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A generalization of the conservation integral

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Starting from the scheme given by Hudson and Parthasarathy [7,11] we extend the conservation integral to the case where the underlying operator does not commute with the time observable. It turns out that there exist two extensions, a left and a right conservation integral. Moreover, Itô's formula demands for a third integral with two integrators. Only the left integral shows similar continuity properties to that derived in [11] used for extending the integral to more than simple integrands. In another approach we extend the previous notions for the integrals to much larger domains of definition and to much more processes, including anticipating ones. Similar to [5,10], we use the Skorohod integral and the Malliavin derivative acting on a symmetric Fock space [3,4]. It appears that this formulation unifies all three integrals in the double integrator one.
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  • Institut für Angewandte Mathematik, Fakultät für Mathematik und Informatik, Friedrich-Schiller-Universität Jena, D-07740 Jena, Germany
  • [1] L. Accardi and F. Fagnola, Stochastic integration, in: L. Accardi and W. von Waldenfels (eds.), Quantum Probability and Applications III. Proceedings, Oberwolfach 1987, volume 1303 of Lecture Notes in Mathematics, Springer-Verlag, Berlin Heidelberg New York 1988, pp. 6-19.
  • [2] L. Accardi, F. Fagnola and J. Quagebeur, A representation free quantum stochastic calculus, J. Funct. Anal. 104 (1992), 149-197.
  • [3] K.-H. Fichtner and W. Freudenberg, Remarks on stochastic calculus on the Fock space, in: L. Accardi (ed.), Quantum Probability and Related Topics VII, World Scientific Publishing Co., Singapore New Jersey London Hong Kong 1991, pp. 305-323.
  • [4] K.-H. Fichtner and G. Winkler, Generalized Brownian Motion, Point Processes and Stochastic Calculus for Random Fields, Math. Nachr. 161 (1993), 291-307.
  • [5] W. Freudenberg, Quantum stochastic integrals on a general Fock space, in: L. Accardi (ed.), Quantum Probability and Related Topics VIII, World Scientific Publishing Co., Singapore 1993, pp. 189-210.
  • [6] B. Gaveau and P. Trauber, L'intégrale stochastique comme opérateur de divergence dans l'espace fonctionnel, J. Funct. Anal. 46 (1982), 230-238.
  • [7] R. L. Hudson and K. R. Parthasarathy, Quantum Itô's formula and stochastic evolution, Commun. Math. Phys. 93 (1984), 301-323.
  • [8] V. Liebscher, Two Limit Theorems for a Class of Quantum Markov Chains associated to Beam Splittings, submitted to Open Systems and Information Dynamics 1996.
  • [9] V. Liebscher, On a central limit theorem for monotone noise, submitted to Infinite Dimensional Analysis, Quantum Probability and Related Topics 1997.
  • [10] J. M. Lindsay, Quantum and Noncausal Stochastic Calculus, Prob. Th. Rel. Fields 97 (1993), 65-80.
  • [11] K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus, Birkhäuser, Basel Boston Berlin 1992.
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