Publication Date:
2015-08-15
Description:
Following the ideas of Hudson [J. Funct. Anal. 34 (2), 266-281 (1979)] and Parthasarathy and Sinha [Probab. Theory Relat. Fields 73 , 317-349 (1987)], we define a quantum stopping time (QST, for short) τ in the interacting Fock space (IFS, for short), Γ, over L 2 (ℝ + ), which is actually a spectral measure in [0, ∞] such that τ ([0, t ]) is an adapted process. Motivated by Parthasarathy and Sinha [Probab. Theory Relat. Fields 73 , 317-349 (1987)] and Applebaum [J. Funct. Anal. 65 , 273-291 (1986)], we also develop a corresponding quantum stopping time stochastic integral (QSTSI, for abbreviations) on the IFS over a subspace of L 2 (ℝ + ) equipped with a filtration. As an application, such integral provides a useful tool for proving that Γ admits a strong factorisation, i.e., Γ = Γ τ ] ⊗ Γ [ τ , where Γ τ ] and Γ [ τ stand for the part “before τ ” and the part “after τ ,” respectively. Additionally, this integral also gives rise to a natural composition operation among QST to make the space of all QSTs a semigroup.
Print ISSN:
0022-2488
Electronic ISSN:
1089-7658
Topics:
Mathematics
,
Physics
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