A Necessary Condition for the Continuity of Linear Functionals of Wick Squares

Abstract

The Dynkin Isomorphism Theorem “establishes a relationship between a Gaussian random field associated with a symmetric Markov process (the free field) and the local times for the process. The free field associated with the Brownian motion plays an important role in the constructive quantum field theory.” [3]. However, to be more precise, Gaussian random fields can be associated with a relatively small class of symmetric Markov processes in this way, essentially a subset of processes in R ¹. In more general cases one can consider linear functionals of Wick powers of the free field. This is explained in Section 8 of [3]. The functionals considered are Gaussian chaoses indexed by measures. We refer to them as {H = H(μ), μ ∈ ℳ} and describe them in detail below. We consider these only for the second Wick power, which is what we mean by Wick squares. These are second order Gaussian chaoses.