Characterization of unitary processes with independent and stationary increments
Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 2, pp. 575-593.

Cet article poursuit la recherche initiée dans (Publ. Res. Inst. Math. Sci. 45 (2009) 745-785) pour caractériser les processus stationnaires unitaires gaussiens à incréments indépendants. L'hypothèse antérieure d'uniforme continuité est remplacée par de la continuité faible. Avec des conditions techniques sur le domaine du générateur, nous montrons que le processus est équivalent unitairement à la solution d'une équation de Hudson-Parthasarathy appropriée.

This is a continuation of the earlier work (Publ. Res. Inst. Math. Sci. 45 (2009) 745-785) to characterize unitary stationary independent increment gaussian processes. The earlier assumption of uniform continuity is replaced by weak continuity and with technical assumptions on the domain of the generator, unitary equivalence of the process to the solution of an appropriate Hudson-Parthasarathy equation is proved.

DOI : 10.1214/09-AIHP327
Classification : 60G51, 81S25
Mots-clés : unitary processes, noise space, Hudson-Parthasarathy equations 
@article{AIHPB_2010__46_2_575_0,
     author = {Sahu, Lingaraj and Sinha, Kalyan B.},
     title = {Characterization of unitary processes with independent and stationary increments},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {575--593},
     publisher = {Gauthier-Villars},
     volume = {46},
     number = {2},
     year = {2010},
     doi = {10.1214/09-AIHP327},
     mrnumber = {2667710},
     zbl = {1203.81094},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/09-AIHP327/}
}
TY  - JOUR
AU  - Sahu, Lingaraj
AU  - Sinha, Kalyan B.
TI  - Characterization of unitary processes with independent and stationary increments
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2010
SP  - 575
EP  - 593
VL  - 46
IS  - 2
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/09-AIHP327/
DO  - 10.1214/09-AIHP327
LA  - en
ID  - AIHPB_2010__46_2_575_0
ER  - 
%0 Journal Article
%A Sahu, Lingaraj
%A Sinha, Kalyan B.
%T Characterization of unitary processes with independent and stationary increments
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2010
%P 575-593
%V 46
%N 2
%I Gauthier-Villars
%U http://www.numdam.org/articles/10.1214/09-AIHP327/
%R 10.1214/09-AIHP327
%G en
%F AIHPB_2010__46_2_575_0
Sahu, Lingaraj; Sinha, Kalyan B. Characterization of unitary processes with independent and stationary increments. Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 2, pp. 575-593. doi : 10.1214/09-AIHP327. http://www.numdam.org/articles/10.1214/09-AIHP327/

[1] L. Accardi, J. L. Journé and J. M. Lindsay. On multi-dimensional Markovian cocycles. In Quantum Probability and Applications, IV (Rome, 1987) 59-67. Lecture Notes in Math. 1396. Springer, Berlin, 1989. | MR | Zbl

[2] I. M. Geĺfand and N. Y. Vilenkin. Generalized functions, Vol. 4: Applications of Harmonic Analysis. Translated from the Russian by Amiel Feinstein. Academic Press, New York, 1964. | MR | Zbl

[3] F. Fagnola. Unitarity of solutions to quantum stochastic differential equations and conservativity of the associated semigroups. In Quantum Probability and Related Topics 139-148. QP-PQ, VII. World Sci. Publ., River Edge, NJ, 1992. | MR | Zbl

[4] D. Goswami and K. B. Sinha. Quantum Stochastic Processes and Geometry. Cambridge Tracts in Mathematics 169. Cambridge Univ. Press, 2007. | MR | Zbl

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

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

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

[8] J. M. Lindsay and S. J. Wills. Construction of some quantum stochastic operator cocycles by the semigroup method. Proc. Indian Acad. Sci. (Math. Sci.) 116 (2006) 519-529. | MR | Zbl

[9] A. Mohari. Quantum stochastic differential equations with unbounded coefficients and dilations of Feller's minimal solution. Sankhyā Ser. A 53 (1991) 255-287. | MR | Zbl

[10] A. Mohari and K. B. Sinha. Stochastic dilation of minimal quantum dynamical semigroup. Proc. Indian Acad. Sci. Math. Sci. 102 (1992) 159-173. | MR | Zbl

[11] K. R. Parthasarathy. An Introduction to Quantum Stochastic Calculus. Monographs in Mathematics 85. Birkhäuser, Basel, 1992. | MR | Zbl

[12] L. Sahu, M. Schürmann and K. B. Sinha. Unitary processes with independent increments and representations of Hilbert tensor algebras. Publ. Res. Inst. Math. Sci. 45 (2009) 745-785. | MR | Zbl

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

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

[15] K. B. Sinha. Quantum dynamical semigroups. In Mathematical Results in Quantum Mechanics 161-169. Oper. Theory Adv. Appl. 70. Birkhäuser, Basel, 1994. | MR | Zbl

Cité par Sources :