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Publications of the Research Institute for Mathematical Sciences

Stopping Unitary Processes in Fock Space

David Applebaum

Published Oct 31, 1988• 3 citations
Publisher

Abstract

Let P T be a stop time in symmetric Fock space over L Z (R + ).We show that to certain unitary operator valued processes U=(U(t),t^Q) which satisfy a stochastic differential equation driven by "non-commutative noise" in Fock space tensored with an "initial" Hilbert space, we can associate a stopped operator U(T).We use these operators to prove a "stopped cocycle relation" whereby for P T finite the process (U(T+t),t^Q) is factorised as the product of U(T) with a new process U T = (U T (t} , f^O) beginning "after time T".

DOI
10.2977/prims/1195174688

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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 L2(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 L2(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 = Γτ] ⊗ Γ[τ, where Γτ] 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.

Journal of Mathematical Physics•Aug 1, 2015•0 citations
Let P T be a stop time in symmetric Fock space over L Z (R + ).We show that to certain unitary operator valued processes U=(U(t),t^Q) which satisfy a stochastic differential equation driven by "non-commutative noise" in Fock space tensored with an "initial" Hilbert space, we can associate a stopped operator U(T).We use these operators to prove a "stopped cocycle relation" whereby for P T finite the process (U(T+t),t^Q) is factorised as the product of U(T) with a new process U T = (U T (t} , f^O) beginning "after time T".
Abstract. A recent characterisation of Fock-adapted contraction operator stochastic cocycles on a Hilbert space, in terms of their associated semigroups, yields a general principle for the construction of such cocycles by approximation of their stochastic generators. This leads to new existence results for quantum stochastic differential equations. We also give necessary and sufficient conditions for a cocycle to satisfy such an equation. In this paper we study the functional equation V0 = I, Vr+t = Vrσr(Vt) for all r, t ≥ 0. for a family of contractions on h ⊗ F adapted to the Fock operator filtration. Here F is the symmetric Fock space over L 2 (R+; k), h and k are fixed but arbitrary
We review Fock space based quantum probability and in particular, the theory of stop times based on it. In the Fock space , a stop time may be defined as a positive self-adjoint operator whose spectral resolution is adapted to the natural filtration based on the splittings Then if K is one of the basic quantum martingales (creation, preservation or annihilation) the corresponding process beginning anew at time T can be defined unambiguosly using functional calculus for the self-adjoint operator T with an operator-valued integrand since the integrand commutes with the integrator. An isometric operator , the forward shift through T, is defined by a similar spectral integral of sure forward shifts and conjugates with . An optional stopping theorem is formulated and proved using similar operator valued spectral integrals.
We develop a rather general framework for constructing quantum Markov processes through Markov operator cocycles (see Chapter 2, Section 3) that satisfy a quantum stochastic differential equation.In order to achieve this goal we first recall the basic facts of Boson Fock quantum stochastic calculus and then give the fundamental results in the theory of quantum stochastic differential equations concerning existence, uniqueness, time reversal, isometricity and coisometricity of solutions. Next we construct the quantum flow associated with a Markov operator cocycle, and give a condition that guarantees that the restriction to a commutative subalgebra is a commutative flow.
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 L2(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 L2(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 = Γτ] ⊗ Γ[τ, where Γτ] 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.