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Stop times in fock space stochastic calculus
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  • Published: July 1987

Stop times in fock space stochastic calculus

  • K. R. Parthasarathy1 &
  • Kalyan B. Sinha1 

Probability Theory and Related Fields volume 75, pages 317–349 (1987)Cite this article

Summary

A stop time S in the boson Fock space ℋ over L 2(ℝ)+ is a spectral measure in [0,∞] such that {S([0,t])} is an adapted process. Following the ideas of Hudson [6], to each stop time S a canonical shift operator U Sis constructed in ℋ. When S({∞}) has the vacuum as a null vector U Sbecomes an isometry. When S({∞})=0 it is shown that ℋ admits a factorisation ℋ S]⊗ℋ{S where ℋ{S is the range of U Sand ℋ S] is a suitable subspace of ℋ called the Fock space upto time S. This, in particular, implies the strong Markov property of quantum Brownian motion in the boson as well as fermion sense and the Dynkin-Hunt property that the classical Brownian motion begins afresh at each stop time. The stopped Weyl and fermion processes are defined and their properties studied. A composition operation is introduced in the space of stop time to make it a semigroup. Stop time integrals are introduced and their properties constitute the basic tools for the subject.

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Authors and Affiliations

  1. Indian Statistical Institute, 110016, New Dehli, India

    K. R. Parthasarathy & Kalyan B. Sinha

Authors
  1. K. R. Parthasarathy
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  2. Kalyan B. Sinha
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Parthasarathy, K.R., Sinha, K.B. Stop times in fock space stochastic calculus. Probab. Th. Rel. Fields 75, 317–349 (1987). https://doi.org/10.1007/BF00318706

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  • Received: 10 June 1986

  • Issue Date: July 1987

  • DOI: https://doi.org/10.1007/BF00318706

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Keywords

  • Brownian Motion
  • Time Integral
  • Statistical Theory
  • Spectral Measure
  • Markov Property
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