The Renormalised Hamilton-Jacobi-Bellman Equation of the Wick-Square Problem

In the setting of The Wick-Square Problem on the Torus: Standing Notation, let $g:H^{-1}\to\mathbb{R}$ be a running cost, and let $\mathcal{D}$ and $F$ be the domain and the renormalised operator for $g$.

¶The renormalised Hamilton-Jacobi-Bellman equation of the Wick-square problem for a function $u:H^{-1}\to\mathbb{R}$ that is twice differentiable along the modes is

$$(u,x)\in\mathcal{D}\quad\text{and}\quad F[u](x)=0\qquad\text{for every }x\in H^{-1}.$$

Formally it reads $\gamma u-Lu+\tfrac12|Du|^{2}=\beta\,{:}x^{2}{:}+g$, with the free-field generator $L$, the gradient energy $|Du|^{2}$ and the Wick square; but the Wick square exists only on the Wick domain, and the equation requires only the limit of the combined cutoff expressions, not of each term separately.