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.
References
Applebaum, D.: The strong Markov property for Fermion Brownian motion. J. Func. Anal. 65, 273–291 (1986)
Barnett, C., Lyons, T.: Stopping non-commuting processes. Math. Proc. Camb. Phil. Soc. 99, 151–161 (1986)
Blumenthal, R.M., Getoor, R.K.: Markov processes and potential theory. New York: Academic Press 1968
Bratteli, O., Robinson, D.W.: Operator algebras and quantum statistical mechanics II. Berlin, Heidelberg, New York: Springer 1981
Chung, K.L.: Lectures from Markov processes to Brownian motion, sect. 4.4. Berlin, Heidelberg, New York: Springer 1980
Hudson, R.L.: The strong Markov property for canonical Wiener processes. J. Func. Anal. 34, 266–281 (1979)
Hudson, R.L., Parthasarathy, K.R.: Quantum Ito's formula and stochastic evolutions. Comm. Math. Phys. 93, 301–323 (1984)
Hudson, R.L., Parthasarathy, K.R.: Unification of Fermion and Boson stochastic calculus. Comm. Math. Phys. 104, 457–470 (1986)
McKean, H.P.: Stochastic integrals. New York: Academic Press 1969
Meyer, P.A.: Eléments de Probabilités quantiques. Séminaire de Probabilités. Univ. de Strassbourg 1984/85
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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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DOI: https://doi.org/10.1007/BF00318706
Keywords
- Brownian Motion
- Time Integral
- Statistical Theory
- Spectral Measure
- Markov Property