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The construction, in [AJN], of a pseudodifferential calculus analogous to theWeyl calculus, in an infinite dimensional setting, required the introduction ofconvenient classes of symbols.  In this article, we proceed with the study of these classes in order toestablish, later on, the properties that a pseudodifferential calculus isexpected to satisfy. The introduction and the study of a new class are renderednecessary in view of applications in QED.  We prove here that the symbols of both classes and the terms of theirTaylorexpansions admit stochastic extensions. We define, in this infinitedimensional setting, a semigroup $H_t$ analogous to the heat semigroup, actingon the symbols belonging to both classes of symbols. The heat operator commuteswith a second order operator similar to the Laplacian, which is itsinfinitesimal generator. For the class defined there, we give an expansion inpowers of $t$ of $H_tf$,according to the classes of symbols.

Stochastic extensions of symbols in Wiener spaces and heat operator