Lévy Laplacians in Hida Calculus and Malliavin Calculus

Abstract

Some connections between different definitions of Levy Laplacians in the stochastic analysis are considered. Two approaches are used to define these operators. The standard one is based on the application of the theory of Sobolev-Schwartz distributions over the Wiener measure (the Hida calculus). One can consider the chain of Levy Laplacians parametrized by a real parameter with the help of this approach. One of the elements of this chain is the classical Levy Laplacian. Another approach to define the Levy Laplacian is based on the application of the theory of Sobolev spaces over the Wiener measure (the Malliavin calculus). It is proved that the Levy Laplacian defined with the help of the second approach coincides with one of the elements of the chain of Levy Laplacians, which is not the classical Levy Laplacian, under the imbedding of the Sobolev space over the Wiener measure into the space of generalized functionals over this measure. It is shown which Levy Laplacian in the stochastic analysis is connected to the gauge fields.