The Renormalised Hamilton-Jacobi-Bellman Operator of the Wick-Square Problem as the Limit of the Wick-Ordered Cutoff Operators

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 $F_{N}$ be the Wick-ordered cutoff operators for $g$.

1. (Domain)¶ The domain $\mathcal{D}$ of the renormalised operator is the set of pairs $(\varphi,x)$ such that $\varphi:H^{-1}\to\mathbb{R}$ is twice differentiable along the modes, $x\in H^{-1}$, and the sequence $(F_{N}[\varphi](x))_{N\in\mathbb{N}}$ converges.

2. (Operator)¶ For $(\varphi,x)\in\mathcal{D}$, the renormalised operator is

$$F[\varphi](x)=\lim_{N\to\infty}F_{N}[\varphi](x),$$

the limit being unique by claim 1 of Uniqueness of Limits and Boundedness of Convergent Real Sequences.