Stochastic extensions of symbols in Wiener spaces and heat operator

Abstract

The construction, in [AJN], of a pseudodifferential calculus analogous to the Weyl calculus, in an infinite dimensional setting, required the introduction of convenient classes of symbols. In this article, we proceed with the study of these classes in order to establish, later on, the properties that a pseudodifferential calculus is expected to satisfy. The introduction and the study of a new class are rendered necessary in view of applications in QED. We prove here that the symbols of both classes and the terms of their Taylorexpansions admit stochastic extensions. We define, in this infinite dimensional setting, a semigroup $H_t$ analogous to the heat semigroup, acting on the symbols belonging to both classes of symbols. The heat operator commutes with a second order operator similar to the Laplacian, which is its infinitesimal generator. For the class defined there, we give an expansion in powers of $t$ of $H_tf$,according to the classes of symbols.