The Counterterm Is Forced at the Level of Galerkin Solutions: Other Counterterms Shift the Solutions by a Constant, and the Bare Solutions Diverge

In the settings of The Wick-Square Problem on the Sobolev Triple of Order Two: Standing Notation and The Galerkin Problems of the Wick-Square Problem: Standing Notation for Cubes of Modes, Coordinates, the Free Galerkin Quadratic and the Riccati Potential, let $C_{g}$, $\ell_{g}$, $g$ and $u$ be as in The Galerkin Solutions of the Wick-Square Problem Converge to the Renormalised Viscosity Solution for Every Bounded Lipschitz Running Cost, and for $N\in\mathbb{N}$ let $U_{N}$ be the Galerkin solution of The Galerkin Solutions of the Wick-Square Problem Converge to the Renormalised Viscosity Solution for Every Bounded Lipschitz Running Cost §galerkin and $u_{N}$ the function of The Galerkin Solutions of the Wick-Square Problem Converge to the Renormalised Viscosity Solution for Every Bounded Lipschitz Running Cost §bounds. Let $b:\mathbb{Z}^{n}\to\mathbb{R}$ be a counterterm, let $\mathcal{F}^{b}_{N}$ be the Galerkin operator at cutoff $N$ with counterterm $b$ for $g$, with viscosity solutions on $\mathbb{R}^{m}$ as in The Galerkin Hamilton-Jacobi-Bellman Equations of the Wick-Square Problem in the Coordinates of a Cube of Modes §equation, and put

$$s_{N}(b)=\beta\sum_{k\in\Gamma_{N}}\bigl(c_{k}-b(k)\bigr),$$

a sum over the finite set $\Gamma_{N}$. Then the following hold.

1. (Other counterterms shift the solution by a constant)¶ For every $N\in\mathbb{N}$, the function $U^{b}_{N}=U_{N}+s_{N}(b)/\gamma$ is the unique viscosity solution of $\mathcal{F}^{b}_{N}$ on $\mathbb{R}^{m}$ for which $U^{b}_{N}-\tilde{u}_{N}$ is bounded.

2. (The counterterm is forced)¶ Let $x\in H^{-1}$ and put $u^{b}_{N}(x)=U^{b}_{N}(\mathrm{pr}_{N}x)=u_{N}(x)+s_{N}(b)/\gamma$. The sequence $(u^{b}_{N}(x))_{N\in\mathbb{N}}$ converges if and only if the family $k\mapsto c_{k}-b(k)$ is cube-summable, and then its limit is $u(x)+\frac{\beta}{\gamma}\sum_{k\in\mathbb{Z}^{n}}\bigl(c_{k}-b(k)\bigr)$.

3. (The bare solutions)¶ Let $C_{N}=\beta\sum_{k\in\Gamma_{N}}c_{k}$, so that $s_{N}(0)=C_{N}$ for the zero counterterm $0$, and by clause 1 the viscosity solution $U^{0}_{N}=U_{N}+C_{N}/\gamma$ of the bare Galerkin operator $\mathcal{F}^{0}_{N}$ is the one differing from $\tilde{u}_{N}$ by a bounded function. If $2\le n$, then for every $x\in H^{-1}$ the sequence $(u^{0}_{N}(x))_{N\in\mathbb{N}}$ is not bounded above, while $(u^{0}_{N}(x)-C_{N}/\gamma)_{N\in\mathbb{N}}$ converges to $u(x)$.