The Cutoff Hamilton-Jacobi-Bellman Operators of the Wick-Square Problem, with a General Counterterm

In the setting of The Wick-Square Problem on the Torus: Standing Notation, let $g:H^{-1}\to\mathbb{R}$ be a function (the running cost). Let $\varphi:H^{-1}\to\mathbb{R}$ be twice differentiable along the modes, let $x\in H^{-1}$ and $N\in\mathbb{N}$, and let $L_{N}\varphi(x)$ and $|D_{N}\varphi(x)|^{2}$ be the free-field generator and the gradient energy at cutoff $N$. Sums over $\Gamma_{N}$ are sums over a finite index set.

1. (With a counterterm)¶ Let $b:\mathbb{Z}^{n}\to\mathbb{R}$ be a family (the counterterm). The cutoff operator at cutoff $N$ with counterterm $b$ is

$$F_{N}^{b}[\varphi](x)=\gamma\,\varphi(x)-L_{N}\varphi(x)+\tfrac12\,|D_{N}\varphi(x)|^{2}-\beta\sum_{k\in\Gamma_{N}}\bigl(x(k)^{2}-b(k)\bigr)-g(x).$$

2. (Wick-ordered)¶ The Wick-ordered cutoff operator $F_{N}$ is $F_{N}^{b}$ with $b(k)=c_{k}$ the free-field variances; thus, with the cutoff Wick square,

$$F_{N}[\varphi](x)=\gamma\,\varphi(x)-L_{N}\varphi(x)+\tfrac12\,|D_{N}\varphi(x)|^{2}-\beta\,{:}x^{2}{:}_{N}-g(x).$$

3. (Bare)¶ The bare cutoff operator $F_{N}^{0}$ is $F_{N}^{b}$ with $b$ the zero family:

$$F_{N}^{0}[\varphi](x)=\gamma\,\varphi(x)-L_{N}\varphi(x)+\tfrac12\,|D_{N}\varphi(x)|^{2}-\beta\sum_{k\in\Gamma_{N}}x(k)^{2}-g(x).$$