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qGaussian processes: Noncommutative and classical aspects
 Commun. Math. Phys
, 1997
"... Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expectation ..."
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Cited by 64 (2 self)
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Abstract. We examine, for −1 < q < 1, qGaussian processes, i.e. families of operators (noncommutative random variables) Xt = at + a ∗ t – where the at fulfill the qcommutation relations asa ∗ t − qa ∗ t as = c(s, t) · 1 for some covariance function c(·, ·) – equipped with the vacuum expectation state. We show that there is a qanalogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on qGaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of qGaussian processes possess a noncommutative kind of Markov property, which ensures that there exist classical versions of these noncommutative processes. This answers an old question of Frisch and Bourret [FB].
Documenta Math. 343 PartitionDependent Stochastic Measures and qDeformed Cumulants
, 2001
"... Abstract. On a qdeformed Fock space, we define multiple qLévy processes. Using the partitiondependent stochastic measures derived from such processes, we define partitiondependent cumulants for their joint distributions, and express these in terms of the cumulant functional using the number of r ..."
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Abstract. On a qdeformed Fock space, we define multiple qLévy processes. Using the partitiondependent stochastic measures derived from such processes, we define partitiondependent cumulants for their joint distributions, and express these in terms of the cumulant functional using the number of restricted crossings of P. Biane. In the single variable case, this allows us to define a qconvolution for a large class of probability measures. We make some comments on the Itô table in this context, and investigate the qBrownian motion and the qPoisson process in more detail.