Lifting Shifted Galerkin Solutions of the Wick-Square Problem to Viscosity Solutions of the Cut-Off Shifted Equation on the Sobolev Triple

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 $g:V\to\mathbb{R}$ be a running cost and let $N\in\mathbb{N}$, with $m$ and the coordinate projection $\mathrm{pr}_{N}$ as fixed in the second setting. Let $\mathcal{F}^{\sharp}_{N}$ be the shifted Galerkin operator for the running cost $g|_{H^{-1}}$, with viscosity sub- and supersolutions 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 let $F^{\sharp}_{N}$ be the cut-off shifted operator at cutoff $N$ for $g$. Let $Z:\mathbb{R}^{m}\to\mathbb{R}$ and put $w(x)=Z(\mathrm{pr}_{N}x)$ for $x\in H$.

1. (Continuity)¶ If $Z$ is continuous on $(\mathbb{R}^{m},d_{E})$, then $w$ is continuous on $H$.

2. (Subsolutions)¶ If $Z$ is a viscosity subsolution of $\mathcal{F}^{\sharp}_{N}$ on $\mathbb{R}^{m}$, then $w$ is a viscosity subsolution of $F^{\sharp}_{N}$ on $H$.

3. (Supersolutions)¶ If $Z$ is a viscosity supersolution of $\mathcal{F}^{\sharp}_{N}$ on $\mathbb{R}^{m}$, then $w$ is a viscosity supersolution of $F^{\sharp}_{N}$ on $H$.