Stop times in fock space stochastic calculus

Parthasarathy, K. R. ; Sinha, Kalyan B. (1987) Stop times in fock space stochastic calculus Probability Theory and Related Fields, 75 (3). pp. 317-349. ISSN 0178-8051

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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 S becomes 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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Deposited On:25 May 2011 13:35
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