Proof of 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

Each result cited is universally quantified over the data in its own statement.

Throughout, $H=H^{-3}$ and $V=H^{-2}$ carry the inner products of orders $-3$ and $-2$ (The Wick-Square Problem on the Sobolev Triple of Order Two: Standing Notation §triple); by The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §domain, $D(A)=H^{-1}$, $Ax$ is the family $k\mapsto\mu_{k}x(k)$ and $|Ax|_{H}=|x|_{H^{-1}}$ for $x\in H^{-1}$; in particular $A$ is linear, since the vector operations on families are modewise. Elementary arithmetic in $\mathbb{R}$ (rearrangement, the triangle inequality, monotonicity of squares of nonnegative numbers, the limit laws and the order of limits) is used without further mention under The Real Numbers: Standing Notation and Background §background, and the Cauchy-Schwarz inequality The Cauchy-Schwarz Inequality in a Real Inner Product Space in $H$, $V$ and $H^{-1}$ under Real Hilbert Spaces: Standing Notation and Background §background. Lattice sums are manipulated with Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear, Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §nonnegative and Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §comparison. We fix two tools. Since $1\le\mu_{k}$ (The Wick-Square Problem on the Torus: Standing Notation §modes), $\mu_{k}^{m}\le\mu_{k}^{m'}$ for natural numbers $m\le m'$; and since $0<q_{k}$ by The Riccati Coefficients of the Wick-Square Problem: Existence, the Gaussian Profile and Summable Constants §root and $q_{k}\le\beta/(2\mu_{k})$ by The Riccati Coefficients of the Wick-Square Problem: Existence, the Gaussian Profile and Summable Constants §profile,

$$0<2q_{k}\le\frac{\beta}{\mu_{k}}\le\beta\qquad\text{for every }k\in\mathbb{Z}^{n}.\qquad\text{(W)}$$

Further, by Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums §sobolev, for $m\in\{1,2,3\}$ a family $c$ lies in $H^{-m}$ exactly when $k\mapsto c(k)^{2}/\mu_{k}^{m}$ is cube-summable, and then $|c|_{H^{-m}}^{2}$ is its lattice sum; we refer to this as (S). For $m=2$ and $m=3$ these are $V$ and $H$ with their norms.

Step 1 (The Riccati drift and the gradient form). (a) Let $y,y'\in V$. By The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §drift, $B_{q}y$ is the family $k\mapsto2q_{k}y(k)$, so $B_{q}y-B_{q}y'=B_{q}(y-y')$. For $m\in\{2,3\}$, (W) gives $(2q_{k}y(k))^{2}/\mu_{k}^{m}\le\beta^{2}y(k)^{2}/\mu_{k}^{m}$, and by (S) the family $k\mapsto y(k)^{2}/\mu_{k}^{m}$ is cube-summable with lattice sum $|y|_{V}^{2}$ for $m=2$ and $|y|_{H}^{2}$ for $m=3$ (recall $V\subseteq H$). By comparison and (S) again, $B_{q}y\in V$ and

$$|B_{q}y|_{V}\le\beta\,|y|_{V},\qquad|B_{q}y|_{H}\le\beta\,|y|_{H}.$$

(b) Let $y\in H^{-1}$ and $v\in V$. By Hilbert Triples: Standing Notation and Background §operator, $\langle Ay,v\rangle_{H}=\langle y,v\rangle_{V}$. Taking $v=y$ and using Cauchy-Schwarz in $H$,

$$|y|_{V}^{2}=\langle Ay,y\rangle_{H}\le|y|_{H^{-1}}\,|y|_{H}.$$

(c) Let $y\in H^{-1}$ and $b\in H$. Let $x$ be the family $k\mapsto b(k)/\mu_{k}^{3}$. Since $x(k)^{2}/\mu_{k}=b(k)^{2}/\mu_{k}^{7}\le b(k)^{2}/\mu_{k}^{3}$ and $b\in H$, (S) and comparison give $x\in H^{-1}$ and $|x|_{H^{-1}}\le|b|_{H}$. By the first statement of The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §pairings, $\langle Ay,e_{k}\rangle_{H}=\mu_{k}y(k)/\mu_{k}^{3}$ and $\langle b,e_{k}\rangle_{H}=b(k)/\mu_{k}^{3}$, so by The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §gradient-form $G(Ay,b)$ is the lattice sum of $k\mapsto y(k)b(k)/\mu_{k}^{5}$; by the last statement of The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §pairings, applied to $x\in H^{-1}$ and $y\in H$, so is $\langle Ax,y\rangle_{H}$. Hence $G(Ay,b)=\langle Ax,y\rangle_{H}$, and by Cauchy-Schwarz

$$|G(Ay,b)|\le|Ax|_{H}\,|y|_{H}=|x|_{H^{-1}}\,|y|_{H}\le|b|_{H}\,|y|_{H}.$$

(d) By The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §gradient-form, $0_{\mathrm{Sym}}\preceq G\preceq I_{H}$, so $0\le G(v,v)\le|v|_{H}^{2}$ for $v\in H$; by Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity §quadratic-norm, $\lVert G\rVert\le1$, whence $|G(v,v')|\le|v|_{H}|v'|_{H}$ for $v,v'\in H$ by Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity §bound. By bilinearity and symmetry, $G(v,v)-G(v',v')=G(v-v',v+v')$, and $G(v+v',v+v')+G(v-v',v-v')=2G(v,v)+2G(v',v')$; as both terms on the left are nonnegative, $G(v\pm v',v\pm v')\le2G(v,v)+2G(v',v')$.

Step 2 (The free solution minus the penalty). Put $A_{0}=\sum_{k\in\mathbb{Z}^{n}}a_{k}$, which exists by The Riccati Coefficients of the Wick-Square Problem: Existence, the Gaussian Profile and Summable Constants §constants, let $\eta_{0}(k)=q_{k}-\beta/(2\mu_{k})$, and let $c_{0}:H^{-3}\to\mathbb{R}$ be the constant function with value $A_{0}$. By The Riccati Coefficients of the Wick-Square Problem: Existence, the Gaussian Profile and Summable Constants §profile, $\mu_{k}^{2}|\eta_{0}(k)|=\mu_{k}^{2}(\beta/(2\mu_{k})-q_{k})\le(\beta\gamma+\beta^{2})/4$, so $\eta_{0}$ is a curvature family with this bound. Since $c_{0}(x)=\langle0,x\rangle_{H^{-3}}+A_{0}$ and $H^{-3}$ is a real Hilbert space (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 §hilbert), Affine and Quadratic Functions on a Real Hilbert Space are of Class $C^2$ §affine gives that $c_{0}$ is of class $C^{2}$ on $H^{-3}$ with $Dc_{0}(x)=0$ for every $x$. Let $x\in H^{-1}$. The families $k\mapsto q_{k}x(k)^{2}+a_{k}$ (The Galerkin Solutions of the Wick-Square Problem Converge to a Classical Solution of the Renormalised Equation, while the Bare Solutions Diverge §limit), $k\mapsto a_{k}$ and $k\mapsto\beta x(k)^{2}/(2\mu_{k})$ (The Gaussian Penalty of the Wick-Square Problem and linearity) are cube-summable, so by linearity so is $k\mapsto\eta_{0}(k)x(k)^{2}$, their combination, and

$$u_{0}(x)-P(x)=\sum_{k\in\mathbb{Z}^{n}}\eta_{0}(k)\,x(k)^{2}+c_{0}(x).$$

Thus $\psi_{0}=u_{0}-P$ is regular (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), with curvature family $\eta_{0}$ and weak part $c_{0}$. 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, $\partial_{k}\psi_{0}(x)=2\eta_{0}(k)x(k)$. By The Cutoff Wick Square is Minus the Free-Field Generator of the Gaussian Penalty, and the Penalised Form of the Cutoff Operators §derivatives, $P$ is twice differentiable along the modes with $\partial_{k}P(x)=\beta x(k)/\mu_{k}$. Since $u_{0}=P+\psi_{0}$ pointwise on $H^{-1}$, Mode Derivatives of Linear Combinations and of Restrictions of Twice Differentiable Functions on a Negative Sobolev Space §linear gives, for all $x\in H^{-1}$ and $k\in\mathbb{Z}^{n}$,

$$\partial_{k}u_{0}(x)=\frac{\beta x(k)}{\mu_{k}}+2q_{k}x(k)-\frac{\beta x(k)}{\mu_{k}}=2q_{k}\,x(k).$$

Step 3 (Clause 1). Let $\chi$ be regular, with a curvature family $\eta$, curvature bound $C_{\chi}$ and weak part $\phi$. By Step 2 and 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 §linear, $u_{0}+\chi-P=\psi_{0}+\chi$ is regular. 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, $\chi$ is twice differentiable along the modes, and so is $u_{0}$ by The Galerkin Solutions of the Wick-Square Problem Converge to a Classical Solution of the Renormalised Equation, while the Bare Solutions Diverge §solution; hence so is $u_{0}+\chi$, with $\partial_{k}(u_{0}+\chi)=\partial_{k}u_{0}+\partial_{k}\chi$ and likewise for the second derivatives and for $\psi_{0}+\chi$ (Mode Derivatives of Linear Combinations and of Restrictions of Twice Differentiable Functions on a Negative Sobolev Space §linear). Fix $x\in H^{-1}$. 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 §generator, $L\chi$ is defined at $x$; 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 §gradient with $\theta=0$, $|D\chi(x)|^{2}$ is defined.

Cross term. Put $z=D\phi(x)\in H^{-3}=H$. 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 and the first statement of The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §pairings, $\partial_{k}\chi(x)=2\eta(k)x(k)+z(k)/\mu_{k}^{3}$, so $2q_{k}x(k)\partial_{k}\chi(x)$ is the sum of $4q_{k}\eta(k)x(k)^{2}$ and $2q_{k}x(k)z(k)/\mu_{k}^{3}$. By (W) and $|\eta(k)|\le C_{\chi}/\mu_{k}^{2}$, the first has absolute value at most $2\beta C_{\chi}x(k)^{2}/\mu_{k}$, a cube-summable family by The Wick-Square Problem on the Torus: Standing Notation §state-space; by (W), the second has absolute value at most $\beta|x(k)z(k)|/\mu_{k}^{2}$, cube-summable by The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §pairings. By comparison and linearity, $k\mapsto2q_{k}x(k)\partial_{k}\chi(x)$ is cube-summable.

The identity. Let $F_{N}$ be the Wick-ordered cutoff operators for the running cost $g|_{H^{-1}}$, and $\widetilde{F}_{N}$, $\widetilde{\mathcal{D}}$, $\widetilde{F}$ the Wick-ordered cutoff operators, the domain and the renormalised operator for the zero running cost. By The Cutoff Wick Square is Minus the Free-Field Generator of the Gaussian Penalty, and the Penalised Form of the Cutoff Operators §penalised, for every $N\in\mathbb{N}$,

$$F_{N}[u_{0}+\chi](x)=\gamma\,(u_{0}+\chi)(x)-L_{N}(\psi_{0}+\chi)(x)+\tfrac12|D_{N}(u_{0}+\chi)(x)|^{2}-g(x),$$ $$\widetilde{F}_{N}[u_{0}](x)=\gamma\,u_{0}(x)-L_{N}\psi_{0}(x)+\tfrac12|D_{N}u_{0}(x)|^{2}.$$

By the additivity of mode derivatives and the definitions The Free-Field Generator with a Mode Cutoff §generator and The Gradient Energy with a Mode Cutoff §gradient (finite sums over $\Gamma_{N}$), $L_{N}(\psi_{0}+\chi)(x)=L_{N}\psi_{0}(x)+L_{N}\chi(x)$, and by Step 2

$$\tfrac12|D_{N}(u_{0}+\chi)(x)|^{2}-\tfrac12|D_{N}u_{0}(x)|^{2}=\sum_{k\in\Gamma_{N}}\tfrac12\bigl(2\,\partial_{k}u_{0}(x)\,\partial_{k}\chi(x)+(\partial_{k}\chi(x))^{2}\bigr)=\sum_{k\in\Gamma_{N}}2q_{k}x(k)\,\partial_{k}\chi(x)+\tfrac12|D_{N}\chi(x)|^{2}.$$

Subtracting,

$$F_{N}[u_{0}+\chi](x)=\widetilde{F}_{N}[u_{0}](x)+\gamma\,\chi(x)-L_{N}\chi(x)+\sum_{k\in\Gamma_{N}}2q_{k}x(k)\,\partial_{k}\chi(x)+\tfrac12|D_{N}\chi(x)|^{2}-g(x).$$

As $N\to\infty$: $\widetilde{F}_{N}[u_{0}](x)\to0$, because $(u_{0},x)\in\widetilde{\mathcal{D}}$ and $\widetilde{F}[u_{0}](x)=0$ by The Galerkin Solutions of the Wick-Square Problem Converge to a Classical Solution of the Renormalised Equation, while the Bare Solutions Diverge §solution (with $u_{0}=R$ by The Wick-Square Problem on the Sobolev Triple of Order Two: Standing Notation §free) and The Renormalised Hamilton-Jacobi-Bellman Operator of the Wick-Square Problem as the Limit of the Wick-Ordered Cutoff Operators §operator; $L_{N}\chi(x)\to L\chi(x)$ and $|D_{N}\chi(x)|^{2}\to|D\chi(x)|^{2}$ by The Free-Field Generator with a Mode Cutoff §limit and The Gradient Energy with a Mode Cutoff §limit; and the cube sums of the cross family converge to its lattice sum. By the limit laws, $(F_{N}[u_{0}+\chi](x))_{N}$ converges, so $(u_{0}+\chi,x)\in\mathcal{D}$ (The Renormalised Hamilton-Jacobi-Bellman Operator of the Wick-Square Problem as the Limit of the Wick-Ordered Cutoff Operators §domain), and its limit $F[u_{0}+\chi](x)$ is the right-hand side of clause 1.

Step 4 (Clause 2). Let $\varphi\in C^{2}(H)$, $c\in\mathbb{R}$ and $\chi=(\varphi+c\,h)|_{H^{-1}}$. As $H$ is $H^{-3}$ with its inner product and $C^{2}(H)$ is the class of The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space §c2 for that space (Hilbert Triples: Standing Notation and Background §open-sets, Real Hilbert Spaces: Series, Products, Orthonormal Bases and Differential Calculus §calculus), $\varphi$ is of class $C^{2}$ on $H^{-3}$, with gradient map $D\varphi$ and Hessian map $D^{2}\varphi$. Put $\eta_{c}(k)=c/(2\mu_{k}^{2})$, so $\mu_{k}^{2}|\eta_{c}(k)|=|c|/2$. For $x\in H^{-1}\subseteq V$, (S) with $m=2$ and linearity give $c\,h(x)=\tfrac{c}{2}|x|_{V}^{2}=\sum_{k\in\mathbb{Z}^{n}}\eta_{c}(k)x(k)^{2}$, so $\chi(x)=\sum_{k}\eta_{c}(k)x(k)^{2}+\varphi(x)$: $\chi$ is regular with curvature family $\eta_{c}$ and weak part $\varphi$.

Fix $x\in H^{-1}=D(A)$, and put $p=D\varphi(x)+c\,Ax\in H$ and $X=D^{2}\varphi(x)|_{V}+c\,I_{V}\in\mathrm{Sym}(V)$. 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,

$$\partial_{k}\chi(x)=\frac{c\,x(k)}{\mu_{k}^{2}}+\langle D\varphi(x),e_{k}\rangle_{H},\qquad\partial_{k}^{2}\chi(x)=\frac{c}{\mu_{k}^{2}}+D^{2}\varphi(x)(e_{k},e_{k}).$$

By The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §pairings, $\langle Ax,e_{k}\rangle_{H}=\mu_{k}x(k)/\mu_{k}^{3}=x(k)/\mu_{k}^{2}$; by linearity of the inner product,

$$\partial_{k}\chi(x)=\langle p,e_{k}\rangle_{H}=\frac{p(k)}{\mu_{k}^{3}}.\qquad\text{(4.1)}$$

Since $e_{k}\in V$ and $I_{V}(e_{k},e_{k})=|e_{k}|_{V}^{2}=1/\mu_{k}^{2}$ (Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums §units with $m=2$), the definitions of restriction and of sums of forms (Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction §restriction, Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction §identity) give

$$\partial_{k}^{2}\chi(x)=X(e_{k},e_{k}).\qquad\text{(4.2)}$$

Gradient energy: by (4.1), $|D_{N}\chi(x)|^{2}$ is the $N$th cube sum of $k\mapsto\langle p,e_{k}\rangle_{H}^{2}$, whose lattice sum is $G(p,p)$ by The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §gradient-form; so $|D\chi(x)|^{2}=G(p,p)$. Generator: by (4.1) and (4.2), $\mu_{k}x(k)\partial_{k}\chi(x)=x(k)p(k)/\mu_{k}^{2}$ and $L_{N}\chi(x)$ is $\tfrac{\nu}{2}$ times the $N$th cube sum of $k\mapsto X(e_{k},e_{k})$ minus that of $k\mapsto x(k)p(k)/\mu_{k}^{2}$; by The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §noise and The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §pairings these converge to $\mathrm{Tr}_{f}X$ and $\langle Ax,p\rangle_{H}$, so $L\chi(x)=\tfrac{\nu}{2}\mathrm{Tr}_{f}X-\langle Ax,p\rangle_{H}$. Cross term: let $x'$ be the family $k\mapsto2q_{k}x(k)/\mu_{k}$. By (W), $x'(k)^{2}/\mu_{k}\le\beta^{2}x(k)^{2}/\mu_{k}$, so $x'\in H^{-1}$ by (S) and comparison, and $Ax'$ is the family $k\mapsto2q_{k}x(k)$, which is $B_{q}x$ (The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §drift). By (4.1), $2q_{k}x(k)\partial_{k}\chi(x)=x'(k)p(k)/\mu_{k}^{2}$, whose lattice sum is $\langle Ax',p\rangle_{H}=\langle B_{q}x,p\rangle_{H}$ by The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §pairings. Substituting in the right-hand side of clause 1 and using $\chi(x)=\varphi(x)+c\,h(x)$, that right-hand side is

$$\gamma\bigl(\varphi(x)+c\,h(x)\bigr)-\tfrac{\nu}{2}\mathrm{Tr}_{f}X+\tfrac12G(p,p)+\langle Ax+B_{q}x,p\rangle_{H}-g(x),$$

which is $F^{\sharp}(x,\varphi(x)+c\,h(x),p,X)$ by The Shifted Hamilton-Jacobi-Bellman Equation of the Wick-Square Problem on the Sobolev Triple of Order Two §operator.

The shifts. Let $\delta>0$. Taking $c=\delta$, the triple form and clause 1 give $F[u_{0}+(\varphi+\delta h)|_{H^{-1}}](x)=F^{\sharp}(x,\varphi(x)+\delta h(x),D\varphi(x)+\delta Ax,D^{2}\varphi(x)|_{V}+\delta I_{V})$, which is $F^{\sharp-}_{\delta}(x,\varphi(x),D\varphi(x),D^{2}\varphi(x))$ by Second-Order Equation Operator on an Open Subset of a Hilbert Triple and Its $\delta$-Shifts §shifted. Taking $c=-\delta$, so that $(\varphi+c\,h)|_{H^{-1}}=(\varphi-\delta h)|_{H^{-1}}$, $c\,Ax=-\delta Ax$ and $c\,I_{V}=-\delta I_{V}$ (Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction §identity), the same argument gives the formula for $F^{\sharp+}_{\delta}$.

Step 5 (Clauses 3 and 4). We apply A Viscous Hamilton-Jacobi Operator with a Monotone Nonlinearity Satisfies the Second-Order Comparison Hypotheses to the triple $(H,V,A)$, which is in the setting of Hilbert Triples: Standing Notation and Background by The Wick-Square Problem on the Sobolev Triple of Order Two: Standing Notation §triple, with: $\lambda_{0}=\gamma$, positive by The Wick-Square Problem on the Torus: Standing Notation §parameters; the given $C_{g}$ and $g$; $\omega_{g}(t)=\ell_{g}t$, a modulus of continuity by Linear Moduli of Continuity §modulus, and for $x,y\in V$ one has $|g(x)-g(y)|\le\ell_{g}|x-y|_{H}\le\ell_{g}|x-y|_{V}$ by Hilbert Triples: Standing Notation and Background §triple; $B=B_{q}$, a monotone nonlinearity by The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §drift; $\ell=0$ and $L$ the zero map of $H$, Lipschitz with constant $0$ since $d_{H}(0_{H},0_{H})=0$, so $c_{L}=0$; $\Gamma=G$, with $0_{\mathrm{Sym}}\preceq G\preceq I_{H}$ by The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §gradient-form; $\nu$, positive; and $f$, square-summable in $V$ by The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §noise. The operator of the proposition is then, at $(x,r,p,X)\in D(A)\times\mathbb{R}\times H\times\mathrm{Sym}(V)$, $\gamma r-\tfrac{\nu}{2}\mathrm{Tr}_{f}X+\tfrac12G(p,p)+\langle Ax+B_{q}x+0_{H},p\rangle_{H}-g(x)=F^{\sharp}(x,r,p,X)$. Hence $F^{\sharp}$ is degenerate elliptic by A Viscous Hamilton-Jacobi Operator with a Monotone Nonlinearity Satisfies the Second-Order Comparison Hypotheses §operator, locally strictly proper by A Viscous Hamilton-Jacobi Operator with a Monotone Nonlinearity Satisfies the Second-Order Comparison Hypotheses §proper, and satisfies the second-order structure condition, the tail-insensitivity condition and the shift-continuity condition by A Viscous Hamilton-Jacobi Operator with a Monotone Nonlinearity Satisfies the Second-Order Comparison Hypotheses §structure, A Viscous Hamilton-Jacobi Operator with a Monotone Nonlinearity Satisfies the Second-Order Comparison Hypotheses §tail (with The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §infinite) and A Viscous Hamilton-Jacobi Operator with a Monotone Nonlinearity Satisfies the Second-Order Comparison Hypotheses §shift: this is clause 3. Since $H$ is not finite-dimensional (The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §infinite), A Viscous Hamilton-Jacobi Operator with a Monotone Nonlinearity Satisfies the Second-Order Comparison Hypotheses §comparison applied to $u=w_{1}$, $v=w_{2}$ and $C$ gives clause 4.

Step 6 (Clause 5: the approximating data). Let $w,\delta,\varphi,\hat{x}$ be as in clause 5; in case (a) put $\tau=1$ and in case (b) $\tau=-1$. Since $w$ is continuous on $H$, Basic Properties of the $\delta$-Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity §semicontinuous-case (with $U=H$, so $V\cap U=V$) shows that $w$ is bounded above and below near each point of $H$ and that $w^{-}_{\delta}(x)=w(x)-\delta h(x)$ and $w^{+}_{\delta}(x)=w(x)+\delta h(x)$ for $x\in V$. Let $e$ be $w^{-}_{\delta}$ in case (a) and $w^{+}_{\delta}$ in case (b); thus $e(x)=w(x)-\tau\delta h(x)$ on $V$, and by hypothesis $e-\varphi$ has at $\hat{x}$ a local maximum relative to $V$ in case (a) and a local minimum relative to $V$ in case (b). Put $\hat{r}=w(\hat{x})-\tau\delta h(\hat{x})$, $\hat{p}=D\varphi(\hat{x})\in H$ and $\hat{Y}=D^{2}\varphi(\hat{x})\in\mathrm{Sym}(H)$. For $y\in H^{-1}$, $s\in\mathbb{R}$, $p\in H$ and $Y\in\mathrm{Sym}(H)$ let $\Phi(y,s,p,Y)$ be $F^{\sharp-}_{\delta}(y,s,p,Y)$ in case (a) and $F^{\sharp+}_{\delta}(y,s,p,Y)$ in case (b); by Second-Order Equation Operator on an Open Subset of a Hilbert Triple and Its $\delta$-Shifts §shifted,

$$\Phi(y,s,p,Y)=F^{\sharp}\bigl(y,\ s+\tau\delta h(y),\ p+\tau\delta Ay,\ Y|_{V}+\tau\delta I_{V}\bigr).$$

By Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple §subsolution in case (a), respectively Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple §supersolution in case (b), applied with $U=H$, $W=D(A)=H^{-1}$, this $\delta$, this $\varphi$, this $\hat{x}$ and $\varepsilon=1/j$, we choose for each $j\in\mathbb{N}$ elements $y_{j}\in H^{-1}$, $s_{j}\in\mathbb{R}$, $p_{j}\in H$ and $Y_{j}\in\mathrm{Sym}(H)$ with

$$|y_{j}-\hat{x}|_{H}<\tfrac1j,\qquad|s_{j}-\hat{r}|<\tfrac1j,\qquad|p_{j}-\hat{p}|_{H}<\tfrac1j,\qquad\lVert Y_{j}-\hat{Y}\rVert<\tfrac1j,\qquad\tau\,\Phi(y_{j},s_{j},p_{j},Y_{j})\le\tfrac1j$$

(in case (b) the last condition is $-\tfrac1j\le F^{\sharp+}_{\delta}(y_{j},s_{j},p_{j},Y_{j})$).

Step 7 (Expansion). Let $y\in H^{-1}$, $s\in\mathbb{R}$, $p\in H$, $Y\in\mathrm{Sym}(H)$, and put $u=p+\tau\delta Ay$. By The Shifted Hamilton-Jacobi-Bellman Equation of the Wick-Square Problem on the Sobolev Triple of Order Two §operator, Elementary Properties of the Trace of a Form along a Square-Summable Sequence §linear, Elementary Properties of the Trace of a Form along a Square-Summable Sequence §identity and bilinearity of $\langle\cdot,\cdot\rangle_{H}$,

$$\Phi(y,s,p,Y)=\Psi(y,s,p,Y)+\tau\delta|Ay|_{H}^{2},$$

where

$$\Psi(y,s,p,Y)=\gamma s+\tau\gamma\delta h(y)-\tfrac{\nu}{2}\mathrm{Tr}_{f}(Y|_{V})-\tfrac{\tau\nu\delta}{2}\sigma(f)+\tfrac12G(u,u)+\langle Ay,p\rangle_{H}+\langle B_{q}y,p\rangle_{H}+\tau\delta\langle B_{q}y,Ay\rangle_{H}-g(y).$$

Step 8 (Coercivity). With $y,s,p,Y,u$ as in Step 7, multiply by $\tau$ (so $\tau^{2}=1$) and bound the terms of $\tau\Psi$ from below: $\gamma\delta h(y)\ge0$; $\delta\langle B_{q}y,Ay\rangle_{H}\ge0$ because $B_{q}$ is a monotone nonlinearity (The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §drift), hence $A$-monotone (Monotone, $A$-Monotone and Locally Bounded Nonlinearities on a Hilbert Triple §nonlinearity, Monotone, $A$-Monotone and Locally Bounded Nonlinearities on a Hilbert Triple §a-monotone); $|\mathrm{Tr}_{f}(Y|_{V})|\le\lVert Y\rVert\sigma(f)$ by Elementary Properties of the Trace of a Form along a Square-Summable Sequence §restriction; $|\langle Ay,p\rangle_{H}|\le|Ay|_{H}|p|_{H}$; $|\langle B_{q}y,p\rangle_{H}|\le\beta|y|_{H}|p|_{H}$ by Step 1(a); $|g(y)|\le C_{g}$; and, by Step 1(d) and Step 1(c) with $b=Ay$,

$$\tfrac{\tau}{2}G(u,u)\ge-\tfrac12G(u,u)\ge-G(p,p)-\delta^{2}G(Ay,Ay)\ge-|p|_{H}^{2}-\delta^{2}|y|_{H}\,|Ay|_{H}.$$

Since $\sigma(f)\ge0$ (Square-Summable Sequences in the Form Space of a Hilbert Triple and the Trace of a Form along Them §square-summable), with $t=|Ay|_{H}$ this gives

$$\tau\,\Phi(y,s,p,Y)\ge\delta t^{2}-\bigl(|p|_{H}+\delta^{2}|y|_{H}\bigr)t-\Bigl(\gamma|s|+\tfrac{\nu}{2}\lVert Y\rVert\sigma(f)+\tfrac{\nu\delta}{2}\sigma(f)+|p|_{H}^{2}+\beta|y|_{H}|p|_{H}+C_{g}\Bigr).$$

For the data of Step 6, $|y_{j}|_{H}\le|\hat{x}|_{H}+1$, $|s_{j}|\le|\hat{r}|+1$, $|p_{j}|_{H}\le|\hat{p}|_{H}+1$ and $\lVert Y_{j}\rVert\le\lVert\hat{Y}\rVert+1$ (Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity §norm-axioms). Put $m_{0}=|\hat{p}|_{H}+1+\delta^{2}(|\hat{x}|_{H}+1)$ and $K_{0}=\gamma(|\hat{r}|+1)+\tfrac{\nu}{2}(\lVert\hat{Y}\rVert+1)\sigma(f)+\tfrac{\nu\delta}{2}\sigma(f)+(|\hat{p}|_{H}+1)^{2}+\beta(|\hat{x}|_{H}+1)(|\hat{p}|_{H}+1)+C_{g}$, and $t_{j}=|Ay_{j}|_{H}=|y_{j}|_{H^{-1}}$. Then $\delta t_{j}^{2}-m_{0}t_{j}-K_{0}\le\tau\Phi(y_{j},s_{j},p_{j},Y_{j})\le1$, i.e. $\delta t_{j}^{2}\le m_{0}t_{j}+K_{0}+1$. If $t_{j}\ge1$ this gives $\delta t_{j}^{2}\le(m_{0}+K_{0}+1)t_{j}$, so $t_{j}\le(m_{0}+K_{0}+1)/\delta$. Hence, with $M=\max\{1,(m_{0}+K_{0}+1)/\delta\}$,

$$|y_{j}|_{H^{-1}}\le M\qquad\text{for every }j\in\mathbb{N}.$$

Step 9 ($\hat{x}\in H^{-1}$). Let $k\in\mathbb{Z}^{n}$. By The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift §pairings and Cauchy-Schwarz in $H$, $|y_{j}(k)-\hat{x}(k)|=\mu_{k}^{3}|\langle y_{j}-\hat{x},e_{k}\rangle_{H}|\le\mu_{k}^{3}|e_{k}|_{H}/j$, so $y_{j}(k)\to\hat{x}(k)$ as $j\to\infty$. Fix $N\in\mathbb{N}$. By The Wick-Square Problem on the Torus: Standing Notation §state-space and Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §nonnegative, $\sum_{k\in\Gamma_{N}}y_{j}(k)^{2}/\mu_{k}\le|y_{j}|_{H^{-1}}^{2}\le M^{2}$ for every $j$; letting $j\to\infty$ in this finite sum gives $\sum_{k\in\Gamma_{N}}\hat{x}(k)^{2}/\mu_{k}\le M^{2}$. By Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §nonnegative, $k\mapsto\hat{x}(k)^{2}/\mu_{k}$ is cube-summable with lattice sum at most $M^{2}$, so by (S) $\hat{x}\in H^{-1}$ and $|\hat{x}|_{H^{-1}}\le M$. In particular $\hat{x}\in D(A)$ and $\Phi(\hat{x},\hat{r},\hat{p},\hat{Y})$ is defined.

Step 10 (Convergence). Put $z_{j}=y_{j}-\hat{x}\in H^{-1}$, so $|z_{j}|_{H}<1/j$, $|z_{j}|_{H^{-1}}\le2M$ and $Az_{j}=Ay_{j}-A\hat{x}$. (i) By Step 1(b), $|z_{j}|_{V}^{2}\le2M/j$, so $|z_{j}|_{V}\to0$. (ii) Let $p\in H$ and $\epsilon>0$. As $V$ is dense in $H$ (Hilbert Triples: Standing Notation and Background §triple), choose $v\in V$ with $|p-v|_{H}<\epsilon$; then by Step 1(b), $|\langle Az_{j},p\rangle_{H}|\le|\langle z_{j},v\rangle_{V}|+|Az_{j}|_{H}|p-v|_{H}\le|z_{j}|_{V}|v|_{V}+2M\epsilon$, which by (i) is below $(1+2M)\epsilon$ for all large $j$; so $\langle Az_{j},p\rangle_{H}\to0$. (iii) We show $\Psi(y_{j},s_{j},p_{j},Y_{j})\to\Psi(\hat{x},\hat{r},\hat{p},\hat{Y})$, term by term. First, $s_{j}\to\hat{r}$; and $||y_{j}|_{V}-|\hat{x}|_{V}|\le|z_{j}|_{V}\to0$, so $h(y_{j})=\tfrac12|y_{j}|_{V}^{2}\to h(\hat{x})$; in particular $(|y_{j}|_{V})_{j}$ is bounded, say by $M_{V}$. Second, by Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity §restriction, Elementary Properties of the Trace of a Form along a Square-Summable Sequence §linear and Elementary Properties of the Trace of a Form along a Square-Summable Sequence §restriction, $|\mathrm{Tr}_{f}(Y_{j}|_{V})-\mathrm{Tr}_{f}(\hat{Y}|_{V})|=|\mathrm{Tr}_{f}((Y_{j}-\hat{Y})|_{V})|\le\sigma(f)/j$. Third, with $u_{j}=p_{j}+\tau\delta Ay_{j}$ and $\hat{u}=\hat{p}+\tau\delta A\hat{x}$, one has $|u_{j}+\hat{u}|_{H}\le2|\hat{p}|_{H}+1+2\delta M$ and $u_{j}-\hat{u}=(p_{j}-\hat{p})+\tau\delta Az_{j}$, so by Step 1(d) and Step 1(c)

$$|G(u_{j},u_{j})-G(\hat{u},\hat{u})|=|G(u_{j}-\hat{u},u_{j}+\hat{u})|\le\bigl(\tfrac1j+\delta|z_{j}|_{H}\bigr)\bigl(2|\hat{p}|_{H}+1+2\delta M\bigr)\to0.$$

Fourth, $\langle Ay_{j},p_{j}\rangle_{H}-\langle A\hat{x},\hat{p}\rangle_{H}=\langle Az_{j},\hat{p}\rangle_{H}+\langle Az_{j},p_{j}-\hat{p}\rangle_{H}+\langle A\hat{x},p_{j}-\hat{p}\rangle_{H}$, whose terms tend to $0$ by (ii) and are bounded by $2M/j$ and $M/j$. Fifth, by Step 1(a), $|\langle B_{q}y_{j},p_{j}\rangle_{H}-\langle B_{q}\hat{x},\hat{p}\rangle_{H}|\le|\langle B_{q}z_{j},p_{j}\rangle_{H}|+|\langle B_{q}\hat{x},p_{j}-\hat{p}\rangle_{H}|\le\beta|z_{j}|_{H}(|\hat{p}|_{H}+1)+\beta|\hat{x}|_{H}/j\to0$. Sixth, by Step 1(a) ($B_{q}z_{j},B_{q}\hat{x}\in V$) and Step 1(b),

$$\langle B_{q}y_{j},Ay_{j}\rangle_{H}-\langle B_{q}\hat{x},A\hat{x}\rangle_{H}=\langle Ay_{j},B_{q}z_{j}\rangle_{H}+\langle Az_{j},B_{q}\hat{x}\rangle_{H}=\langle y_{j},B_{q}z_{j}\rangle_{V}+\langle z_{j},B_{q}\hat{x}\rangle_{V},$$

of absolute value at most $\beta M_{V}|z_{j}|_{V}+\beta|\hat{x}|_{V}|z_{j}|_{V}\to0$ by (i). Seventh, $|g(y_{j})-g(\hat{x})|\le\ell_{g}/j$. By the limit laws, $\Psi(y_{j},s_{j},p_{j},Y_{j})\to\Psi(\hat{x},\hat{r},\hat{p},\hat{Y})$.

Step 11 (Conclusion of clause 5). Since $Ay_{j}=A\hat{x}+Az_{j}$, $|Ay_{j}|_{H}^{2}=|A\hat{x}|_{H}^{2}+2\langle Az_{j},A\hat{x}\rangle_{H}+|Az_{j}|_{H}^{2}\ge|A\hat{x}|_{H}^{2}+2\langle Az_{j},A\hat{x}\rangle_{H}$. By Steps 6 and 7,

$$\tfrac1j\ge\tau\,\Phi(y_{j},s_{j},p_{j},Y_{j})=\tau\,\Psi(y_{j},s_{j},p_{j},Y_{j})+\delta|Ay_{j}|_{H}^{2}\ge\tau\,\Psi(y_{j},s_{j},p_{j},Y_{j})+\delta|A\hat{x}|_{H}^{2}+2\delta\langle Az_{j},A\hat{x}\rangle_{H}.$$

By Step 10(iii) and Step 10(ii) with $p=A\hat{x}$, the right-hand side converges to $\tau\Psi(\hat{x},\hat{r},\hat{p},\hat{Y})+\delta|A\hat{x}|_{H}^{2}=\tau\Phi(\hat{x},\hat{r},\hat{p},\hat{Y})$ (Step 7), and the left-hand side to $0$; by the order of limits, $\tau\Phi(\hat{x},\hat{r},\hat{p},\hat{Y})\le0$. Together with $\hat{x}\in H^{-1}$ (Step 9), in case (a), where $\tau=1$ and $\hat{r}=w(\hat{x})-\delta h(\hat{x})$, this is $F^{\sharp-}_{\delta}(\hat{x},w(\hat{x})-\delta h(\hat{x}),D\varphi(\hat{x}),D^{2}\varphi(\hat{x}))\le0$; in case (b), where $\tau=-1$ and $\hat{r}=w(\hat{x})+\delta h(\hat{x})$, it is $0\le F^{\sharp+}_{\delta}(\hat{x},w(\hat{x})+\delta h(\hat{x}),D\varphi(\hat{x}),D^{2}\varphi(\hat{x}))$. This proves clause 5 and completes the proof.