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Convolutions of generalized white noise functionals

Ji U. C., Kim Y. Y., Park Y. J.

Probability and Mathematical Statistics

2013

Vol. 33, Fasc. 2

435--450

We study a general definition of convolution products of test white noise functionals, of which the consistency property is examined. As an application of the consistency property of the convolution product we study an extension of the convolution to generalized white noise functionals. We also study relations between the convolution and generalized Fourier-Gauss and generalized Fourier-Mehler transforms.

Quantum Laplacians on generalized operators on boson fock space

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

Probability and Mathematical Statistics

2011

Vol. 31, Fasc. 2

203--225

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.