A similarity between the Gross Laplacian and the Levy Laplacian

In this paper we present a construction of an infinite dimensional separable Hilbert space associated with a norm induced from the Levy trace. The space is slightly different from the Cesaro Hilbert space introduced in Ref. 1. The Levy Laplacian is discussed with a suitable domain which is constructed by a rigging of Fock spaces based on a rigging of Hilbert spaces with the Levy trace. Then the Levy Laplacian can be considered as the Gross Laplacian acting on a certain countable Hilbert space. By constructing one-parameter group of operators of which the infinitesimal generator is the Levy Laplacian, we study the existence and uniqueness of solution of heat equation associated with the Levy Laplacian. Moreover we give an infinite dimensional stochastic process generated by the Levy Laplacian.