On Markovian cocycle perturbations in classical and quantum probability
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...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2003-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171203211200 |
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| Summary: | 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. |
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| ISSN: | 0161-1712 1687-0425 |