  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.
