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Contractions and central extensions of quantum white noise Lie algebras

Accardi L., Boukas A.

Probability and Mathematical Statistics

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2015

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Vol. 35, Fasc. 1

41--72

EN

We show that the Renormalized Powers of Quantum White Noise Lie algebra RPQWN*, with the convolution type renormalization δn(t − s) = δ(s)δ(t − s) of the n ≥ 2 powers of the Dirac delta function, can be obtained through a contraction of the Renormalized Powers of Quantum White Noise Lie algebra RPQWNc with the scalar renormalization δn(t) = cn−1δ(t), c > 0. Using this renormalization, we also obtain a Lie algebra W∞(c) which contains the w∞ Lie algebra of Bakas and the Witt algebra as contractions. Motivated by the W∞ algebra of Pope, Romans and Shen, we show that W∞(c) can also be centrally extended in a non-trivial fashion. In the case of the Witt subalgebra of W∞, the central extension coincides with that of the Virasoro algebra.

2

The quantum decomposition associated with the Lévy white noise processes without moments

Accardi L., Riahi A., Rebei H.

Probability and Mathematical Statistics

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2014

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Vol. 34, Fasc. 2

337--362

EN

The theory of one-mode type Interacting Fock Space (IFS) allows us to construct the quantum decomposition associated with stochastic processes on R with moments of any order. The problem to extend this result to processes without moments of any order is still open but the Araki-Woods-Parthasarathy-Schmidt characterization of Lévy processes in terms of boson Fock spaces, canonically associated with the Lévy-Khintchine functions of these processes, provides a quantum decomposition for tchem which is based on boson creations, annihilation and preservation operators rather than on their IFS counterparts. In order to compare the two quantum decompositions in their common domain of application (i.e., the Lévy processes with moments of all orders) the first step is to give a precise formulation of the quantum decomposition for these processes and the analytical conditions of its validity. We show that these conditions distinguish three different notions of quantum decomposition of a Lévy process on R according to the existence of second or only first moments, or no moments at all. For the last class a multiplicative renormalization procedure is needed.

3

Quantum Laplacians on generalized operators on boson fock space

Accardi L., Barhoumi A., Ji U. C.

Probability and Mathematical Statistics

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2011

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Vol. 31, Fasc. 2

203--225

EN

By adapting the white noise theory, the quantum analogues of the (classical) Gross Laplacian and Lévy Laplacian, so called the quantum Gross Laplacian and quantum Lévy Laplacian, respectively, are introduced as the Laplacians acting on the spaces of generalized operators. Then the integral representations of the quantum Laplacians in terms of quantum white noise derivatives are studied. Correspondences of the classical Laplacians and quantum Laplacians are studied. The solutions of heat equations associated with the quantum Laplacians are obtained from a normal-ordered white noise differential equation.

4

Stochastic evolutions driven by non-linear white noise

Accardi L., Boukas A.

Probability and Mathematical Statistics

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2002

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Vol. 22, Fasc. 1

141--154

EN

We prove the existence and uniqueness theorem for stochastic differential equations with bounded coefficients driven by the renormalized square of white noise.These equations are interpreted as sesquilinear forms on the linear span of the exponential vectors (of the first order white noise) and the existence theorem is establishedon the space of these forms.