The Renormalised Operator of the Wick-Square Problem: the Penalised Form on the Whole State Space, the Wick Domain without the Penalty, and the Forced Counterterm

In the setting of The Wick-Square Problem on the Torus: Standing Notation, let $g:H^{-1}\to\mathbb{R}$ be a running cost, let $\mathcal{D}$ and $F$ be the domain and the renormalised operator for $g$, let $F_{N}^{b}$ be the cutoff operators with counterterm $b$ for $g$, and $F_{N}^{0}$ the bare ones. Let $P$ be the Gaussian penalty, $c_{k}$ the free-field variances, $\mathcal{W}$ the Wick domain, and $L$ and $|D\cdot|^{2}$ the uncut free-field generator and gradient energy. Then the following hold.

1. (With the penalty: the whole state space, in penalised form)¶ Let $\varphi:H^{-1}\to\mathbb{R}$ be such that $\varphi-P$ is regular. Then $\varphi$ is twice differentiable along the modes, and for every $x\in H^{-1}$: $L(\varphi-P)$ is defined at $x$, $|D\varphi(x)|^{2}$ is defined, $(\varphi,x)\in\mathcal{D}$, and

$$F[\varphi](x)=\gamma\,\varphi(x)-L(\varphi-P)(x)+\tfrac12\,|D\varphi(x)|^{2}-g(x).$$

2. (Without the penalty: only the Wick domain)¶ Let $\theta\in\mathbb{R}$ with $\theta\ne1$, and let $\varphi:H^{-1}\to\mathbb{R}$ be such that $\varphi-\theta P$ is regular. Then $\varphi$ is twice differentiable along the modes, and for every $x\in H^{-1}$

$$(\varphi,x)\in\mathcal{D}\iff x\in\mathcal{W}.$$

3. (The Wick domain is dense)¶ For every $y\in H^{-1}$ and every real $\varepsilon>0$ there is $z\in\mathcal{W}$ with $d_{H^{-1}}(y,z)<\varepsilon$.

4. (The Wick domain contains no ball)¶ For every $x\in H^{-1}$ and every real $\varepsilon>0$ there is $z\in H^{-1}$ with $z\notin\mathcal{W}$ and $d_{H^{-1}}(x,z)<\varepsilon$.

5. (The counterterm is forced)¶ Let $b:\mathbb{Z}^{n}\to\mathbb{R}$, let $\varphi:H^{-1}\to\mathbb{R}$ be such that $\varphi-P$ is regular, and let $x\in H^{-1}$. Then the sequence $(F_{N}^{b}[\varphi](x))_{N\in\mathbb{N}}$ converges if and only if the family $k\mapsto c_{k}-b(k)$ is cube-summable. If moreover $2\le n$, the sequence $(F_{N}^{0}[\varphi](x))_{N\in\mathbb{N}}$ of bare cutoff operators is not bounded below.