Mollified massive two-dimensional Gaussian Wick powers

Assume the Axiom of Choice, its countable-family consequence, and $m>0$. Let $\mathrm{\Phi }$ be the real centered Gaussian random tempered distribution on ${\mathbb{R}}^2$ with covariance $C_m(f,g)=(2\pi {)}^{-2}{\int }_{{\mathbb{R}}^2}\overline{\widehat{f}(p)}\widehat{g}(p)(\mathrm{\mid }p{\mathrm{\mid }}^2+m^2{)}^{-1}{d}^{2}p$ for real Schwartz tests and $\widehat{f}(p)=\int e^{-ip\cdot x}f(x)d^{2}x$. Write ${\mu }_C$ for the Gaussian probability measure on the cylinder $\sigma$-algebra of ${\mathcal{S}}_{\mathbb{R}}^{\mathrm{\prime }}({\mathbb{R}}^2)$ supplied by A massive Euclidean Gaussian generalized field exists, equivalently the law of $\mathrm{\Phi }$; all expectations below are with respect to ${\mu }_C$. Fix one nonnegative $\rho \in C_c^{\mathrm{\infty }}({\mathbb{R}}^2)$ with $\int \rho (x)d^{2}x=1$, set ${\rho }_{ϵ}(x)={ϵ}^{-2}\rho (x\mathrm{/}ϵ)$ for $ϵ>0$, and define ${\mathrm{\Phi }}_{ϵ}(x)=\mathrm{\Phi }(u\mapsto {\rho }_{ϵ}(x-u))$. For each $n\ge 0$, define the finite-cutoff Wick polynomial $:\text{\hspace{0.17em}⁣}{\mathrm{\Phi }}_{ϵ}(x{)}^n\text{\hspace{0.17em}⁣}:$ by

$\exp \text{\hspace{0.17em}⁣}(t{\mathrm{\Phi }}_{ϵ}(x)-\frac{1}{2}{t}^2\mathbb{E}[{\mathrm{\Phi }}_{ϵ}(x{)}^2])={\sum }_{n=0}^{\mathrm{\infty }}:\text{\hspace{0.17em}⁣}{\mathrm{\Phi }}_{ϵ}(x{)}^n\text{\hspace{0.17em}⁣}:\frac{t^n}{n!}$

as an identity of formal power series. For a bounded measurable $f:{\mathbb{R}}^2\to \mathbb{C}$ that vanishes outside a bounded set, define its cutoff smearing by the samplewise Lebesgue integral $W_{ϵ,n}(f)=\int f(x):\text{\hspace{0.17em}⁣}{\mathrm{\Phi }}_{ϵ}(x{)}^n\text{\hspace{0.17em}⁣}:d^{2}x$. Every tempered sample gives a smooth $x\mapsto {\mathrm{\Phi }}_{ϵ}(x)$, so this integral is finite for each sample; its $L^2({\mu }_C)$ limit as $ϵ\downarrow 0$ is a separate theorem, not part of this definition.

A massive Euclidean Gaussian generalized field exists

Finite covariance Wick normal powers