The Riccati Shift: the Renormalised Operator at the Free Solution plus a Regular Function is the Shifted Operator, Which Satisfies the Hilbert-Triple Comparison Hypotheses, and Its Touching Points Lie in the State Space

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

$$|g(x)|\le C_{g}\quad\text{and}\quad|g(x)-g(y)|\le\ell_{g}\,|x-y|_{H}\qquad\text{for all }x,y\in V .$$

Let $F^{\sharp}$ be the shifted operator for $g$, with $\delta$-shifts $F^{\sharp-}_{\delta}$ and $F^{\sharp+}_{\delta}$. Let $\mathcal{D}$ and $F$ be the domain and the renormalised operator for the running cost $g|_{H^{-1}}$, let $P$ be the Gaussian penalty, and let $L$ and $|D\cdot|^{2}$ be the uncut free-field generator and gradient energy. 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, and the mode derivatives of a regular function are given 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 §derivatives. Then the following hold.

1. (The shift identity)¶ Let $\chi:H^{-1}\to\mathbb{R}$ be regular. Then $u_{0}+\chi-P$ is regular, and for every $x\in H^{-1}$: $L\chi$ is defined at $x$, $|D\chi(x)|^{2}$ is defined, the family $k\mapsto2q_{k}x(k)\,\partial_{k}\chi(x)$ is cube-summable, $(u_{0}+\chi,x)\in\mathcal{D}$, and

$$F[u_{0}+\chi](x)=\gamma\,\chi(x)-L\chi(x)+\sum_{k\in\mathbb{Z}^{n}}2q_{k}x(k)\,\partial_{k}\chi(x)+\tfrac12\,|D\chi(x)|^{2}-g(x).$$

2. (The triple form)¶ Let $\varphi\in C^{2}(H)$ and $c\in\mathbb{R}$, and let $\chi=(\varphi+c\,h)|_{H^{-1}}$. Then $\chi$ is regular, and for every $x\in H^{-1}$ the right-hand side of the identity in clause 1 equals

$$F^{\sharp}\bigl(x,\ \varphi(x)+c\,h(x),\ D\varphi(x)+c\,Ax,\ D^{2}\varphi(x)|_{V}+c\,I_{V}\bigr).$$

¶In particular, for every real $\delta>0$ and $x\in H^{-1}$,

$$F\bigl[u_{0}+(\varphi+\delta h)|_{H^{-1}}\bigr](x)=F^{\sharp-}_{\delta}\bigl(x,\varphi(x),D\varphi(x),D^{2}\varphi(x)\bigr),\qquad F\bigl[u_{0}+(\varphi-\delta h)|_{H^{-1}}\bigr](x)=F^{\sharp+}_{\delta}\bigl(x,\varphi(x),D\varphi(x),D^{2}\varphi(x)\bigr).$$

3. (The comparison hypotheses)¶ $F^{\sharp}$ is degenerate elliptic and locally strictly proper, and satisfies the second-order structure condition, the shift-continuity condition and the tail-insensitivity condition.

4. (Comparison for the shifted equation)¶ If $w_{1},w_{2}:H\to\mathbb{R}$ and $C\in\mathbb{R}$ satisfy $w_{1}\le C$ and $-C\le w_{2}$ on $H$, $w_{1}$ is a viscosity subsolution and $w_{2}$ a viscosity supersolution of $F^{\sharp}$ on $H$, then $w_{1}(x)\le w_{2}(x)$ for every $x\in V$.

5. (Touching points lie in the state space)¶ Let $w:H\to\mathbb{R}$ be bounded and continuous on $H$, let $\delta\in\mathbb{R}$ be positive, let $\varphi\in C^{2}(H)$ and let $\hat{x}\in V$.

(a)¶ If $w$ is a viscosity subsolution of $F^{\sharp}$ on $H$ and the function $V\to\mathbb{R}$, $x\mapsto w(x)-\delta h(x)-\varphi(x)$, has a local maximum relative to $V$ at $\hat{x}$, then $\hat{x}\in H^{-1}$ and

$$F^{\sharp-}_{\delta}\bigl(\hat{x},\ w(\hat{x})-\delta h(\hat{x}),\ D\varphi(\hat{x}),\ D^{2}\varphi(\hat{x})\bigr)\le0 .$$

(b)¶ If $w$ is a viscosity supersolution of $F^{\sharp}$ on $H$ and the function $V\to\mathbb{R}$, $x\mapsto w(x)+\delta h(x)-\varphi(x)$, has a local minimum relative to $V$ at $\hat{x}$, then $\hat{x}\in H^{-1}$ and

$$0\le F^{\sharp+}_{\delta}\bigl(\hat{x},\ w(\hat{x})+\delta h(\hat{x}),\ D\varphi(\hat{x}),\ D^{2}\varphi(\hat{x})\bigr).$$