Quantum stochastic convolution cocycles I

Lindsay, J. Martin; Skalski, Adam G.

doi:10.1016/j.anihpb.2004.10.002

Lindsay, J. Martin ; Skalski, Adam G.

Annales de l'I.H.P. Probabilités et statistiques, Tome 41 (2005) no. 3, pp. 581-604.

EuDML MR Zbl

DOI : 10.1016/j.anihpb.2004.10.002

@article{AIHPB_2005__41_3_581_0,
     author = {Lindsay, J. Martin and Skalski, Adam G.},
     title = {Quantum stochastic convolution cocycles {I}},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {581--604},
     publisher = {Elsevier},
     volume = {41},
     number = {3},
     year = {2005},
     doi = {10.1016/j.anihpb.2004.10.002},
     mrnumber = {2139034},
     zbl = {1074.81044},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpb.2004.10.002/}
}

TY  - JOUR
AU  - Lindsay, J. Martin
AU  - Skalski, Adam G.
TI  - Quantum stochastic convolution cocycles I
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2005
SP  - 581
EP  - 604
VL  - 41
IS  - 3
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpb.2004.10.002/
DO  - 10.1016/j.anihpb.2004.10.002
LA  - en
ID  - AIHPB_2005__41_3_581_0
ER  -

%0 Journal Article
%A Lindsay, J. Martin
%A Skalski, Adam G.
%T Quantum stochastic convolution cocycles I
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2005
%P 581-604
%V 41
%N 3
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpb.2004.10.002/
%R 10.1016/j.anihpb.2004.10.002
%G en
%F AIHPB_2005__41_3_581_0

Lindsay, J. Martin; Skalski, Adam G. Quantum stochastic convolution cocycles I. Annales de l'I.H.P. Probabilités et statistiques, Tome 41 (2005) no. 3, pp. 581-604. doi : 10.1016/j.anihpb.2004.10.002. http://www.numdam.org/articles/10.1016/j.anihpb.2004.10.002/

Bibliographie
Cité par

[1] L. Accardi, On the quantum Feynman-Kac formula, Rend. Sem. Mat. Fis. Milano 48 (1978) 135-180. | MR | Zbl

[2] L. Accardi, J.-L. Journé, J.M. Lindsay, On multidimensional Markovian cocycles, in: Accardi L., Von Waldenfels W. (Eds.), Quantum Probability and Applications IV, Lecture Notes in Math., vol. 1396, Springer-Verlag, Berlin, 1989, pp. 59-67. | MR | Zbl

[3] L. Accardi, A. Mohari, On the structure of classical and quantum flows, J. Funct. Anal. 135 (2) (1996) 421-455. | MR | Zbl

[4] W. Arveson, Noncommutative Dynamics and E-Semigroups, Springer-Verlag, New York, 2003. | MR | Zbl

[5] L. Accardi, M. Schürmann, W. Von Waldenfels, Quantum independent increment processes on superalgebras, Math. Z. 198 (4) (1988) 451-477. | MR | Zbl

[6] D. Applebaum, B.V.R. Bhat, J. Kustermans, J.M. Lindsay, Quantum Independent Increment Processes I: From Classical Probability to Quantum Stochastics, Lecture Notes in Math., vol. 1865, Springer-Verlag, Heidelberg, 2005. | MR | Zbl

[7] O.E. Barndorff-Nielsen, U. Franz, G. Gohm, B. Krümmerer, S. Thorbjørsen, Quantum Independent Increment Processes II: Structure of Quantum Lévy Processes, Classical Probability and Physics, U. Franz, M. Schürmann (Eds.), Lecture Notes in Math., vol. 1866, Springer-Verlag, Heidelberg, in press. | MR

[8] W.S. Bradshaw, Stochastic cocycles as a characterisation of quantum flows, Bull. Sci. Math. 116 (1) (1992) 1-34. | MR | Zbl

[9] M.P. Evans, R.L. Hudson, Perturbations of quantum diffusions, J. London Math. Soc. (2) 41 (2) (1990) 373-384. | MR | Zbl

[10] F. Fagnola, Characterization of isometric and unitary weakly differentiable cocycles in Fock space, in: Accardi L. (Ed.), Quantum Probability and Related Topics VIII, World Scientific, Singapore, 1993, pp. 143-164. | MR

[11] U. Franz, Lévy processes on quantum groups and dual groups, in [7]. | Zbl

[12] U. Franz, R. Schott, Stochastic Processes and Operator Calculus on Quantum Groups, Math. Appl., vol. 490, Kluwer Academic, Dordrecht, 1999. | MR | Zbl

[13] P. Glockner, Quantum stochastic differential equations on *-bialgebras, Math. Proc. Cambridge Philos. Soc. 109 (3) (1991) 571-595. | MR | Zbl

[14] D. Goswami, J.M. Lindsay, S.J. Wills, A stochastic Stinespring theorem, Math. Ann. 319 (4) (2001) 647-673. | MR | Zbl

[15] A. Guichardet, Symmetric Hilbert Spaces and Related Topics, Lecture Notes in Math., vol. 267, Springer, Heidelberg, 1970. | MR | Zbl

[16] J. Hellmich, C. Köstler, B. Kümmerer, Noncommutative continuous Bernoulli shifts, Preprint, Queen's University, Kingston, 2004.

[17] R.L. Hudson, Unitarity and multiplicativity via higher Itô product formula, Tatra Mt. Math. Publ. 10 (1997) 95-108. | MR | Zbl

[18] R.L. Hudson, J.M. Lindsay, On characterizing quantum stochastic evolutions, Math. Proc. Cambridge Philos. Soc. 102 (2) (1987) 363-369. | MR | Zbl

[19] R.L. Hudson, K.R. Parthasarathy, Quantum Itô's formula and stochastic evolutions, Comm. Math. Phys. 93 (3) (1984) 301-323. | MR | Zbl

[20] J.-L. Journé, Structure des cocycles markoviens sur l'espace de Fock, Probab. Theory Related Fields 75 (2) (1987) 291-316. | MR | Zbl

[21] J. Kustermans, Locally compact quantum groups, in [6]. | Zbl

[22] J. Kustermans, S. Vaes, Locally compact quantum groups, Ann. Sci. École Norm. Sup. (4) 33 (6) (2000) 837-934. | Numdam | MR | Zbl

[23] J.M. Lindsay, Integral-sum kernel operators, in: Attal S., Lindsay J.M. (Eds.), Quantum Probability Communications XII, World Scientific, Singapore, 2003, pp. 1-21. | MR | Zbl

[24] J.M. Lindsay, Quantum stochastic analysis - an introduction, in [6]. | Zbl

[25] J.M. Lindsay, A.G. Skalski, Quantum stochastic convolution cocycles-algebraic and $C^{*}$ -algebraic, in: M. Bożejko, R. Lenczewski, W. Młotkowski, J. Wysoczański (Eds.), Quantum Probability and Related Topics, Banach Center Publications, Polish Academy of Sciences, Warsaw, 2005, in press. | Zbl

[26] J.M. Lindsay, S.J. Wills, Existence, positivity, and contractivity for quantum stochastic flows with infinite dimensional noise, Probab. Theory Related Fields 116 (4) (2000) 505-543. | MR | Zbl

[27] J.M. Lindsay, S.J. Wills, Markovian cocycles on operator algebras, adapted to a Fock filtration, J. Funct. Anal. 178 (2) (2000) 269-305. | MR | Zbl

[28] J.M. Lindsay, S.J. Wills, Homomorphic Feller cocycles on a $C^{*}$ -algebra, J. London Math. Soc. (2) 68 (1) (2003) 255-272. | Zbl

[29] J.M. Lindsay, S.J. Wills, Operator Markovian cocycles via associated semigroups, Preprint, 2004.

[30] H. Maassen, Quantum Markov processes on Fock space described by integral kernels, in: Accardi L., Von Waldenfels W. (Eds.), Quantum Probability and Applications II, Lecture Notes in Math., vol. 1136, Springer-Verlag, Berlin, 1985, pp. 361-374. | MR

[31] P.-A. Meyer, Quantum Probability for Probabilists, Lecture Notes in Math., vol. 1538, Springer-Verlag, Berlin, 1995. | MR | Zbl

[32] K.R. Parthasarathy, Introduction to Quantum Stochastic Calculus, Birkhäuser, Basel, 1992. | MR | Zbl

[33] K. Schmüdgen, Unbounded Operator Algebras and Representation Theory, Akademie-Verlag, Berlin, 1990. | MR | Zbl

[34] M. Schürmann, Noncommutative stochastic processes with independent and stationary increments satisfy quantum stochastic differential equations, Probab. Theory Related Fields 84 (4) (1990) 473-490. | MR | Zbl

[35] M. Schürmann, White noise on involutive bialgebras, in: Quantum Probability & Related Topics VI, World Scientific, Singapore, 1991, pp. 401-419. | MR | Zbl

[36] M. Schürmann, White Noise on Bialgebras, Lecture Notes in Math., vol. 1544, Springer, Heidelberg, 1993. | MR | Zbl

[37] M. Schürmann, Operator processes majorizing their quadratic variation, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3 (1) (2000) 99-120. | MR | Zbl

[38] A.V. Skorohod, Operator stochastic differential equations and stochastic semigroups, Uspekhi Mat. Nauk 37 (6) (1982) 157-183, (228) (in Russian), Russian Math. Surveys 37 (6) (1982) 177-204. | MR | Zbl

[39] M.E. Sweedler, Hopf Algebras, Benjamin, New York, 1969. | MR | Zbl

[40] S.L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (4) (1987) 613-665. | MR | Zbl

[41] S.L. Woronowicz, Compact quantum groups, in: Connes A., Gawedzki K., Zinn-Justin J. (Eds.), Symétries Quantiques, Proceedings, Les Houches, 1995, North-Holland, Amsterdam, 1998, pp. 845-884. | MR | Zbl

Cité par Sources :