Characterization of unitary processes with independent and stationary increments

Lingaraj Sahu; Kalyan B. Sinha

Annales de l'I.H.P. Probabilités et statistiques (2010)

  • Volume: 46, Issue: 2, page 575-593
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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Sahu, Lingaraj, and Sinha, Kalyan B.. "Characterization of unitary processes with independent and stationary increments." Annales de l'I.H.P. Probabilités et statistiques 46.2 (2010): 575-593. <http://eudml.org/doc/240601>.

@article{Sahu2010,
abstract = {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.},
author = {Sahu, Lingaraj, Sinha, Kalyan B.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {unitary processes; noise space; Hudson–Parthasarathy equations; quantum stochastic calculus; quantum unitary process with i.i.d. stationary increments; quantum Gaussian process},
language = {eng},
number = {2},
pages = {575-593},
publisher = {Gauthier-Villars},
title = {Characterization of unitary processes with independent and stationary increments},
url = {http://eudml.org/doc/240601},
volume = {46},
year = {2010},
}

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
PB - Gauthier-Villars
VL - 46
IS - 2
SP - 575
EP - 593
AB - 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.
LA - eng
KW - unitary processes; noise space; Hudson–Parthasarathy equations; quantum stochastic calculus; quantum unitary process with i.i.d. stationary increments; quantum Gaussian process
UR - http://eudml.org/doc/240601
ER -

References

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  1. [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. Zbl0674.60064MR1019566
  2. [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. Zbl0144.17202MR173945
  3. [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. Zbl0789.47027MR1186660
  4. [4] D. Goswami and K. B. Sinha. Quantum Stochastic Processes and Geometry. Cambridge Tracts in Mathematics 169. Cambridge Univ. Press, 2007. Zbl1144.81002MR2299106
  5. [5] R. L. Hudson and J. M. Lindsay. On characterizing quantum stochastic evolutions. Math. Proc. Cambridge Philos. Soc. 102 (1987) 363–369. Zbl0644.46046MR898155
  6. [6] R. L. Hudson and K. R. Parthasarathy. Quantum Ito’s formula and stochastic evolutions. Comm. Math. Phys. 93 (1984) 301–323. Zbl0546.60058MR745686
  7. [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. Zbl0969.60066MR1802896
  8. [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. Zbl1123.47052MR2349207
  9. [9] A. Mohari. Quantum stochastic differential equations with unbounded coefficients and dilations of Feller’s minimal solution. Sankhyā Ser. A 53 (1991) 255–287. Zbl0751.60062MR1189771
  10. [10] A. Mohari and K. B. Sinha. Stochastic dilation of minimal quantum dynamical semigroup. Proc. Indian Acad. Sci. Math. Sci. 102 (1992) 159–173. Zbl0766.60119MR1213635
  11. [11] K. R. Parthasarathy. An Introduction to Quantum Stochastic Calculus. Monographs in Mathematics 85. Birkhäuser, Basel, 1992. Zbl0751.60046MR1164866
  12. [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. Zbl1194.81138MR2569566
  13. [13] M. Schürmann. Noncommutative stochastic processes with independent and stationary increments satisfy quantum stochastic differential equations. Probab. Theory Related Fields 84, (1990) 473–490. Zbl0685.60070MR1042061
  14. [14] M. Schürmann. White Noise on Bialgebras. Lecture Notes in Math. 1544. Springer, Berlin, 1993. Zbl0773.60100MR1238942
  15. [15] K. B. Sinha. Quantum dynamical semigroups. In Mathematical Results in Quantum Mechanics 161–169. Oper. Theory Adv. Appl. 70. Birkhäuser, Basel, 1994. Zbl0831.46077MR1309019

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