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, 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]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047

[8]

Chuan Ding, Da-Hai Li. Angel capitalists exit decisions under information asymmetry: IPO or acquisitions. Journal of Industrial & Management Optimization, 2021, 17 (1) : 369-392. doi: 10.3934/jimo.2019116

[9]

Jian Zhang, Tony T. Lee, Tong Ye, Liang Huang. An approximate mean queue length formula for queueing systems with varying service rate. Journal of Industrial & Management Optimization, 2021, 17 (1) : 185-204. doi: 10.3934/jimo.2019106

[10]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

[11]

Chao Wang, Qihuai Liu, Zhiguo Wang. Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 281-300. doi: 10.3934/cpaa.2020266

[12]

Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018

[13]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276

[14]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076

[15]

Yuanfen Xiao. Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $ \beta $-transformation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 525-536. doi: 10.3934/dcds.2020267

2019 Impact Factor: 1.311

Metrics

  • PDF downloads (8)
  • HTML views (35)
  • Cited by (0)

Other articles
by authors

[Back to Top]