The Cutoff Hamilton-Jacobi-Bellman Equations of the Wick-Square Problem, Wick-Ordered and Bare

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 $N\in\mathbb{N}$, and let $u:H^{-1}\to\mathbb{R}$ be twice differentiable along the modes.

1. (Wick-ordered)¶ The Wick-ordered Hamilton-Jacobi-Bellman equation at cutoff $N$ for $u$ is: $F_{N}[u](x)=0$ for every $x\in H^{-1}$, with $F_{N}$ the Wick-ordered cutoff operator for $g$; that is,

$$\gamma\,u(x)-L_{N}u(x)+\tfrac12\,|D_{N}u(x)|^{2}=\beta\,{:}x^{2}{:}_{N}+g(x)\qquad(x\in H^{-1}).$$

2. (Bare)¶ The bare Hamilton-Jacobi-Bellman equation at cutoff $N$ for $u$ is: $F_{N}^{0}[u](x)=0$ for every $x\in H^{-1}$, with $F_{N}^{0}$ the bare cutoff operator for $g$; that is,

$$\gamma\,u(x)-L_{N}u(x)+\tfrac12\,|D_{N}u(x)|^{2}=\beta\sum_{k\in\Gamma_{N}}x(k)^{2}+g(x)\qquad(x\in H^{-1}).$$