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arXiv:math/0608380

Functional spaces and operators connected with some Lévy noises

E. Lytvynov

Published Aug 15, 2006• 2 citations
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Abstract

We review some recent developments in white noise analysis and quantum probability. We pay a special attention to spaces of test and generalized functionals of some Lévy white noises, as well as as to the structure of quantum white noise on these spaces.
DOI
10.48550/arxiv.math/0608380

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IntroductionLet ata_{t}at​ and at∗a_{t}^{*}at∗​ be respectively annihilation and creation operators at apoint t∈Rt\in \mathrm{R}t∈R .In the theory of white noise operators [23][24], it is known that ata_{t}at​ is acontinuous linear operator from Γu(Ec)\Gamma_{u}(E\mathrm{c})Γu​(Ec) , the (Fock) space of test functions over acomplexified nuclear space EcE_{\mathrm{c}}Ec​ , into itself, and at is acontinuous linear operator from Γu(EC)∗\Gamma_{u}(E_{\mathbb{C}})^{*}Γu​(EC​)∗ , the dual space of Γu(EC)\Gamma_{u}(E_{\mathbb{C}})Γu​(EC​) , into itself.In paticular, at+at∗a_{t}+a_{t}^{*}at​+at∗​ and at+at∗+at∗at+Ia_{t}+a_{t}^{*}+a_{t}^{*}a_{t}+Iat​+at∗​+at∗​at​+I are cffied the quantum Gaussian white noise and the quantum Poisson white noise, respectively (cf.[11] [24]).These are known to be elemental generalzed qunatum stochastic processes.The main purpose of this work is to examine relationships between classical and quantum white noises from the point of the classical white noise theory of Gaussian [19][20] and Poisson types [13].Then it will be clarified that quantum Gaussian and Poisson white noises are the Fock space realzations of classical Gaussian and Poisson white noises, respectively.We shall discuss them in Section 4. As amatter of fact, one can see that such facts can be proved not only through JJJ -transform given by (2.4) and the holomorphy, but also through Sx-transform gllven by (3.3) depending on the exponential function ϕξX\phi_{\xi}^{X}ϕξX​ .The function ϕξX\phi_{\xi}^{X}ϕξX​ determines aunitary isomorphism between Boson Fock space and L2(E∗,μx)L^{2}(E^{*},\mu x)L2(E∗,μx) , X=GX=GX=G, PPP , where μG\mu_{G}μG​ and μP\mu_{P}μP​ are a Gaussian measure and aPoisson measure on E∗E^{*}E∗ , respectively.As is pointed out in Remark 4.3, the choice of ϕξX\phi_{\xi}^{X}ϕξX​ (kernel function of SXS_{X}SX​ -transform)is the essential part to represent the above different quantum noises acting on the same space Γu(Ec)∗\Gamma_{u}(E\mathrm{c})^{*}Γu​(Ec)∗ .In Appendix, some connections between [6][7][6][7][6][7] and [10] wiU\mathrm{w}\mathrm{i}\mathrm{U}wiU be noted.