Renormalised Viscosity Subsolutions, Supersolutions and Solutions of the Wick-Square Problem, Tested by the Gaussian Penalty plus a Regular Function

In the setting of The Wick-Square Problem on the Torus: Standing Notation, let $g:H^{-1}\to\mathbb{R}$ be a running cost, let $F$ be the renormalised operator for $g$, let $P$ be the Gaussian penalty, and let $u:H^{-1}\to\mathbb{R}$. Regular functions are those of Regular Functions for the Wick-Square Problem: Diagonal Quadratics of Curvature of Order the Inverse Squared Weight, plus Twice Differentiable Functions on the Sobolev Space of Order -3; for a regular $\psi$ and every $x\in H^{-1}$ the value $F[P+\psi](x)$ is defined by Regular Functions: Mode Derivatives, the Free-Field Generator, the Gradient Energy, the Renormalised Operator at the Gaussian Penalty plus a Regular Function, and Linear Combinations §operator. The set $H^{-1}$ is regarded as a subset of the metric space $(H^{-3},d_{H^{-3}})$, where $H^{-3}=H^{-3}(\mathbb{T}^{n})$ is the Sobolev space of order $-3$ with its distance $d_{H^{-3}}$, which contains $H^{-1}$ by The Negative-Order Sobolev Spaces of the Torus are Hilbert Spaces: the Embedding of the Square-Integrable Classes, the Rescaled Trigonometric Basis, the Series Form of the Inner Product and the Inclusion of the Scale §inclusion (applied with $m=1$ and $m=2$); local maxima and minima relative to $H^{-1}$ of a function on $H^{-1}$ refer to this metric space.

1. (Subsolution)¶ $u$ is a renormalised viscosity subsolution of the Wick-square problem with running cost $g$ if for every regular $\psi:H^{-1}\to\mathbb{R}$ and every $\hat{x}\in H^{-1}$ at which $u-P-\psi$ has a local maximum relative to $H^{-1}$ and $u(\hat{x})=P(\hat{x})+\psi(\hat{x})$,

$$F[P+\psi](\hat{x})\le0 .$$

2. (Supersolution)¶ $u$ is a renormalised viscosity supersolution of the Wick-square problem with running cost $g$ if for every regular $\psi:H^{-1}\to\mathbb{R}$ and every $\hat{x}\in H^{-1}$ at which $u-P-\psi$ has a local minimum relative to $H^{-1}$ and $u(\hat{x})=P(\hat{x})+\psi(\hat{x})$,

$$0\le F[P+\psi](\hat{x}) .$$

3. (Solution)¶ $u$ is a renormalised viscosity solution of the Wick-square problem with running cost $g$ if it is both a renormalised viscosity subsolution and a renormalised viscosity supersolution.