Stop times in fock space stochastic calculus

Abstract

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.