Geodesic
Asymptotics of small deviations of the Bogoliubov processes with respect to a quadratic norm
Teoretičeskaâ i matematičeskaâ fizika, Tome 165 (2010) no. 1, pp. 134-144 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We obtain results on small deviations of Bogoliubov's Gaussian measure occurring in the theory of the statistical equilibrium of quantum systems. For some random processes related to Bogoliubov processes, we find the exact asymptotic probability of their small deviations with respect to a Hilbert norm.
Keywords: Bogoliubov measure, small deviation. 
@article{TMF_2010_165_1_a7,
     author = {R. S. Pusev},
     title = {Asymptotics of small deviations of {the~Bogoliubov} processes with respect to a~quadratic norm},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {134--144},
     year = {2010},
     volume = {165},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2010_165_1_a7/}
}
TY  - JOUR
AU  - R. S. Pusev
TI  - Asymptotics of small deviations of the Bogoliubov processes with respect to a quadratic norm
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2010
SP  - 134
EP  - 144
VL  - 165
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_2010_165_1_a7/
LA  - ru
ID  - TMF_2010_165_1_a7
ER  - 
%0 Journal Article
%A R. S. Pusev
%T Asymptotics of small deviations of the Bogoliubov processes with respect to a quadratic norm
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2010
%P 134-144
%V 165
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_2010_165_1_a7/
%G ru
%F TMF_2010_165_1_a7
R. S. Pusev. Asymptotics of small deviations of the Bogoliubov processes with respect to a quadratic norm. Teoretičeskaâ i matematičeskaâ fizika, Tome 165 (2010) no. 1, pp. 134-144. http://geodesic.mathdoc.fr/item/TMF_2010_165_1_a7/

[1] D. P. Sankovich, TMF, 119:2 (1999), 345–352 | DOI | MR | Zbl

[2] D. P. Sankovich, TMF, 126:1 (2001), 149–163 | DOI | MR | Zbl

[3] N. N. Bogolyubov, Dokl. AN SSSR, 99:2 (1954), 225–226 | MR | Zbl

[4] D. P. Sankovich, TMF, 127:1 (2001), 125–142 | DOI | MR | Zbl

[5] V. R. Fatalov, TMF, 157:2 (2008), 286–308 | DOI | MR | Zbl

[6] W. V. Li, Q.-M. Shao, “Gaussian processes: inequalities, small ball probabilities and applications”, Stochastic Processes: Theory and Methods, Handbook Statist., 19, eds. D. N. Shanbhag, C. R. Rao, North-Holland, Amsterdam, 2001, 533–597 | DOI | MR | Zbl

[7] M. A. Lifshits, Bibliography of small deviation probabilities\par http://www.proba.jussieu.fr/pageperso/smalldev/biblio.pdf

[8] F. Ferraty, P. Vieu, Nonparametric Functional Data Analysis, Springer Ser. Statist., Springer, New York, 2006 | MR | Zbl

[9] A. W. van der Vaart, J. H. van Zanten, Ann. Statist., 36:3 (2008), 1435–1463 | DOI | MR | Zbl

[10] F. Aurzada, I. A. Ibragimov, M. A. Lifshits, H. J. van Zanden, TVP, 53:4 (2008), 788–798 | DOI | MR | Zbl

[11] L. Beghin, Ya. Yu. Nikitin, E. Orsingher, “Exact small ball constants for some Gaussian processes under $L^2$-norm”, Veroyatnost i statistika. 6, Zap. nauchn. sem. POMI, 298, POMI, SPb., 2003, 5–21 | DOI | MR | Zbl

[12] Ya. Yu. Nikitin, P. A. Kharinskii, “Tochnaya asimptotika malykh uklonenii v $L_2$-norme dlya odnogo klassa Gaussovskikh protsessov”, Veroyatnost i statistika. 7, Zap. nauchn. sem. POMI, 311, POMI, SPb., 2004, 214–221 | DOI | MR | Zbl

[13] A. I. Nazarov, Ya. Yu. Nikitin, Probab. Theory Related Fields, 129:4 (2004), 469–494 | DOI | MR | Zbl

[14] A. I. Nazarov, R. S. Pusev, “Tochnaya asimptotika malykh uklonenii v $L_2$-norme s vesom dlya nekotorykh gaussovskikh protsessov”, Veroyatnost i statistika. 14–2, Zap. nauchn. sem. POMI, 364, POMI, SPb., 2009, 166–199 | DOI | Zbl

[15] A. I. Nazarov, Problemy matem. analiza, 26 (2003), 179–213 | DOI | MR | Zbl

[16] A. I. Nazarov, J. Theoret. Probab., 22:3 (2009), 640–665 | DOI | MR | Zbl

[17] R. J. Adler, An Introduction to Continuity, Extrema and Related Topics for General Gaussian Processes, IMS Lecture Notes – Monograph Ser., 12, Institute of Mathematical Statistics, Hayward, California, 1990 | MR | Zbl

[18] V. A. Zorich, Matematicheskii analiz, Chast 2, MTsNMO, M., 1998 | MR | MR | Zbl

[19] E. Kamke, Spravochnik po obyknovennym differentsialnym uravneniyam, Lan, SPb., 2003 | MR | MR | Zbl

[20] I. S. Gradshtein, I. M. Ryzhik, Tablitsy integralov, summ, ryadov i proizvedenii, Fizmatlit, M., 1971 | MR | MR | Zbl

[21] E. Titchmarsh, Teoriya funktsii, Nauka, M., 1980 | MR | Zbl

[22] F. Gao, J. Hannig, F. Torcaso, Ann. Probab., 31:3 (2003), 1320–1337 | DOI | MR | Zbl

[23] A. Lachal, Math. Meth. Statist., 10 (2001), 73–104 | MR | Zbl

[24] P. Groeneboom, G. Jongbloed, J. A. Wellner, Ann. Statist., 29:6 (2001), 1620–1652 | DOI | MR | Zbl

[25] Ya. G. Sinai, TMF, 90:3 (1992), 323–353 | DOI | MR | Zbl

[26] G. Molchan, A. Khokhlov, J. Stat. Phys., 114:3–4 (2004), 923–946 | DOI | MR | Zbl

[27] L. Galleani, L. Sacerdote, P. Tavella, C. Zucca, Metrologia, 40:3 (2003), S257–S264 | DOI

[28] M. A. Naimark, Lineinye differentsialnye operatory, Nauka, M., 1969 | MR | MR | MR | Zbl