A massive Euclidean Gaussian generalized field exists

Let $I={\mathbb{N}}_0^2$, enumerated by total degree and lexicographic order. For each finite $F\subset I$, let ${\mu }_F$ have density ${\prod }_{\alpha \in F}(2\pi {)}^{-1\mathrm{/}2}{e}^{-x_{\alpha }^2\mathrm{/}2}$ on ${\mathbb{R}}^F$. [F5] makes these probability laws; integrating out coordinates with [F6] proves their consistency. Each coordinate space $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ is standard Borel by [F4]. Applying [F1] with [A1] yields $\mathrm{\Omega }={\mathbb{R}}^I$ with its cylinder $\sigma$-algebra and a probability $\mathbb{P}$ having these finite marginals. The coordinate variables ${\xi }_{\alpha }(\omega )={\omega }_{\alpha }$ are therefore independent standard normals.

For $p\in {\mathbb{R}}^2$ put $w_m(p)=(\mathrm{\mid }p{\mathrm{\mid }}^2+m^2{)}^{-1\mathrm{/}2}$. Since $m>0$, $w_m$ is smooth, and every derivative is bounded: its derivatives are finite sums of polynomial factors divided by positive powers of $\mathrm{\mid }p{\mathrm{\mid }}^2+m^2$, bounded near the origin and decaying at infinity. The product rule therefore shows multiplication by $w_m$ maps Schwartz space continuously to itself. Define $T_{m}f={\mathcal{F}}_{2\pi }^{-1}(w_m(2\pi \cdot ){\mathcal{F}}_{2\pi }f)$. By [F8], $T_{m}f$ is Schwartz; because $w_m$ is real and even, $T_{m}f$ is real for real $f$. Under the exact convention in the Statement, $\widehat{f}(p)={\mathcal{F}}_{2\pi }f(p\mathrm{/}(2\pi ))$, and the change of variables $p=2\pi \xi$ gives $\widehat{T_{m}f}(p)=w_m(p)\widehat{f}(p)$.

For a finite initial segment $I_N$ of the fixed enumeration set $S_N={\sum }_{\alpha \in I_N}(1+\mathrm{\mid }\alpha \mathrm{\mid }{)}^{-4}{\xi }_{\alpha }^2$. The density in [F5] is normalized, and integration by parts using $(-x{e}^{-x^2\mathrm{/}2}{)}^{\mathrm{\prime }}=(x^2-1)e^{-x^2\mathrm{/}2}$ gives $\mathbb{E}{\xi }_{\alpha }^2=1$. Hence $\mathbb{E}{S}_N={\sum }_{\alpha \in I_N}(1+\mathrm{\mid }\alpha \mathrm{\mid }{)}^{-4}\le {\sum }_{n\ge 0}(n+1)(n+1{)}^{-4}={\sum }_{n\ge 0}(n+1{)}^{-3}=:C<\mathrm{\infty }$: grouping $n+1$ in dyadic blocks $[2^j,2^{j+1})$ bounds each block by $2^j{2}^{-3j}=2^{-2j}$, a summable geometric series. Markov's inequality, which follows from ${\mathbf{1}}_{\{S_N>M\}}\le S_N\mathrm{/}M$, gives $\mathbb{P}(S_N>M)\le C\mathrm{/}M$. Since the events increase to $\{S:={\sum }_{\alpha }(1+\mathrm{\mid }\alpha \mathrm{\mid }{)}^{-4}{\xi }_{\alpha }^2>M\}$, countable additivity gives $\mathbb{P}(S>M)\le C\mathrm{/}M$; letting integer $M\to \mathrm{\infty }$ shows $S<\mathrm{\infty }$ on a common event ${\mathrm{\Omega }}_0$ of probability one.

For $f\in {\mathcal{S}}_{\mathbb{R}}({\mathbb{R}}^2)$ put $c_{\alpha }(f)=⟨f,h_{\alpha }⟩$. On ${\mathrm{\Omega }}_0$, [F11] and [F3] give ${\sum }_{\alpha }\mathrm{\mid }{\xi }_{\alpha }{c}_{\alpha }(f)\mathrm{\mid }\le S^{1\mathrm{/}2}({\sum }_{\alpha }(1+\mathrm{\mid }\alpha \mathrm{\mid }{)}^4\mathrm{\mid }{c}_{\alpha }(f){\mathrm{\mid }}^2{)}^{1\mathrm{/}2}<\mathrm{\infty }$. Thus $W_0(f):={\sum }_{\alpha }{\xi }_{\alpha }{c}_{\alpha }(f)$ converges absolutely for every $f$ on the same event, is real-linear, and satisfies $\mathrm{\mid }{W}_0(f)\mathrm{\mid }\le S^{1\mathrm{/}2}\mathrm{\parallel }{H}^{2}f{\mathrm{\parallel }}_2$. The last quantity is a continuous Schwartz seminorm by [F3], so each sample is a tempered distribution. Set $W_0=0$ outside ${\mathrm{\Omega }}_0$. Every evaluation $W_0(f)$ is measurable as the almost-sure limit of measurable finite sums, hence $W_0$ is measurable for the cylinder $\sigma$-algebra.

The partial sums for $W_0(f)$ converge in $L^2(\mathrm{\Omega },\mathbb{P})$ as well: independence, unit variance, and Parseval in [F10] make their squared $L^2$ tail equal to the corresponding tail of ${\sum }_{\alpha }\mathrm{\mid }{c}_{\alpha }(f){\mathrm{\mid }}^2=\mathrm{\parallel }f{\mathrm{\parallel }}_2^2$. For any finite real combination ${\sum }_j{t}_j{W}_0(f_j)$, a finite partial sum is a linear combination of independent standard normals, with characteristic function ${\prod }_{\alpha }{e}^{-t^2{c}_{\alpha }^2\mathrm{/}2}=e^{-t^2{\sum }_{\alpha }{c}_{\alpha }^2\mathrm{/}2}$; by [F10] its variance converges to $\mathrm{\parallel }{\sum }_j{t}_j{f}_j{\mathrm{\parallel }}_2^2$. If $X_N\to X$ in $L^2$, then $\mathrm{\mid }\mathbb{E}{e}^{it{X}_N}-\mathbb{E}{e}^{itX}\mathrm{\mid }\le \mathrm{\mid }t\mathrm{\mid }\mathbb{E}\mathrm{\mid }{X}_N-X\mathrm{\mid }\le \mathrm{\mid }t\mathrm{\mid }\mathrm{\parallel }{X}_N-X{\mathrm{\parallel }}_2\to 0$, so the limit has the centered normal characteristic function with that variance. For positive variance this is the characteristic function of the density defining $N(0,v)$, by [F5] and scaling; at zero variance it is the characteristic function of the point mass at zero. Applying [F13] to these Borel probability laws identifies the limit law. Hence the family is jointly Gaussian, centered, and $\mathbb{E}[W_0(f)W_0(g)]=\int f(x)g(x)d^{2}x$.

Set $\mathrm{\Phi }(f)=W_0(T_{m}f)$. Each sample in ${\mathrm{\Omega }}_0$ is a continuous real-linear functional of $f$ because both $W_0$ and $T_m$ are continuous on Schwartz space; outside ${\mathrm{\Omega }}_0$ it is zero. Thus $\mathrm{\Phi }$ is a measurable ${\mathcal{S}}_{\mathbb{R}}^{\mathrm{\prime }}$-valued random variable for the cylinder $\sigma$-algebra. From step 4.1 its covariance is $\mathbb{E}[\mathrm{\Phi }(f)\mathrm{\Phi }(g)]=\int T_{m}f(x)T_{m}g(x)d^{2}x$. Applying [F9] and changing variables $p=2\pi \xi$ gives $(2\pi {)}^{-2}\int \widehat{T_{m}f}(p)\overline{\widehat{T_{m}g}(p)}{d}^{2}p$. Since $f,g$ are real, $\widehat{f}(-p)=\overline{\widehat{f}(p)}$ and $\widehat{g}(-p)=\overline{\widehat{g}(p)}$; the substitution $p\mapsto -p$ changes this to $(2\pi {)}^{-2}\int \overline{\widehat{f}(p)}\widehat{g}(p)w_m(p{)}^2{d}^{2}p=C_m(f,g)$ by step 1.2. The finite-dimensional distributions remain centered Gaussian by step 4.1 and linearity of $T_m$.