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Operator-Valued Stochastic Differential Equations Arising from Unitary Group Representations

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  • David Applebaum

    (Nottingham Trent University)

Abstract

Let π be a unitary representation of a Lie group G and (φ(t), t≥0) be a Lévy process in G. Using analytic vector techniques it is shown that the unitary process U(t)=π(φ(t)) satisfies an operator-valued stochastic differential equation. The prescription J(t) π(f)=U(t) π(f) U(t)* gives rise to an algebraic stochastic flow on the algebra generated by operators of the form π(f)=∫ f(g) π(g) dg where f is in the group algebra and dg is a left Haar measure. J(t) itself satisfies an operator-valued stochastic differential equation of a type which has been previously studied within the context of quantum stochastic calculus.

Suggested Citation

  • David Applebaum, 2001. "Operator-Valued Stochastic Differential Equations Arising from Unitary Group Representations," Journal of Theoretical Probability, Springer, vol. 14(1), pages 61-76, January.
  • Handle: RePEc:spr:jotpro:v:14:y:2001:i:1:d:10.1023_a:1007816930696
    DOI: 10.1023/A:1007816930696
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    References listed on IDEAS

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    1. Applebaum, D., 1994. "Unitary Actions of Lévy Flows of Diffeomorphisms," Journal of Multivariate Analysis, Elsevier, vol. 49(2), pages 266-277, May.
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