2013, 20: 12-30. doi: 10.3934/era.2013.20.12

Infinite determinantal measures

1. 

Laboratoire d'Analyse, Topologie, Probabilités, Aix-Marseille Université, CNRS, Marseille, France

Received  July 2012 Revised  November 2012 Published  February 2013

Infinite determinantal measures introduced in this note are inductive limits of determinantal measures on an exhausting family of subsets of the phase space. Alternatively, an infinite determinantal measure can be described as a product of a determinantal point process and a convergent, but not integrable, multiplicative functional.
    Theorem 4.1, the main result announced in this note, gives an explicit description for the ergodic decomposition of infinite Pickrell measures on the spaces of infinite complex matrices in terms of infinite determinantal measures obtained by finite-rank perturbations of Bessel point processes.
Citation: Alexander I. Bufetov. Infinite determinantal measures. Electronic Research Announcements, 2013, 20: 12-30. doi: 10.3934/era.2013.20.12
References:
[1]

Vol. II, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.  Google Scholar

[2]

Oxford University Press, Oxford, (2011), 231-249.  Google Scholar

[3]

J. Amer. Math. Soc., 13 (2000), 481-515. doi: 10.1090/S0894-0347-00-00337-4.  Google Scholar

[4]

Comm. Math. Phys., 223 (2001), 87-123. doi: 10.1007/s002200100529.  Google Scholar

[5]

J. Stat. Phys., 121 (2005), 291-317. doi: 10.1007/s10955-005-7583-z.  Google Scholar

[6]

arXiv:1105.0664, 2011. Google Scholar

[7]

to appear in Annales de l'Institut Fourier, arXiv:1108.2737, 2011. Google Scholar

[8]

Uspekhi Mat. Nauk, 67 (2012), 177-178; translation in Russian Math. Surveys, 67 (2012), 181-182. doi: 10.1070/RM2012v067n01ABEH004779.  Google Scholar

[9]

Probab. Surv., 3 (2006), 206-229. doi: 10.1214/154957806000000078.  Google Scholar

[10]

Springer-Verlag, Berlin-New York, 1933.  Google Scholar

[11]

Arch. Rational Mech. Anal., 59 (1975), 219-239.  Google Scholar

[12]

Publ. Math. Inst. Hautes Études Sci., 98 (2003), 167-212. doi: 10.1007/s10240-003-0016-0.  Google Scholar

[13]

Duke Math. J., 120 (2003), 515-575. doi: 10.1215/S0012-7094-03-12032-3.  Google Scholar

[14]

Rev. Math. Phys., 14 (2002), 1073-1098. doi: 10.1142/S0129055X02001533.  Google Scholar

[15]

Advances in Appl. Probability, 7 (1975), 83-122.  Google Scholar

[16]

Duke Math. J., 114 (2002), 239-266. doi: 10.1215/S0012-7094-02-11423-9.  Google Scholar

[17]

Adv. Math., 226 (2011), 2305-2350. doi: 10.1016/j.aim.2010.09.015.  Google Scholar

[18]

in "Representation of Lie Groups and Related Topics," Adv. Stud. Contemp. Math., 7, Gordon and Breach, NY, 1990, 269-463. Available from: http://www.iitp.ru/upload/userpage/52/HoweForm.pdf.  Google Scholar

[19]

(Russian), D. Sci Thesis, Institute of Geography of the Russian Academy of Sciences, 1989. Available from: http://www.iitp.ru/upload/userpage/52/Olshanski_thesis.pdf. Google Scholar

[20]

in "Contemporary Mathematical Physics," Amer. Math. Soc. Transl. Ser. 2, 175, Amer. Math. Soc., Providence, RI, (1996), 137-175.  Google Scholar

[21]

Pacific J. Math., 150 (1991), 139-166.  Google Scholar

[22]

J. Funct. Anal., 90 (1990), 1-26. doi: 10.1016/0022-1236(90)90078-Y.  Google Scholar

[23]

J. Funct. Anal., 70 (1987), 323-356. doi: 10.1016/0022-1236(87)90116-9.  Google Scholar

[24]

J. Lie Theory, 18 (2008), 645-670.  Google Scholar

[25]

Ann. Inst. Fourier (Grenoble), 58 (2008), 1551-1573.  Google Scholar

[26]

Second edition, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980.  Google Scholar

[27]

in "Stochastic Analysis on Large Scale Interacting Systems," Adv. Stud. Pure Math., 39, Math. Soc. Japan, Tokyo, (2004), 345-354.  Google Scholar

[28]

J. Funct. Anal., 205 (2003), 414-463. doi: 10.1016/S0022-1236(03)00171-X.  Google Scholar

[29]

Ann. Probab., 31 (2003), 1533-1564. doi: 10.1214/aop/1055425789.  Google Scholar

[30]

(Russian) Uspekhi Mat. Nauk, 55 (2000), 107-160; translation in Russian Math. Surveys, 55 (2000), 923-975. doi: 10.1070/rm2000v055n05ABEH000321.  Google Scholar

[31]

AMS, 1969. Google Scholar

[32]

Comm. Math. Phys., 161 (1994), 289-309.  Google Scholar

[33]

(Russian) Dokl. Akad. Nauk SSSR, 218 (1974), 749-752.  Google Scholar

show all references

References:
[1]

Vol. II, Springer-Verlag, Berlin, 2007. doi: 10.1007/978-3-540-34514-5.  Google Scholar

[2]

Oxford University Press, Oxford, (2011), 231-249.  Google Scholar

[3]

J. Amer. Math. Soc., 13 (2000), 481-515. doi: 10.1090/S0894-0347-00-00337-4.  Google Scholar

[4]

Comm. Math. Phys., 223 (2001), 87-123. doi: 10.1007/s002200100529.  Google Scholar

[5]

J. Stat. Phys., 121 (2005), 291-317. doi: 10.1007/s10955-005-7583-z.  Google Scholar

[6]

arXiv:1105.0664, 2011. Google Scholar

[7]

to appear in Annales de l'Institut Fourier, arXiv:1108.2737, 2011. Google Scholar

[8]

Uspekhi Mat. Nauk, 67 (2012), 177-178; translation in Russian Math. Surveys, 67 (2012), 181-182. doi: 10.1070/RM2012v067n01ABEH004779.  Google Scholar

[9]

Probab. Surv., 3 (2006), 206-229. doi: 10.1214/154957806000000078.  Google Scholar

[10]

Springer-Verlag, Berlin-New York, 1933.  Google Scholar

[11]

Arch. Rational Mech. Anal., 59 (1975), 219-239.  Google Scholar

[12]

Publ. Math. Inst. Hautes Études Sci., 98 (2003), 167-212. doi: 10.1007/s10240-003-0016-0.  Google Scholar

[13]

Duke Math. J., 120 (2003), 515-575. doi: 10.1215/S0012-7094-03-12032-3.  Google Scholar

[14]

Rev. Math. Phys., 14 (2002), 1073-1098. doi: 10.1142/S0129055X02001533.  Google Scholar

[15]

Advances in Appl. Probability, 7 (1975), 83-122.  Google Scholar

[16]

Duke Math. J., 114 (2002), 239-266. doi: 10.1215/S0012-7094-02-11423-9.  Google Scholar

[17]

Adv. Math., 226 (2011), 2305-2350. doi: 10.1016/j.aim.2010.09.015.  Google Scholar

[18]

in "Representation of Lie Groups and Related Topics," Adv. Stud. Contemp. Math., 7, Gordon and Breach, NY, 1990, 269-463. Available from: http://www.iitp.ru/upload/userpage/52/HoweForm.pdf.  Google Scholar

[19]

(Russian), D. Sci Thesis, Institute of Geography of the Russian Academy of Sciences, 1989. Available from: http://www.iitp.ru/upload/userpage/52/Olshanski_thesis.pdf. Google Scholar

[20]

in "Contemporary Mathematical Physics," Amer. Math. Soc. Transl. Ser. 2, 175, Amer. Math. Soc., Providence, RI, (1996), 137-175.  Google Scholar

[21]

Pacific J. Math., 150 (1991), 139-166.  Google Scholar

[22]

J. Funct. Anal., 90 (1990), 1-26. doi: 10.1016/0022-1236(90)90078-Y.  Google Scholar

[23]

J. Funct. Anal., 70 (1987), 323-356. doi: 10.1016/0022-1236(87)90116-9.  Google Scholar

[24]

J. Lie Theory, 18 (2008), 645-670.  Google Scholar

[25]

Ann. Inst. Fourier (Grenoble), 58 (2008), 1551-1573.  Google Scholar

[26]

Second edition, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980.  Google Scholar

[27]

in "Stochastic Analysis on Large Scale Interacting Systems," Adv. Stud. Pure Math., 39, Math. Soc. Japan, Tokyo, (2004), 345-354.  Google Scholar

[28]

J. Funct. Anal., 205 (2003), 414-463. doi: 10.1016/S0022-1236(03)00171-X.  Google Scholar

[29]

Ann. Probab., 31 (2003), 1533-1564. doi: 10.1214/aop/1055425789.  Google Scholar

[30]

(Russian) Uspekhi Mat. Nauk, 55 (2000), 107-160; translation in Russian Math. Surveys, 55 (2000), 923-975. doi: 10.1070/rm2000v055n05ABEH000321.  Google Scholar

[31]

AMS, 1969. Google Scholar

[32]

Comm. Math. Phys., 161 (1994), 289-309.  Google Scholar

[33]

(Russian) Dokl. Akad. Nauk SSSR, 218 (1974), 749-752.  Google Scholar

[1]

Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021066

[2]

Masashi Wakaiki, Hideki Sano. Stability analysis of infinite-dimensional event-triggered and self-triggered control systems with Lipschitz perturbations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021021

[3]

Kiyoshi Igusa, Gordana Todorov. Picture groups and maximal green sequences. Electronic Research Archive, , () : -. doi: 10.3934/era.2021025

[4]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109

[5]

Françoise Demengel. Ergodic pairs for degenerate pseudo Pucci's fully nonlinear operators. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3465-3488. doi: 10.3934/dcds.2021004

[6]

Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212

[7]

Marzia Bisi, Maria Groppi, Giorgio Martalò, Romina Travaglini. Optimal control of leachate recirculation for anaerobic processes in landfills. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2957-2976. doi: 10.3934/dcdsb.2020215

[8]

Tao Wu, Yu Lei, Jiao Shi, Maoguo Gong. An evolutionary multiobjective method for low-rank and sparse matrix decomposition. Big Data & Information Analytics, 2017, 2 (1) : 23-37. doi: 10.3934/bdia.2017006

[9]

Jean-François Biasse. Improvements in the computation of ideal class groups of imaginary quadratic number fields. Advances in Mathematics of Communications, 2010, 4 (2) : 141-154. doi: 10.3934/amc.2010.4.141

[10]

Xiaochen Mao, Weijie Ding, Xiangyu Zhou, Song Wang, Xingyong Li. Complexity in time-delay networks of multiple interacting neural groups. Electronic Research Archive, , () : -. doi: 10.3934/era.2021022

[11]

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, 2021, 41 (7) : 3021-3029. doi: 10.3934/dcds.2020395

[12]

Charlene Kalle, Niels Langeveld, Marta Maggioni, Sara Munday. Matching for a family of infinite measure continued fraction transformations. Discrete & Continuous Dynamical Systems, 2020, 40 (11) : 6309-6330. doi: 10.3934/dcds.2020281

[13]

Seung-Yeal Ha, Myeongju Kang, Bora Moon. Collective behaviors of a Winfree ensemble on an infinite cylinder. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2749-2779. doi: 10.3934/dcdsb.2020204

[14]

Rafael G. L. D'Oliveira, Marcelo Firer. Minimum dimensional Hamming embeddings. Advances in Mathematics of Communications, 2017, 11 (2) : 359-366. doi: 10.3934/amc.2017029

[15]

Matheus C. Bortolan, José Manuel Uzal. Upper and weak-lower semicontinuity of pullback attractors to impulsive evolution processes. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3667-3692. doi: 10.3934/dcdsb.2020252

[16]

Mingshang Hu, Shige Peng. G-Lévy processes under sublinear expectations. Probability, Uncertainty and Quantitative Risk, 2021, 6 (1) : 1-22. doi: 10.3934/puqr.2021001

[17]

Xiaomao Deng, Xiao-Chuan Cai, Jun Zou. A parallel space-time domain decomposition method for unsteady source inversion problems. Inverse Problems & Imaging, 2015, 9 (4) : 1069-1091. doi: 10.3934/ipi.2015.9.1069

[18]

Mao Okada. Local rigidity of certain actions of solvable groups on the boundaries of rank-one symmetric spaces. Journal of Modern Dynamics, 2021, 17: 111-143. doi: 10.3934/jmd.2021004

[19]

Lars Grüne, Luca Mechelli, Simon Pirkelmann, Stefan Volkwein. Performance estimates for economic model predictive control and their application in proper orthogonal decomposition-based implementations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021013

[20]

Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881

2019 Impact Factor: 0.5

Metrics

  • PDF downloads (67)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]