Transformation Groups on White Noise Functionals and Their Applications

Transformation Groups on White Noise Functionals and Their Applications In this paper we first construct a two-parameter transformation group G on the space of test white noise functionals in which the adjoints of Kuo's Fourier and Kuo's Fourier—Mehler transforms are included. Next we show that the group G is a two-dimensional complex Lie group whose infinitesimal generators are the Gross Laplacian Δ G and the number operator N , and then find an explicit description of a differentiable one-parameter subgroup of G whose infinitesimal generator is aΔ G +bN . As an application, we study the solution and fundamental solution for the Cauchy problem associated with aΔ G +bN . Finally we show that each element of the adjoint group G * of G can be characterized in terms of differentiation and multiplication operators. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

Transformation Groups on White Noise Functionals and Their Applications

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Publisher
Springer-Verlag
Copyright
Copyright © Inc. by 1998 Springer-Verlag New York
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s002459900074
Publisher site
See Article on Publisher Site

Abstract

In this paper we first construct a two-parameter transformation group G on the space of test white noise functionals in which the adjoints of Kuo's Fourier and Kuo's Fourier—Mehler transforms are included. Next we show that the group G is a two-dimensional complex Lie group whose infinitesimal generators are the Gross Laplacian Δ G and the number operator N , and then find an explicit description of a differentiable one-parameter subgroup of G whose infinitesimal generator is aΔ G +bN . As an application, we study the solution and fundamental solution for the Cauchy problem associated with aΔ G +bN . Finally we show that each element of the adjoint group G * of G can be characterized in terms of differentiation and multiplication operators.

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Apr 1, 2023

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