Lévy Laplacians and annihilation process

The Lévy Laplacians are infinite-dimensional Laplace operators defined as the Cesaro mean of the second-order directional derivatives. In the theory of Sobolev–Schwarz distributions over a Gaussian measure on an infinite-dimensional space (the Hida calculus), we can consider two canonical Lévy Laplacians. The first Laplacian, the so-called classical Lévy Laplacian, has been well studied. The interest in the second Laplacian is due to its connection with the Malliavin calculus (the theory of Sobolev spaces over the Wiener measure) and the Yang–Mills gauge theory. The representation in the form of the quadratic function of the annihilation process for the classical Lévy-Laplacian is known. This representation can be obtained using the $S$-transform (the Segal–Bargmann transform). In the paper, we show, by analogy, that the representation in the form of the quadratic function of the derivative of the annihilation process exists for the second Lévy-Laplacian. The obtained representation can be used for studying the gauge fields and the Lévy Laplacian in the Malliavin calculus.