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24
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 65 (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].
Noncommutative martingale inequalities
, 1997
"... We prove the analogue of the classical BurkholderGundy inequalites for noncommutative martingales. As applications we give a characterization for an ItoClifford integral to be an Lpmartingale via its integrand, and then extend the ItoClifford integral theory in L2, developed by Barnett, Streater ..."
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Cited by 41 (9 self)
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We prove the analogue of the classical BurkholderGundy inequalites for noncommutative martingales. As applications we give a characterization for an ItoClifford integral to be an Lpmartingale via its integrand, and then extend the ItoClifford integral theory in L2, developed by Barnett, Streater and Wilde, to Lp for all 1 < p < ∞. We include an appendix on the noncommutative analogue of the classical Fefferman duality between H¹ and BMO.
Transient and Recurrent Spectrum
, 1981
"... We deal primarily with spectral analysis of an abstract selfadjoint operator. H, on a Hilbert space, X”. We propose a further refinement of the absolutely continuous subspace,;F”a,, into the transient absolutely continuous subspace, &‘a,. which is the closure of those cp with (cp, ej/Ho) = O(t“) ..."
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Cited by 11 (2 self)
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We deal primarily with spectral analysis of an abstract selfadjoint operator. H, on a Hilbert space, X”. We propose a further refinement of the absolutely continuous subspace,;F”a,, into the transient absolutely continuous subspace, &‘a,. which is the closure of those cp with (cp, ej/Ho) = O(t“) for all N and the recurrent absolutely continuous subspace, 2 & = ‘qc nX&. We discuss general features of this breakup. In a subsequent paper, we construct analytic almost periodic functions, V, on (03, 03) so that H =d*/dx * + V(x) on L2(co, co) has only recurrent absolutely continuous spectrum in the sense that qa, =.;Y.
Holomorphic functions and the heat kernel measure on an infinite dimensional complex orthogonal group
 Potential Analysis
"... Abstract. The heat kernel measure µt is constructed on an infinite dimensional complex group using a diffusion in a Hilbert space. Then it is proved that holomorphic polynomials on the group are square integrable with respect to the heat kernel measure. The closure of these polynomials, HL2 (SOHS, µ ..."
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Cited by 11 (7 self)
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Abstract. The heat kernel measure µt is constructed on an infinite dimensional complex group using a diffusion in a Hilbert space. Then it is proved that holomorphic polynomials on the group are square integrable with respect to the heat kernel measure. The closure of these polynomials, HL2 (SOHS, µt), is one of two spaces of holomorphic functions we consider. The second space, HL2 (SO(∞)), consists of functions which are holomorphic on an analog of the CameronMartin subspace for the group. It is proved that there is an isometry from the first space to the second one. The main theorem is that an infinite dimensional nonlinear analog of the Taylor expansion defines an isometry from HL2 (SO(∞)) into the Hilbert space associated with a Lie algebra of the infinite dimensional group. This is an extension to infinite dimensions of an isometry of B. Driver and L. Gross for complex Lie groups. All the results of this paper are formulated for one concrete group, the HilbertSchmidt complex orthogonal group, though our methods can be applied in more general situations. 1.
qLévy processes
 J. Reine Angew. Math
, 2004
"... ABSTRACT. We continue the investigation of the Lévy processes on a qdeformed full Fock space started in [1]. First, we show that the vacuum vector is cyclic and separating for the algebra generated by such a process. Next, we describe a chaotic representation property in terms of multiple integrals ..."
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Cited by 10 (0 self)
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ABSTRACT. We continue the investigation of the Lévy processes on a qdeformed full Fock space started in [1]. First, we show that the vacuum vector is cyclic and separating for the algebra generated by such a process. Next, we describe a chaotic representation property in terms of multiple integrals with respect to diagonal measures, in the style of Nualart and Schoutens. We define stochastic integration with respect to these processes, and calculate their combinatorial stochastic measures. Finally, we show that they generate infinite von Neumann algebras. 1.
Hypercontractivity in noncommutative holomorphic spaces
 Commun. Math. Phys
, 2005
"... ABSTRACT. We prove an analog of Janson’s strong hypercontractivity inequality in a class of noncommutative “holomorphic ” algebras. Our setting is the qGaussian algebras Γq associated to the qFock spaces of Bozejko, Kümmerer and Speicher, for q ∈ [−1, 1]. We construct subalgebras Hq ⊂ Γq, a qSeg ..."
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Cited by 8 (6 self)
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ABSTRACT. We prove an analog of Janson’s strong hypercontractivity inequality in a class of noncommutative “holomorphic ” algebras. Our setting is the qGaussian algebras Γq associated to the qFock spaces of Bozejko, Kümmerer and Speicher, for q ∈ [−1, 1]. We construct subalgebras Hq ⊂ Γq, a qSegalBargmann transform, and prove Janson’s strong hypercontractivity L 2 (Hq) → L r (Hq) for r an even integer. 1.
ON THE CONVERGENCE OF GENERALIZED POLYNOMIAL CHAOS EXPANSIONS ∗
"... Abstract. A number of approaches for discretizing partial differential equations with random data are based on generalized polynomial chaos expansions of random variables. These constitute generalizations of the polynomial chaos expansions introduced by Norbert Wiener to expansions in polynomials or ..."
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Cited by 7 (1 self)
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Abstract. A number of approaches for discretizing partial differential equations with random data are based on generalized polynomial chaos expansions of random variables. These constitute generalizations of the polynomial chaos expansions introduced by Norbert Wiener to expansions in polynomials orthogonal with respect to nonGaussian probability measures. We present conditions on such measures which imply meansquare convergence of generalized polynomial chaos expansions to the correct limit and complement these with illustrative examples.
2000]: “Classical and quantum probability
 Journ. Math. Phys
"... We follow the development of probability theory from the beginning of the last century, emphasising that quantum theory is really a generalisation of this theory. The great achievements of probability theory, such as the theory of processes, generalised random fields, estimation theory and informati ..."
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Cited by 5 (0 self)
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We follow the development of probability theory from the beginning of the last century, emphasising that quantum theory is really a generalisation of this theory. The great achievements of probability theory, such as the theory of processes, generalised random fields, estimation theory and information geometry, are reviewed. Their quantum versions are then described.
Regular generalized functions in Gaussian analysis
 Infinite Dim. Anal. Quantum Prob
, 1997
"... The concepts of regular generalized functions in Gaussian analysis are presented. Spaces of regular generalized functions are characterized and their probabilistic structure is worked out. Finally, these concepts are applied to a nonlinear (Verhulst type) equation. Its solution is shown to be a regu ..."
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Cited by 4 (2 self)
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The concepts of regular generalized functions in Gaussian analysis are presented. Spaces of regular generalized functions are characterized and their probabilistic structure is worked out. Finally, these concepts are applied to a nonlinear (Verhulst type) equation. Its solution is shown to be a regular generalized process with martingale property.
Holomorphic Functions and Subelliptic Heat Kernels over Lie groups. ∗
, 2008
"... A Hermitian form q on the dual space, g ∗ , of the Lie algebra, g, of a Lie group, G, determines a subLaplacian, ∆, on G. It will be shown that Hörmander’s condition for hypoellipticity of the subLaplacian holds if and only if the associated Hermitian form, induced by q on the dual of the universa ..."
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Cited by 4 (1 self)
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A Hermitian form q on the dual space, g ∗ , of the Lie algebra, g, of a Lie group, G, determines a subLaplacian, ∆, on G. It will be shown that Hörmander’s condition for hypoellipticity of the subLaplacian holds if and only if the associated Hermitian form, induced by q on the dual of the universal enveloping algebra, U ′ , is nondegenerate. The subelliptic heat semigroup, e t∆/4, is given by convolution by a C ∞ probability density ρt. When G is complex and u: G → C is a holomorphic function, the collection of derivatives of u at the identity in G gives rise to an element, û(e) ∈ ∗Key words and phrases. Subelliptic, heat kernel, complex groups, universal enveloping algebra, Taylor map.