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In this paper, we consider the Lévy Laplacian acting on multiple Wiener integrals by the stochastic process given as a difference of two independent Lévy processes, and give a necessary and sufficient condition for eigenfunctions of the Lévy Laplacian. Moreover we give a decomposition of the L2-space on Lévy noise probability space by eigenspaces consisting of multiple Wiener integrals by the above process in terms of the Lévy Laplacian. By this decomposition, we obtain an expression of the semigroup generated by the Lévy Laplacian related to the semigroup generated by the number operator.

A Decomposition of a Space of Multiple Wiener Integrals by the Difference of Two Independent Lévy Processes in Terms of the Lévy Laplacian