Analysis, 12 (1978), 7-17.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24] B. Simon, Trace Ideals and Their Applications, vol. 35 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1979.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[27]

G. Toscani, Score functions, generalized relative Fisher information and applications, Ricerche mat., 66 (2017) 15–26. doi: 10.1007/s11587-016-0281-0.  Google Scholar

[1]

Illés Horváth, Kristóf Attila Horváth, Péter Kovács, Miklós Telek. Mean-field analysis of a scaling MAC radio protocol. Journal of Industrial & Management Optimization, 2021, 17 (1) : 279-297. doi: 10.3934/jimo.2019111

[2]

Jingrui Sun, Hanxiao Wang. Mean-field stochastic linear-quadratic optimal control problems: Weak closed-loop solvability. Mathematical Control & Related Fields, 2021, 11 (1) : 47-71. doi: 10.3934/mcrf.2020026

[3]

Laura Aquilanti, Simone Cacace, Fabio Camilli, Raul De Maio. A Mean Field Games model for finite mixtures of Bernoulli and categorical distributions. Journal of Dynamics & Games, 2020  doi: 10.3934/jdg.2020033

[4]

Daniele Bartolucci, Changfeng Gui, Yeyao Hu, Aleks Jevnikar, Wen Yang. Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3093-3116. doi: 10.3934/dcds.2020039

[5]

Alain Bensoussan, Xinwei Feng, Jianhui Huang. Linear-quadratic-Gaussian mean-field-game with partial observation and common noise. Mathematical Control & Related Fields, 2021, 11 (1) : 23-46. doi: 10.3934/mcrf.2020025

[6]

Wenya Qi, Padmanabhan Seshaiyer, Junping Wang. A four-field mixed finite element method for Biot's consolidation problems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020127

[7]

Andrew D. Lewis. Erratum for "nonholonomic and constrained variational mechanics". Journal of Geometric Mechanics, 2020, 12 (4) : 671-675. doi: 10.3934/jgm.2020033

[8]

Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020374

[9]

Jie Li, Xiangdong Ye, Tao Yu. Mean equicontinuity, complexity and applications. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 359-393. doi: 10.3934/dcds.2020167

[10]

Wai-Ki Ching, Jia-Wen Gu, Harry Zheng. On correlated defaults and incomplete information. Journal of Industrial & Management Optimization, 2021, 17 (2) : 889-908. doi: 10.3934/jimo.2020003

[11]

Kalikinkar Mandal, Guang Gong. On ideal $ t $-tuple distribution of orthogonal functions in filtering de bruijn generators. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020125

[12]

Yutong Chen, Jiabao Su. Nontrivial solutions for the fractional Laplacian problems without asymptotic limits near both infinity and zero. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021007

[13]

Shengxin Zhu, Tongxiang Gu, Xingping Liu. AIMS: Average information matrix splitting. Mathematical Foundations of Computing, 2020, 3 (4) : 301-308. doi: 10.3934/mfc.2020012

[14]

Xiaoming Wang. Upper semi-continuity of stationary statistical properties of dissipative systems. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 521-540. doi: 10.3934/dcds.2009.23.521

[15]

Hongfei Yang, Xiaofeng Ding, Raymond Chan, Hui Hu, Yaxin Peng, Tieyong Zeng. A new initialization method based on normed statistical spaces in deep networks. Inverse Problems & Imaging, 2021, 15 (1) : 147-158. doi: 10.3934/ipi.2020045

[16]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323

[17]

Patrick Martinez, Judith Vancostenoble. Lipschitz stability for the growth rate coefficients in a nonlinear Fisher-KPP equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 695-721. doi: 10.3934/dcdss.2020362

[18]

Isabeau Birindelli, Françoise Demengel, Fabiana Leoni. Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020395

[19]

Jean-Claude Saut, Yuexun Wang. Long time behavior of the fractional Korteweg-de Vries equation with cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1133-1155. doi: 10.3934/dcds.2020312

[20]

Hong Fu, Mingwu Liu, Bo Chen. Supplier's investment in manufacturer's quality improvement with equity holding. Journal of Industrial & Management Optimization, 2021, 17 (2) : 649-668. doi: 10.3934/jimo.2019127

2019 Impact Factor: 1.311

Metrics

  • PDF downloads (12)
  • HTML views (76)
  • Cited by (0)

Other articles
by authors

[Back to Top]