Renormalised Viscosity Solutions versus Viscosity Solutions of the Shifted Equation on the Sobolev Triple of Order Two

In the setting of The Wick-Square Problem on the Sobolev Triple of Order Two: Standing Notation, let $C_{g},\ell_{g}\in\mathbb{R}$ be nonnegative and let $g:V\to\mathbb{R}$ satisfy $|g(x)|\le C_{g}$ and $|g(x)-g(y)|\le\ell_{g}|x-y|_{H}$ for all $x,y\in V$. Let $F^{\sharp}$ be the shifted operator for $g$, and let renormalised viscosity sub- and supersolutions be those of Renormalised Viscosity Subsolutions, Supersolutions and Solutions of the Wick-Square Problem, Tested by the Gaussian Penalty plus a Regular Function for the running cost $g|_{H^{-1}}$. Let $w:H\to\mathbb{R}$ be bounded and continuous on $H$, and let $u=u_{0}+w|_{H^{-1}}$.

1. (From the shifted equation)¶ If $w$ is a viscosity subsolution of $F^{\sharp}$ on $H$, then $u$ is a renormalised viscosity subsolution; if $w$ is a viscosity supersolution of $F^{\sharp}$ on $H$, then $u$ is a renormalised viscosity supersolution.

2. (To the shifted equation)¶ Assume moreover that $w$ is Lipschitz from $(H,d_{H})$ to $\mathbb{R}$. If $u$ is a renormalised viscosity subsolution, then $w$ is a viscosity subsolution of $F^{\sharp}$ on $H$; if $u$ is a renormalised viscosity supersolution, then $w$ is a viscosity supersolution of $F^{\sharp}$ on $H$.