The Galerkin Solutions of the Wick-Square Problem Converge to a Classical Solution of the Renormalised Equation, while the Bare Solutions Diverge

In the setting of The Wick-Square Problem on the Torus: Standing Notation, take the running cost $g$ to be the zero function on $H^{-1}$, and let $F_{N}$, $F_{N}^{0}$, $\mathcal{D}$ and $F$ be the Wick-ordered and bare cutoff operators, and the domain and renormalised operator, for this $g$. Let $q_{k}$ and $a_{k}$ be the Riccati coefficients and renormalised constants, $c_{k}$ the free-field variances, and $P$ the Gaussian penalty. For $N\in\mathbb{N}$ and $x\in H^{-1}$ put

$$u_{N}(x)=\sum_{k\in\Gamma_{N}}\bigl(q_{k}\,x(k)^{2}+a_{k}\bigr),\qquad C_{N}=\beta\sum_{k\in\Gamma_{N}}c_{k},\qquad v_{N}(x)=u_{N}(x)+\frac{C_{N}}{\gamma}.$$

Then the following hold.

1. (Galerkin solutions)¶ For every $N\in\mathbb{N}$, $u_{N}$ is twice differentiable along the modes and solves the Wick-ordered cutoff equation: $F_{N}[u_{N}](x)=0$ for every $x\in H^{-1}$.

2. (Bare solutions diverge)¶ For every $N\in\mathbb{N}$, $v_{N}$ is twice differentiable along the modes and solves the bare cutoff equation: $F_{N}^{0}[v_{N}](x)=0$ for every $x\in H^{-1}$. If $2\le n$, the sequence $(v_{N}(x))_{N\in\mathbb{N}}$ is not bounded above, for every $x\in H^{-1}$.

3. (Renormalised limit)¶ For every $x\in H^{-1}$ the family $k\mapsto q_{k}\,x(k)^{2}+a_{k}$ is cube-summable, and $(u_{N}(x))_{N\in\mathbb{N}}$ converges to

$$R(x)=\sum_{k\in\mathbb{Z}^{n}}\bigl(q_{k}\,x(k)^{2}+a_{k}\bigr).$$

4. (The limit carries the Gaussian profile)¶ With $R$ as in clause 3, $R-P$ is regular.

5. (The limit solves the renormalised equation)¶ With $R$ as in clause 3, $R$ is twice differentiable along the modes and solves the renormalised equation: for every $x\in H^{-1}$, $(R,x)\in\mathcal{D}$ and $F[R](x)=0$.