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Noncommutative stochastic processes with independent and stationary additive increments

  • Michael Schürmann
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1396)

Abstract

We prove that a noncommutative stochastic process with independent and stationary additive increments (in the sense of [3]) can be embedded into a sum of annihilation, creation and second quantisation processes on Fock space. As a corollary we prove an embedding theorem for infinitely divisible representations of tensor algebras and Lie algebras.

Keywords

Convolution Product Stationary Increment Cyclic Representation Infinite Divisibility Hermitian Element 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Abe, E., Hopf Algebras, Cambridge University Press, 1980Google Scholar
  2. [2]
    Accardi, L., Frigerio, A. and Lewis, J. T., Quantum stochastic processes, Publ. RIMS, Kyoto Univ. 18, 97–133 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Accardi, L., Schürmann, M. and von Waldenfels, W., Quantum independent increment processes on superalgebras, to appear in Math. ZeitschriftGoogle Scholar
  4. [4]
    Araki, H., Factorizable representation of current algebra, Publ. RIMS, Kyoto Univ. 5, 361–422 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Bourbaki, N., Elements of Mathematics, Algebra I, Chapter III, § 6, Addison-Wesley, Reading, 1974zbMATHGoogle Scholar
  6. [6]
    Bratelli, O. and Robinson, D. W., Operator Algebras and Quantum Statistical Mechanics II, Texts and Monographs in Physics, Springer, New York Heidelberg Berlin, 1981CrossRefGoogle Scholar
  7. [7]
    Canisius, J., Algebraische Grenzwertsatze und unbegrenzt teilbare Funktionale, Diplomarbeit, Heidelberg, 1978Google Scholar
  8. [8]
    Cockroft, A. M. and Hudson, R. L., Quantum mechanical Wiener processes, J. Mult. Anal. 7, 107–124 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Giri, N. and von Waldenfels, W., An algebraic version of the central limit theorem, Z. Wahrscheinlichkeitstheorie verw. Gebiete 42, 129–134 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Guichardet, A., Symmetric Hilbert spaces and related topics, Lect. Notes Math. 261, Springer, Berlin Heidelberg New York, 1972zbMATHGoogle Scholar
  11. [11]
    Hegerfeld, G. C., Noncommutative analogs of probabilistic notions and results, J. Funct. Anal. 64, 436–456 (1985)MathSciNetCrossRefGoogle Scholar
  12. [12]
    Hudson, R. L. and Parthasarathy, K. R., Quantum Ito's formula and stochastic evolutions, Comm. Math. Phys. 93, 301–323 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Jacobson, N., Lie Algebras, Wiley, New York London, 1962.zbMATHGoogle Scholar
  14. [14]
    Milnor, J. W. and Moore, J. C., On the structure of Hopf algebras, Ann. of Math. 81, 211–264 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Parthasarathy, K. R. and Schmidt, K., Positive definite kernels, continuous tensor products, and central limit theorems of probability theory, Lect. Notes Math. 272, Springer, Berlin Heidelberg New York, 1972zbMATHGoogle Scholar
  16. [16]
    Schürmann, M., Positive and conditionally positive linear functionals on coalgebras, in: Accardi, L. and von Waldenfels, W. (Eds.), Quantum probability and applications II, Proceedings, Heidelberg 1984, Lect. Notes Math. 1136, Springer, Berlin Heidelberg New York Tokyo, 1985Google Scholar
  17. [17]
    Schürmann, M., Noncommutative stochastic processes with independent and stationary increments satisfy quantum stochastic differential equations, submitted for publicationGoogle Scholar
  18. [18]
    Schürmann, M. and von Waldenfels, W., A central limit theorem on the free Lie group, in: Accardi, L. and von Waldenfels, W. (Eds), Quantum probability and applications III, Proceedings, Oberwolfach 1987, Lect. Notes Math. 1303, Springer, Berlin Heidelberg New York London Paris Tokyo, 1988Google Scholar
  19. [19]
    Streater, R., Infinitely divisible representations of Lie algebras, Z. Wahrscheinlichkeitstheorie verw. Geb. 19, 67–80 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Streater, R., Current commutation relations, continuous tensor products and infinitely divisible group representations, Jost, R. (Ed), Local quantum theory, Academic Press, London New York, 1969Google Scholar
  21. [21]
    Sweedler, M. E., Hopf Algebras, Benjamin, New York, 1969zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • Michael Schürmann
    • 1
  1. 1.Institut für Angewandte MathematikUniversität HeidelbergHeidelberg 1Federal Republic of Germany

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