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arXiv:math/0204327 · International Journal of Mathematics and Mathematical Sciences

On Markovian cocycle perturbations in classical and quantum probability

Grigori Gennadievich Amosov

Published Jan 1, 2003• 13 citations
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Abstract

We introduce Markovian cocycle perturbations of the groups of transformations associated with classical and quantum stochastic processes with stationary increments, which are characterized by a localization of the perturbation to the algebra of events of the past. The Markovian cocycle perturbations of the Kolmogorov flows associated with the classical and quantum noises result in the perturbed group of transformations which can be decomposed into the sum of two parts. One part in the decomposition is associated with a deterministic stochastic process lying in the past of the initial process, while another part is associated with the noise isomorphic to the initial one. This construction can be considered as some analog of the Wold decomposition for classical stationary processes excluding a nondeterministic part of the process in the case of the stationary quantum stochastic processes on the von Neumann factors which are the Markovian perturbations of the quantum noises. For the classical stochastic process with noncorrelated increments, the model of Markovian perturbations describing all Markovian cocycles up to a unitary equivalence of the perturbations has been constructed. Using this model, we construct Markovian cocycles transforming the Gaussian state ρ to the Gaussian states equivalent to ρ.

DOI
10.1155/s0161171203211200

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We introduce Markovian cocycle perturbations of the groups of transformations associated with classical and quantum stochastic processes with stationary increments, which are characterized by a localization of the perturbation to the algebra of events of the past. The Markovian cocycle perturbations of the Kolmogorov flows associated with the classical and quantum noises result in the perturbed group of transformations which can be decomposed into the sum of two parts. One part in the decomposition is associated with a deterministic stochastic process lying in the past of the initial process, while another part is associated with the noise isomorphic to the initial one. This construction can be considered as some analog of the Wold decomposition for classical stationary processes excluding a nondeterministic part of the process in the case of the stationary quantum stochastic processes on the von Neumann factors which are the Markovian perturbations of the quantum noises. For the classical stochastic process with noncorrelated increments, the model of Markovian perturbations describing all Markovian cocycles up to a unitary equivalence of the perturbations has been constructed. Using this model, we construct Markovian cocycles transforming the Gaussian state ρ to the Gaussian states equivalent to ρ.
We introduce “Markovian” cocycle perturbations of quantum stochastic processes with stationary increments and the Kolmogorov flows generated by them, which are characterized by a localization of the perturbation to the algebra of events of the past. The Markovian perturbations of the Kolmogorov flow generated by the quantum white noise result in the groups of automorphisms on the algebras of events (the von Neumann algebras in the quantum case) possessing the restrictions being isomorphic to the initial Kolmogorov flow. The possibility of obtaining this restriction can be interpreted as some analogue (in the quantum case) of the Wold decomposition, which allows us to exclude the “nondeterministic” part of the process.
Stationary quantum stochastic process j is introduced as a *-homomorphism embedding an involutive graded algebra K~=⊕i=1∞Ki\tilde K=\oplus_{i=1}^{\infty}K_iK~=⊕i=1∞​Ki​ into a ring of (abelian) cohomologies of the one-parameter group ααα consisting of *-automorphisms of certain operator algebra in a Hilbert space such that every x from KiK_iKi​ is translated into an additive i−αi-αi−α-cocycle j(x). It is shown that (noncommutative) multiplicative markovian cocycle defines a perturbation of the stationary quantum stochastic process in the sense of such definition. The E0E_0E0​-semigroup β~\tilde ββ~​ on the von Neumann algebra N\cal NN associated with the markovian perturbation of K-flow j posseses the restriction β~∣N0,N0⊂N\tilde β|_{{\cal N}_0}, {\cal N}_0\subset {\cal N}β~​∣N0​​,N0​⊂N, which is conjugate to the flow of Powers shifts βββ associated with j. It yields for β~\tilde ββ~​ an analogue of the Wold decomposition for classical stochastic process on completely nondeterministic and deterministic parts. The examples of quantum stationary stochastic processes on the algebras of canonical commutation, anticommutation and square of white noise relations are considered. In the model situation of the space L2(R)L^2(\mathbb R)L2(R) all markovian cocycles of the group of shifts are described up to unitary equivalence of perturbations.
We develop fully noncommutative Feynman–Kac formulae by employing quantum stochastic processes. To this end, we establish some theory for perturbing quantum stochastic flows on von Neumann algebras by multiplier cocycles. Multiplier cocycles are constructed via quantum stochastic differential equations whose coefficients are driven by the flow. The resulting class of cocycles is characterized under alternative assumptions of separability or Markov regularity. Our results generalize those obtained using classical Brownian motion on the one hand, and results for unitarily implemented flows on the other.
Every Markov-regular quantum Levy process on a multiplier C*-bialgebra is shown to be equivalent to one governed by a quantum stochastic differential equation, and the generating functionals of norm-continuous convolution semigroups on a multiplier C*-bialgebra are then completely characterised. These results are achieved by extending the theory of quantum Levy processes on a compact quantum group, and more generally quantum stochastic convolution cocycles on a C*-bialgebra, to locally compact quantum groups and multiplier C*-bialgebras. Strict extension results obtained by Kustermans, together with automatic strictness properties developed here, are exploited to obtain existence and uniqueness for coalgebraic quantum stochastic differential equations in this setting. Then, working in the universal enveloping von Neumann bialgebra, we characterise the stochastic generators of Markov-regular, *-homomorphic (respectively completely positive and contractive), quantum stochastic convolution cocycles.
A concept of quantum stochastic convolution cocycle is introduced and studied in two different contexts -- purely algebraic and operator space theoretic. A quantum stochastic convolution cocycle is a quantum stochastic process on a coalgebra satisfying the convolution cocycle relation and the initial condition given by the counit. The notion generalises that of quantum Levy process, which in turn is a noncommutative probability counterpart of classical Levy process on a group. Convolution cocycles arise as solutions of quantum stochastic differential equations. In turn every sufficiently regular cocycle satisfies an equation of that type. This is proved along with the corresponding existence and uniqueness of solutions for coalgebraic quantum stochastic differential equations. The stochastic generators of unital *-homomorphic cocycles are characterised in terms of structure maps on a *-bialgebra. This yields a simple proof of the Schurmann Reconstruction Theorem for a quantum Levy process; it also yields a topological version for a quantum Levy process on a C*-bialgebra. Precise characterisation of the stochastic generators of completely positive and contractive quantum stochastic convolution cocycles in the C*-algebraic context is given, leading to some dilation results. A few examples are presented and some interpretations offered for quantum stochastic convolution cocycles and their stochastic generators on different types of *-bialgebra.