Operators of Gamma white noise analysis

Details Operators of Gamma white noise analysis Operators of Gamma white noise analysis

2006-08-14

arxiv.org/abs/math/0608340v1

Infin. Dimens. anal. Quantum Probab. Relat. Top. 3 (2000), 303-335

Mathematics

Probability

Scientific paper

The paper is devoted to the study of Gamma white noise analysis. We define an extended Fock space $\Gama(\Ha)$ over $\Ha=L^2(\R^d, d\sigma)$, and show how to include the usual Fock space ${\cal F} (\Ha)$ in it as a subspace. We introduce in $\Gama(\Ha)$ operators $a(\xi)=\int_{\R^d} dx \xi(x)a(x)$, $\xi\in S$, with $a(x)=\dig_x+2\dig_x\di_x+1+\di_x +\dig_x\di_x\di_x$, where $\dig_x$ and $\di_x$ are the creation and annihilation operators at $x$. We show that $(a(\xi))_{\xi\in S}$ is a family of commuting selfadjoint operators in $\Gama(\Ha)$ and construct the Fourier transform in generalized joint eigenvectors of this family. This transform is a unitary $I$ between $\Gama(\Ha)$ and the $L^2$-space $L^2(S',d\mu_{\mathrm G})$, where $\mu_{\mathrm G}$ is the measure of Gamma white noise with intensity $\sigma$. The image of $a(\xi)$ under $I$ is the operator of multiplication by $\la\cdot,\xi\ra$, so that $a(\xi)$'s are Gamma field operators. The Fock structure of the Gamma space determined by $I$ coincides with that discovered in {\bf [}{\it Infinite Dimensional Analysis, Quantum Probability and Related Topics} {\bf 1} (1998), 91--117{\bf ]}. We note that $I$ extends in a natural way the multiple stochastic integral (chaos) decomposition of the ``chaotic'' subspace of the Gamma space. Next, we introduce and study spaces of test and generalized functions of Gamma white noise and derive explicit formulas for the action of the creation, neutral, and Gamma annihilation operators on these spaces.

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