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

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

The notation of The Wick-Square Problem on the Sobolev Triple of Order Two: Standing Notation, Hilbert Triples: Standing Notation and Background and The Galerkin Problems of the Wick-Square Problem: Standing Notation for Cubes of Modes, Coordinates, the Free Galerkin Quadratic and the Riccati Potential is in force, and elementary facts about real numbers, finite sums, reindexing of finite sums along bijections, absolute values and squares are those carried by The Real Numbers: Standing Notation and Background. The items Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple, The $\delta$-Envelopes $u^-_\delta$ and $u^+_\delta$ of a Function on an Open Subset of a Hilbert Triple, Second-Order Equation Operator on an Open Subset of a Hilbert Triple and Its $\delta$-Shifts, Basic Properties of the $\delta$-Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity, First- and Second-Order Conditions at a Local Extremum of a $C^2$ Function Penalised by $h$ on the Small Space and The Penalty Function $h=\tfrac12|\cdot|_V^2$ of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds are stated in the setting of Hilbert Triples: Standing Notation and Background; their notation ($H$, $V$, $A$, $D(A)$, $h$, $C^{2}(H)$, $\mathrm{Sym}(H)$, $\mathrm{Sym}(V)$, $Y|_{V}$, $I_{V}$) is the notation fixed for the present triple by Hilbert Triples: Standing Notation and Background, and we apply them with $U=H$, so that $V\cap U=V$ and $W=D(A)$, as recorded in Hilbert Triples: Standing Notation and Background §open-sets.

Since $\Gamma_{N}$ is nonempty, $m\ge1$. For $i\in[m]$ we write $k_{i}=\kappa_{N}(i)$ and $E_{i}=e_{k_{i}}$. Since $\kappa_{N}$ is a bijection from $[m]$ onto $\Gamma_{N}$ (The Galerkin Problems of the Wick-Square Problem: Standing Notation for Cubes of Modes, Coordinates, the Free Galerkin Quadratic and the Riccati Potential §cutoff), $\sum_{k\in\Gamma_{N}}f(k)=\sum_{i=1}^{m}f(k_{i})$ for every real-valued $f$ on $\Gamma_{N}$, and $k_{i}\neq k_{j}$ for $i\neq j$. Recall $1\le\mu_{k}$ for every mode $k$ (The Wick-Square Problem on the Torus: Standing Notation §modes).

Step 1 (Coordinate identities). Every unit family $e_{k}$ lies in $H^{-1}=D(A)\subseteq V\subseteq H$ (The Wick-Square Problem on the Torus: Standing Notation §units, 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, Hilbert Triples: Standing Notation and Background §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 §pairings, for every $z\in H$ and every mode $k$,

$$\langle z,e_{k}\rangle_{H}=\frac{z(k)}{\mu_{k}^{3}},\qquad\text{and, if }z\in V,\qquad \langle z,e_{k}\rangle_{V}=\frac{z(k)}{\mu_{k}^{2}}.\qquad(1)$$

Taking $z=e_{k}$ gives $|e_{k}|_{H}^{2}=\mu_{k}^{-3}\le1$, hence $|e_{k}|_{H}\le1$; and $\langle e_{l},e_{k}\rangle_{V}=e_{l}(k)/\mu_{k}^{2}$, which is $\mu_{k}^{-2}$ for $l=k$ and $0$ for $l\neq k$.

(a) By The Galerkin Problems of the Wick-Square Problem: Standing Notation for Cubes of Modes, Coordinates, the Free Galerkin Quadratic and the Riccati Potential §coordinates, $\iota_{N}\zeta=\sum_{i=1}^{m}\zeta_{i}E_{i}$ has value $\zeta_{i}$ at $k_{i}$ and $0$ off $\Gamma_{N}$, so $\mathrm{pr}_{N}\iota_{N}\zeta=\zeta$ for $\zeta\in\mathbb{R}^{m}$; and for every family $x$, $\Pi_{N}x=\iota_{N}(\mathrm{pr}_{N}x)=\sum_{i=1}^{m}x(k_{i})E_{i}$ agrees with $x$ at each $k_{i}$ and vanishes off $\Gamma_{N}$. Hence $\mathrm{pr}_{N}\Pi_{N}=\mathrm{pr}_{N}$ and $\Pi_{N}\Pi_{N}=\Pi_{N}$, and $\mathrm{pr}_{N}$, $\iota_{N}$, $\Pi_{N}$ are linear, the operations on families being pointwise. For $x\in H$, $\Pi_{N}x\in H^{-1}\subseteq V$ by The Shifted Hamilton-Jacobi-Bellman Equation of the Wick-Square Problem with Noise, Control and Cost Cut Off to a Cube of Modes §data. For a mode $k\notin\Gamma_{N}$, $e_{k}$ vanishes at every $k_{i}$, so $\Pi_{N}e_{k}=0_{H}$.

(b) Let $\zeta,\zeta'\in\mathbb{R}^{m}$ and let $c\in H$ vanish at every $k_{i}$. By Inner Products Against Finite Sums, and Orthonormal Families, in a Real Inner Product Space §combinations and (1),

$$\langle\iota_{N}\zeta,c\rangle_{H}=\sum_{i=1}^{m}\zeta_{i}\frac{c(k_{i})}{\mu_{k_{i}}^{3}}=0,\qquad \langle\iota_{N}\zeta,\iota_{N}\zeta'\rangle_{V}=\sum_{i=1}^{m}\frac{\zeta_{i}\zeta'_{i}}{\mu_{k_{i}}^{2}},$$

and, if moreover $c\in V$, $\langle\iota_{N}\zeta,c\rangle_{V}=\sum_{i}\zeta_{i}c(k_{i})/\mu_{k_{i}}^{2}=0$. With The Penalty Function $h=\tfrac12|\cdot|_V^2$ of a Hilbert Triple: Expansion Identities, Closed Sublevel Sets, Density and Local Bounds §expansion-v, applied to the points $\iota_{N}\zeta$ and $\iota_{N}\zeta+c$ of $V$, this gives, for $c\in V$ vanishing at every $k_{i}$,

$$h(\iota_{N}\zeta+c)=h(\iota_{N}\zeta)+h(c),\qquad h(\iota_{N}\zeta)=\tfrac12\sum_{i=1}^{m}\frac{\zeta_{i}^{2}}{\mu_{k_{i}}^{2}}.\qquad(2)$$

(c) Let $a,a'\in H$. By Inner Products Against Finite Sums, and Orthonormal Families, in a Real Inner Product Space §combinations and (1), $\langle\Pi_{N}a,a'\rangle_{H}=\sum_{i=1}^{m}a(k_{i})a'(k_{i})/\mu_{k_{i}}^{3}$, which is symmetric in $a,a'$; so $\langle\Pi_{N}a,a'\rangle_{H}=\langle a,\Pi_{N}a'\rangle_{H}$. The family $a-\Pi_{N}a$ vanishes at every $k_{i}$, so by (b) $\langle\Pi_{N}a,a-\Pi_{N}a\rangle_{H}=0$, and Elementary Identities in a Real Inner Product Space §expansion gives $|a|_{H}^{2}=|\Pi_{N}a|_{H}^{2}+|a-\Pi_{N}a|_{H}^{2}\ge|a-\Pi_{N}a|_{H}^{2}$, whence $|a-\Pi_{N}a|_{H}\le|a|_{H}$, both sides being nonnegative. For $y\in V$, (2) with $\zeta=\mathrm{pr}_{N}y$ and $c=y-\Pi_{N}y$ gives

$$h(y)=h(\Pi_{N}y)+h(y-\Pi_{N}y).\qquad(3)$$

(d) For $a\in H$ and a mode $k$, (1) and The Cauchy-Schwarz Inequality in a Real Inner Product Space give $|a(k)|=\mu_{k}^{3}|\langle a,e_{k}\rangle_{H}|\le\mu_{k}^{3}|a|_{H}|e_{k}|_{H}\le\mu_{k}^{3}|a|_{H}$. For $\zeta\in\mathbb{R}^{m}$ the family $\iota_{N}\zeta$ has value $\zeta_{i}$ at $k_{i}$ by (a), so Inner Products Against Finite Sums, and Orthonormal Families, in a Real Inner Product Space §combinations and (1) give, as in (b),

$$|\iota_{N}\zeta|_{H}^{2}=\langle\iota_{N}\zeta,\iota_{N}\zeta\rangle_{H}=\sum_{i=1}^{m}\zeta_{i}\frac{\zeta_{i}}{\mu_{k_{i}}^{3}}=\sum_{i=1}^{m}\frac{\zeta_{i}^{2}}{\mu_{k_{i}}^{3}}\le\sum_{i=1}^{m}\zeta_{i}^{2}=\lVert\zeta\rVert^{2},$$

the inequality termwise because $\mu_{k_{i}}^{-3}\le1$, and the last equality by Euclidean Norm on $\mathbb{R}^n$. Both $|\iota_{N}\zeta|_{H}$ and $\lVert\zeta\rVert$ being nonnegative,

$$|\iota_{N}\zeta|_{H}\le\lVert\zeta\rVert .\qquad(4)$$

Put $C_{N}=\sum_{i=1}^{m}\mu_{k_{i}}^{3}$; its terms are at least $1$ since $1\le\mu_{k}$, so $\mu_{k_{i}}^{3}\le C_{N}$ for each $i$, and $1\le\mu_{k_{1}}^{3}\le C_{N}$ because $1\le m$. For $a\in H$, $\iota_{N}(\mathrm{pr}_{N}a)=\Pi_{N}a$, and the identity $|a|_{H}^{2}=|\Pi_{N}a|_{H}^{2}+|a-\Pi_{N}a|_{H}^{2}$ of (c) gives $|\Pi_{N}a|_{H}^{2}\le|a|_{H}^{2}$; so, by Euclidean Norm on $\mathbb{R}^n$ and the computation above with $\zeta=\mathrm{pr}_{N}a$,

$$\lVert\mathrm{pr}_{N}a\rVert^{2}=\sum_{i=1}^{m}\mu_{k_{i}}^{3}\,\frac{a(k_{i})^{2}}{\mu_{k_{i}}^{3}}\le C_{N}\sum_{i=1}^{m}\frac{a(k_{i})^{2}}{\mu_{k_{i}}^{3}}=C_{N}|\Pi_{N}a|_{H}^{2}\le C_{N}|a|_{H}^{2}\le C_{N}^{2}|a|_{H}^{2},$$

whence, both sides being nonnegative,

$$\lVert\mathrm{pr}_{N}a\rVert\le C_{N}|a|_{H}\qquad(a\in H).\qquad(4')$$

Step 2 (Clause 1, and semicontinuity of $w$). For $a,a'\in H$, $\mathrm{pr}_{N}a-\mathrm{pr}_{N}a'=\mathrm{pr}_{N}(a-a')$ by linearity of $\mathrm{pr}_{N}$ (Step 1(a)), so (4') gives $\lVert\mathrm{pr}_{N}a-\mathrm{pr}_{N}a'\rVert\le C_{N}|a-a'|_{H}$. Given a real $\varepsilon>0$, every $a'\in H$ with $|a-a'|_{H}<\varepsilon/C_{N}$ thus satisfies $d_{E}(\mathrm{pr}_{N}a,\mathrm{pr}_{N}a')<\varepsilon$, so $\mathrm{pr}_{N}:H\to\mathbb{R}^{m}$ is continuous on $H$ relative to $H$ in the sense of Continuous Map Between Metric Spaces. By Semicontinuity and Continuity Under Composition with a Continuous Map, applied with the metric spaces $(\mathbb{R}^{m},d_{E})$ and $(H,d_{H})$, with its map taken to be $\mathrm{pr}_{N}$ on $H$ and its function taken to be $Z$ on $\mathbb{R}^{m}$: if $Z$ is continuous on $(\mathbb{R}^{m},d_{E})$ then $w=Z\circ\mathrm{pr}_{N}$ is continuous on $H$ (claim 3), which is continuity on $H$ in the sense of Real Hilbert Spaces: Standing Notation and Background §topology; this proves clause 1. Likewise, by claims 1 and 2 there, $w$ is upper semicontinuous on $H$ when $Z$ is upper semicontinuous on $\mathbb{R}^{m}$, and lower semicontinuous on $H$ when $Z$ is lower semicontinuous on $\mathbb{R}^{m}$.

Step 3 (Composition with an affine contraction). For $a\in H$ put $La=a-\Pi_{N}a$. By Step 1(a),(c), $L:H\to H$ is linear, $|La|_{H}\le|a|_{H}$, and $\langle La,a'\rangle_{H}=\langle a,a'\rangle_{H}-\langle\Pi_{N}a,a'\rangle_{H}=\langle a,La'\rangle_{H}$ for $a,a'\in H$. Let $c\in H$, let $\chi\in C^{2}(H)$, and define $\chi^{T}:H\to\mathbb{R}$ by $\chi^{T}(y)=\chi(c+Ly)$. We claim that $\chi^{T}\in C^{2}(H)$, with

$$D\chi^{T}(y)=L\,D\chi(c+Ly),\qquad D^{2}\chi^{T}(y)=B_{y},\quad B_{y}(a,a')=D^{2}\chi(c+Ly)(La,La').$$

Fix $y\in H$ and write $x=c+Ly$, so $c+L(y+v)=x+Lv$ for $v\in H$. Differentiability, in the sense of Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space §differentiable: given a real $\varepsilon>0$, let $\eta>0$ be a radius for $\chi$ at $x$ and $\varepsilon$; for $v\in H$ with $|v|_{H}<\eta$ one has $|Lv|_{H}\le|v|_{H}<\eta$, hence

$$\bigl|\chi^{T}(y+v)-\chi^{T}(y)-\langle LD\chi(x),v\rangle_{H}\bigr|=\bigl|\chi(x+Lv)-\chi(x)-\langle D\chi(x),Lv\rangle_{H}\bigr|\le\varepsilon|Lv|_{H}\le\varepsilon|v|_{H}.$$

So $\chi^{T}$ is differentiable at $y$ with gradient $LD\chi(x)$, and by uniqueness of the gradient (Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space §gradient) $D\chi^{T}(y)=LD\chi(x)$; as $y$ was arbitrary, $\chi^{T}$ is differentiable on $H$ and $D\chi^{T}(y')=LD\chi(c+Ly')$ for every $y'\in H$. Continuity of the gradient map: $|LD\chi(c+Ly')-LD\chi(c+Ly)|_{H}\le|D\chi(c+Ly')-D\chi(c+Ly)|_{H}$ and $|(c+Ly')-(c+Ly)|_{H}\le|y'-y|_{H}$, so continuity of $D\chi$ at $x$ (part of The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space §c1) yields continuity of $D\chi^{T}$ at $y$ with the same radius. The form $B_{y}$: symmetry, additivity and homogeneity follow from those of $D^{2}\chi(x)$ and linearity of $L$, and $|B_{y}(a,a')|\le\lVert D^{2}\chi(x)\rVert|La|_{H}|La'|_{H}\le\lVert D^{2}\chi(x)\rVert|a|_{H}|a'|_{H}$ by Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity §bound; so $B_{y}\in\mathrm{Sym}(H)$ by Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction §form. Second derivative, in the sense of The Second Derivative and the Hessian on an Open Subset of a Real Inner Product Space §expansion with $\rho=1$ (every ball lies in $H$ and $\chi^{T}$ is differentiable on $H$): given $\varepsilon>0$, let $\delta''\le1$ be a radius witnessing that $D^{2}\chi(x)$ is a second derivative of $\chi$ at $x$ (with $\rho=1$, admissible since the condition does not depend on $\rho$). For $v,a\in H$ with $|v|_{H}<\delta''$, $|Lv|_{H}<\delta''$, and by self-adjointness of $L$,

$$\bigl|\langle D\chi^{T}(y+v)-D\chi^{T}(y),a\rangle_{H}-B_{y}(v,a)\bigr|=\bigl|\langle D\chi(x+Lv)-D\chi(x),La\rangle_{H}-D^{2}\chi(x)(Lv,La)\bigr|\le\varepsilon|Lv|_{H}|La|_{H}\le\varepsilon|v|_{H}|a|_{H}.$$

So $B_{y}$ is a second derivative of $\chi^{T}$ at $y$, and by uniqueness of the Hessian (The Second Derivative and the Hessian on an Open Subset of a Real Inner Product Space §hessian) $D^{2}\chi^{T}(y)=B_{y}$. Continuity of the Hessian map: for $y'\in H$ and $a,a'\in H$, $|(B_{y'}-B_{y})(a,a')|=|(D^{2}\chi(c+Ly')-D^{2}\chi(x))(La,La')|\le\lVert D^{2}\chi(c+Ly')-D^{2}\chi(x)\rVert\,|a|_{H}|a'|_{H}$ by Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity §bound, so by the second part of that claim $d_{\mathrm{Sym}}(B_{y'},B_{y})\le d_{\mathrm{Sym}}(D^{2}\chi(c+Ly'),D^{2}\chi(x))$, and continuity of $D^{2}\chi$ at $x$ (part of The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space §c2) gives continuity of $y\mapsto B_{y}$. This proves the claim.

Step 4 (Restriction of a $C^{2}$ function to a coordinate slice). Let $y_{0}\in H$ and $\chi\in C^{2}(H)$, put $S(z)=\iota_{N}z+y_{0}$ for $z\in\mathbb{R}^{m}$ and $\chi_{0}(z)=\chi(S(z))$. By (4) and linearity of $\iota_{N}$, $|S(z')-S(z)|_{H}\le\lVert z'-z\rVert$. We claim that $\chi_{0}$ is of class $C^{2}$ on $\mathbb{R}^{m}$ (C^k Maps on a Euclidean Open Set), with

$$\partial_{i}\chi_{0}(z)=\langle D\chi(S(z)),E_{i}\rangle_{H},\qquad \partial_{l}\partial_{i}\chi_{0}(z)=D^{2}\chi(S(z))(E_{l},E_{i})\qquad(i,l\in[m]).$$

Continuity: if $\Theta:H\to\mathbb{R}$ is continuous at $S(z)$ with radius $\eta$ for a given $\varepsilon$, then $\Theta\circ S$ is continuous at $z$ with the same radius $\eta$. This applies to $\chi$ (continuous by Basic Properties of Differentiability on an Open Subset of a Real Inner Product Space §continuous), to $x\mapsto\langle D\chi(x),E_{i}\rangle_{H}$ (as $|\langle D\chi(x')-D\chi(x),E_{i}\rangle_{H}|\le|D\chi(x')-D\chi(x)|_{H}$ by The Cauchy-Schwarz Inequality in a Real Inner Product Space and $|E_{i}|_{H}\le1$, and $D\chi$ is continuous) and to $x\mapsto D^{2}\chi(x)(E_{l},E_{i})$ (as $|(D^{2}\chi(x')-D^{2}\chi(x))(E_{l},E_{i})|\le\lVert D^{2}\chi(x')-D^{2}\chi(x)\rVert$ by Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity §continuity, and $D^{2}\chi$ is continuous). First partials, in the sense of Partial Derivative on a Euclidean Open Set: moving the $i$th coordinate of $z$ by $t$ moves $S(z)$ to $S(z)+tE_{i}$. Given $\varepsilon>0$, let $\eta>0$ be a radius for differentiability of $\chi$ at $S(z)$ with gradient $D\chi(S(z))$ and $\varepsilon/2$, the gradient $D\chi(S(z))$ being by definition the unique vector with this property (Fréchet Differentiability and the Gradient on an Open Subset of a Real Inner Product Space §gradient); for $0<|t|<\eta$, $|tE_{i}|_{H}\le|t|<\eta$, so $|\chi(S(z)+tE_{i})-\chi(S(z))-t\langle D\chi(S(z)),E_{i}\rangle_{H}|\le\tfrac{\varepsilon}{2}|t|$, and dividing by $|t|$ the difference quotient of $\chi_{0}$ is within $\tfrac{\varepsilon}{2}<\varepsilon$ of $\langle D\chi(S(z)),E_{i}\rangle_{H}$. Second partials: with $\omega_{i}(z)=\langle D\chi(S(z)),E_{i}\rangle_{H}$ one has $\omega_{i}(z')-\omega_{i}(z)=\langle D\chi(S(z)+tE_{l})-D\chi(S(z)),E_{i}\rangle_{H}$ when $z'$ is $z$ with $l$th coordinate moved by $t$; given $\varepsilon>0$, a radius $\delta''$ witnessing The Second Derivative and the Hessian on an Open Subset of a Real Inner Product Space §expansion for $\chi$ at $S(z)$ with $\varepsilon/2$ and the form $D^{2}\chi(S(z))$, which is by definition the unique second derivative of $\chi$ at $S(z)$ (The Second Derivative and the Hessian on an Open Subset of a Real Inner Product Space §hessian), used with its increment $tE_{l}$ and direction $E_{i}$, gives for $0<|t|<\delta''$ the bound $|\omega_{i}(z')-\omega_{i}(z)-t\,D^{2}\chi(S(z))(E_{l},E_{i})|\le\tfrac{\varepsilon}{2}|t|$, so the difference quotient of $\omega_{i}$ is within $\tfrac{\varepsilon}{2}<\varepsilon$ of $D^{2}\chi(S(z))(E_{l},E_{i})$. Thus $\chi_{0}$ is continuous with continuous first partials, i.e. of class $C^{1}$, and each $\partial_{i}\chi_{0}=\omega_{i}$ is continuous with continuous first partials, i.e. of class $C^{1}$; by clause 2 of C^k Maps on a Euclidean Open Set (with $k=1$) and its scalar convention, $\chi_{0}$ is of class $C^{2}$ on $\mathbb{R}^{m}$, and the formulas hold.

Step 5 (Set-up for clauses 2 and 3). Let $\sigma\in\{1,-1\}$: in case $\sigma=1$ assume that $Z$ is a viscosity subsolution of $\mathcal{F}^{\sharp}_{N}$ on $\mathbb{R}^{m}$, and in case $\sigma=-1$ that it is a viscosity supersolution, both in the sense of The Galerkin Hamilton-Jacobi-Bellman Equations of the Wick-Square Problem in the Coordinates of a Cube of Modes §equation, that is of Viscosity Subsolution and Supersolution of a Second-Order Equation for the open set $\mathbb{R}^{m}$. Then $Z$ is upper semicontinuous on $\mathbb{R}^{m}$ if $\sigma=1$ and lower semicontinuous if $\sigma=-1$, so by Step 2 $w$ is upper, respectively lower, semicontinuous on $H$. By Basic Properties of the $\delta$-Envelopes: Semicontinuity, Duality, Closed Superlevel Sets, Bounds and Monotonicity §semicontinuous-case, $w$ is bounded above (if $\sigma=1$), respectively below (if $\sigma=-1$), near each point of $H$; this one-sided local bound comes from the semicontinuity of $w$ alone, and no bound on $Z$ is assumed or used anywhere in this proof. So the standing hypothesis of Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple §subsolution, respectively Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple §supersolution, holds; and for every real $\delta>0$ the envelope $w^{\sigma}_{\delta}$, meaning $w^{-}_{\delta}$ if $\sigma=1$ and $w^{+}_{\delta}$ if $\sigma=-1$ (The $\delta$-Envelopes $u^-_\delta$ and $u^+_\delta$ of a Function on an Open Subset of a Hilbert Triple §minus, The $\delta$-Envelopes $u^-_\delta$ and $u^+_\delta$ of a Function on an Open Subset of a Hilbert Triple §plus), satisfies

$$w^{\sigma}_{\delta}(x)=w(x)-\sigma\delta h(x)\qquad(x\in V).$$

The operator $F^{\sharp}_{N}$ is a second-order equation operator on $H$ relative to $(H,V,A)$ by The Shifted Hamilton-Jacobi-Bellman Equation of the Wick-Square Problem with Noise, Control and Cost Cut Off to a Cube of Modes §operator. Write $F^{\sigma}$ for its shift $(F^{\sharp}_{N})^{-}_{\delta}$ if $\sigma=1$ and $(F^{\sharp}_{N})^{+}_{\delta}$ if $\sigma=-1$; by Second-Order Equation Operator on an Open Subset of a Hilbert Triple and Its $\delta$-Shifts §shifted, in both cases

$$F^{\sigma}(x,r,p,Y)=F^{\sharp}_{N}\bigl(x,\ r+\sigma\delta h(x),\ p+\sigma\delta Ax,\ Y|_{V}+\sigma\delta I_{V}\bigr)$$

for $x\in D(A)$, $r\in\mathbb{R}$, $p\in H$ and $Y\in\mathrm{Sym}(H)$, where $Y|_{V}+(-\delta)I_{V}$ is the form $Y|_{V}-\delta I_{V}$.

Fix a real $\delta>0$, $\varphi\in C^{2}(H)$, a point $\hat{x}\in V$ at which $w^{\sigma}_{\delta}-\varphi$ has a local maximum (if $\sigma=1$), respectively a local minimum (if $\sigma=-1$), relative to $V$, and a real $\varepsilon>0$. By Local Maximum of a Function Relative to a Subset of a Metric Space and Local Minimum of a Function Relative to a Subset of a Metric Space, with the metric $d_{H}$ (Hilbert Triples: Standing Notation and Background §open-sets), there is a real $r>0$ such that every $x\in V$ with $|x-\hat{x}|_{H}<r$ satisfies

$$\sigma\Bigl[\bigl(w(x)-\sigma\delta h(x)-\varphi(x)\bigr)-\bigl(w(\hat{x})-\sigma\delta h(\hat{x})-\varphi(\hat{x})\bigr)\Bigr]\le0.\qquad(5)$$

Put $\hat{z}=\mathrm{pr}_{N}\hat{x}\in\mathbb{R}^{m}$ and $y_{0}=\hat{x}-\Pi_{N}\hat{x}=L\hat{x}\in V$; then $\hat{x}=\iota_{N}\hat{z}+y_{0}$, $y_{0}$ vanishes at every $k_{i}$, and $\hat{z}_{i}=\hat{x}(k_{i})$.

Step 6 (The point $\hat{x}$ lies in $D(A)$, and the shifted gradient has no high modes). Define $T:H\to H$ by $Ty=\Pi_{N}\hat{x}+Ly$. For $y\in V$: $Ty\in V$; $\mathrm{pr}_{N}Ty=\mathrm{pr}_{N}\hat{x}+\mathrm{pr}_{N}y-\mathrm{pr}_{N}y=\hat{z}$ by Step 1(a), so $w(Ty)=Z(\hat{z})=w(\hat{x})$; $Ty=\iota_{N}\hat{z}+Ly$ with $Ly\in V$ vanishing at every $k_{i}$, so (2) gives $h(Ty)=h(\Pi_{N}\hat{x})+h(Ly)$; $T\hat{x}=\hat{x}$; and $Ty-\hat{x}=L(y-\hat{x})$, so $|Ty-\hat{x}|_{H}\le|y-\hat{x}|_{H}$.

Let $b_{N}$ be the map assigning to $y,y'\in H$ the number $b_{N}(y,y')=\delta\sum_{i=1}^{m}y(k_{i})y'(k_{i})/\mu_{k_{i}}^{2}$. It is symmetric, additive and homogeneous in each argument (the operations being pointwise), and by Step 1(d) $|b_{N}(y,y')|\le\bigl(\delta\sum_{i=1}^{m}\mu_{k_{i}}^{4}\bigr)|y|_{H}|y'|_{H}$; so $b_{N}\in\mathrm{Sym}(H)$. Let $Q(y)=\tfrac12 b_{N}(y,y)$. By Affine and Quadratic Functions on a Real Hilbert Space are of Class $C^2$ §form, $Q\in C^{2}(H)$ with $DQ(y)=T_{b_{N}}y$, where $\langle T_{b_{N}}y,a\rangle_{H}=b_{N}(y,a)$ for $a\in H$ (The Bounded Symmetric Operator Represented by a Bounded Symmetric Bilinear Form on a Real Hilbert Space §representation). By (2) with $\zeta=\mathrm{pr}_{N}y$, $Q(y)=\delta h(\Pi_{N}y)$ for $y\in H$. Put

$$\psi_{\sigma}(y)=-\varphi(Ty)+\sigma Q(y)\qquad(y\in H).$$

By Step 3 (with $c=\Pi_{N}\hat{x}$, $\chi=\varphi$) and Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space §scalar, Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space §sum, $\psi_{\sigma}\in C^{2}(H)$ and, as $T\hat{x}=\hat{x}$,

$$D\psi_{\sigma}(\hat{x})=-L\,D\varphi(\hat{x})+\sigma T_{b_{N}}\hat{x}.\qquad(6)$$

For $y\in V$, $Q(y)=\delta h(\Pi_{N}y)$ and (3) give $\psi_{\sigma}(y)-\sigma\delta h(y)=-\varphi(Ty)-\sigma\delta h(Ly)$, and by the properties of $T$ above this equals $\bigl(w(Ty)-\sigma\delta h(Ty)-\varphi(Ty)\bigr)-w(\hat{x})+\sigma\delta h(\Pi_{N}\hat{x})$. Subtracting the same identity at $y=\hat{x}$ (where $T\hat{x}=\hat{x}$),

$$\bigl(\psi_{\sigma}(y)-\sigma\delta h(y)\bigr)-\bigl(\psi_{\sigma}(\hat{x})-\sigma\delta h(\hat{x})\bigr)=\bigl(w-\sigma\delta h-\varphi\bigr)(Ty)-\bigl(w-\sigma\delta h-\varphi\bigr)(\hat{x}).$$

If $y\in V$ and $|y-\hat{x}|_{H}<r$, then $Ty\in V$ and $|Ty-\hat{x}|_{H}<r$, so (5) at $x=Ty$ shows that $\sigma$ times the left side is at most $0$. Thus for $\sigma=1$ the function $\psi_{1}-\delta h$ has a local maximum at $\hat{x}$ relative to $V$, and First- and Second-Order Conditions at a Local Extremum of a $C^2$ Function Penalised by $h$ on the Small Space §maximum (with $\lambda=\delta$) gives $\hat{x}\in D(A)$ and $\delta A\hat{x}=D\psi_{1}(\hat{x})$; for $\sigma=-1$ the function $\psi_{-1}+\delta h$ has a local minimum at $\hat{x}$ relative to $V$, and First- and Second-Order Conditions at a Local Extremum of a $C^2$ Function Penalised by $h$ on the Small Space §minimum gives $\hat{x}\in D(A)$ and $\delta A\hat{x}=-D\psi_{-1}(\hat{x})$. In both cases $\hat{x}\in D(A)$ and $\sigma\delta A\hat{x}=D\psi_{\sigma}(\hat{x})$. Put

$$p_{\sigma}=D\varphi(\hat{x})+\sigma\delta A\hat{x}\in H .$$

Let $k\notin\Gamma_{N}$. Then $Le_{k}=e_{k}$ (Step 1(a)), so $\langle LD\varphi(\hat{x}),e_{k}\rangle_{H}=\langle D\varphi(\hat{x}),e_{k}\rangle_{H}$ by self-adjointness of $L$, and $\langle T_{b_{N}}\hat{x},e_{k}\rangle_{H}=b_{N}(\hat{x},e_{k})=0$ because $e_{k}$ vanishes at every $k_{i}$. Pairing $\sigma\delta A\hat{x}=D\psi_{\sigma}(\hat{x})$ with $e_{k}$ and using (6) gives $\sigma\delta\langle A\hat{x},e_{k}\rangle_{H}=-\langle D\varphi(\hat{x}),e_{k}\rangle_{H}$, i.e. $\langle p_{\sigma},e_{k}\rangle_{H}=0$, so by (1)

$$p_{\sigma}(k)=0\qquad(k\notin\Gamma_{N}).\qquad(7)$$

Moreover, 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 and (1), $\langle A\hat{x},E_{i}\rangle_{H}=\mu_{k_{i}}\hat{x}(k_{i})/\mu_{k_{i}}^{3}=\hat{z}_{i}/\mu_{k_{i}}^{2}$ for $i\in[m]$, hence

$$\langle D\varphi(\hat{x}),E_{i}\rangle_{H}+\sigma\delta\frac{\hat{z}_{i}}{\mu_{k_{i}}^{2}}=\langle p_{\sigma},E_{i}\rangle_{H}=\frac{p_{\sigma}(k_{i})}{\mu_{k_{i}}^{3}}.\qquad(8)$$

Step 7 (The Euclidean test function). With $S(z)=\iota_{N}z+y_{0}\in V$ as in Step 4, define $\Psi:\mathbb{R}^{m}\to\mathbb{R}$ by $\Psi(z)=\varphi(S(z))+\sigma\delta h(S(z))$. Since $y_{0}\in V$ vanishes at every $k_{i}$, (2) gives

$$\Psi(z)=\varphi_{0}(z)+\sigma\delta h(y_{0})+\frac{\sigma\delta}{2}\sum_{i=1}^{m}\frac{z_{i}^{2}}{\mu_{k_{i}}^{2}},\qquad \varphi_{0}=\varphi\circ S .$$

By Step 4, $\varphi_{0}$ is of class $C^{2}$ on $\mathbb{R}^{m}$; the remaining terms form a function built from constants and coordinate functions by sums, scalar multiples and products, which is smooth on $\mathbb{R}^{m}$ by claims 2 and 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, and hence of class $C^{2}$ by Smooth Map on a Euclidean Open Set; so $\Psi$ is of class $C^{2}$ on $\mathbb{R}^{m}$ by claim 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set. By claim 1 there (the $i$th partial of the $l$th coordinate function being $1$ if $l=i$ and $0$ otherwise, directly from Partial Derivative on a Euclidean Open Set) and Step 4, for $z\in\mathbb{R}^{m}$ and $i,l\in[m]$,

$$\partial_{i}\Psi(z)=\langle D\varphi(S(z)),E_{i}\rangle_{H}+\sigma\delta\frac{z_{i}}{\mu_{k_{i}}^{2}},\qquad \partial_{l}\partial_{i}\Psi(z)=D^{2}\varphi(S(z))(E_{l},E_{i})+\begin{cases}\sigma\delta\mu_{k_{i}}^{-2}&(l=i),\\0&(l\neq i).\end{cases}$$

For $z\in\mathbb{R}^{m}$, $\mathrm{pr}_{N}S(z)=z+\mathrm{pr}_{N}y_{0}=z$, so $w(S(z))=Z(z)$ and

$$Z(z)-\Psi(z)=\bigl(w-\sigma\delta h-\varphi\bigr)(S(z)),\qquad S(\hat{z})=\hat{x},\qquad |S(z)-\hat{x}|_{H}\le\lVert z-\hat{z}\rVert$$

by (4). Hence every $z$ with $d_{E}(z,\hat{z})<r$ satisfies $\sigma[(Z-\Psi)(z)-(Z-\Psi)(\hat{z})]\le0$ by (5): $Z-\Psi$ has a local maximum (if $\sigma=1$), respectively minimum (if $\sigma=-1$), at $\hat{z}$ relative to $\mathbb{R}^{m}$. The defining property in Viscosity Subsolution and Supersolution of a Second-Order Equation, applied to the test function $\Psi$, gives

$$\sigma\,\mathcal{F}^{\sharp}_{N}\bigl(\hat{z},Z(\hat{z}),D\Psi(\hat{z}),D^{2}\Psi(\hat{z})\bigr)\le0.\qquad(9)$$

Step 8 (Translation term by term). Let $s=w^{\sigma}_{\delta}(\hat{x})=w(\hat{x})-\sigma\delta h(\hat{x})$ and $X_{\sigma}=D^{2}\varphi(\hat{x})|_{V}+\sigma\delta I_{V}$. Since $\hat{x}\in D(A)$ (Step 6), Step 5 gives

$$F^{\sigma}\bigl(\hat{x},s,D\varphi(\hat{x}),D^{2}\varphi(\hat{x})\bigr)=F^{\sharp}_{N}\bigl(\hat{x},w(\hat{x}),p_{\sigma},X_{\sigma}\bigr),$$

and $w(\hat{x})=Z(\hat{z})$. We compare the terms of The Shifted Hamilton-Jacobi-Bellman Equation of the Wick-Square Problem with Noise, Control and Cost Cut Off to a Cube of Modes §operator with those of The Galerkin Hamilton-Jacobi-Bellman Equations of the Wick-Square Problem in the Coordinates of a Cube of Modes §shifted, using $S(\hat{z})=\hat{x}$.

Gradient. By Step 7 and (8), $\partial_{i}\Psi(\hat{z})=\langle p_{\sigma},E_{i}\rangle_{H}$ for every $i$. The gradient $D\Psi(\hat{z})$ is the vector of the partials $\partial_{i}\Psi(\hat{z})$ (Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives, Gradient of a Real-Valued Function on a Euclidean Open Set), and the squared Euclidean norm of a vector is the sum of the squares of its coordinates (Euclidean Norm on $\mathbb{R}^n$). Hence $\tfrac12\lVert D\Psi(\hat{z})\rVert^{2}=\tfrac12\sum_{i=1}^{m}\langle p_{\sigma},E_{i}\rangle_{H}^{2}=\tfrac12\sum_{k\in\Gamma_{N}}\langle p_{\sigma},e_{k}\rangle_{H}^{2}=\tfrac12 G_{N}(p_{\sigma},p_{\sigma})$, with $G_{N}$ as in The Shifted Hamilton-Jacobi-Bellman Equation of the Wick-Square Problem with Noise, Control and Cost Cut Off to a Cube of Modes §data.

Drift. By The Galerkin Problems of the Wick-Square Problem: Standing Notation for Cubes of Modes, Coordinates, the Free Galerkin Quadratic and the Riccati Potential §functions and claim 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, $\partial_{i}\Phi_{N}(\hat{z})=(\mu_{k_{i}}+2q_{k_{i}})\hat{z}_{i}$; since $D\Phi_{N}(\hat{z})$ and $D\Psi(\hat{z})$ are the vectors of partials (Gradient of a Real-Valued Function on a Euclidean Open Set) and the dot product is the sum of the coordinatewise products (Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$), and $\hat{z}_{i}=\hat{x}(k_{i})$,

$$D\Phi_{N}(\hat{z})\cdot D\Psi(\hat{z})=\sum_{i=1}^{m}\bigl(\mu_{k_{i}}+2q_{k_{i}}\bigr)\hat{x}(k_{i})\,\langle p_{\sigma},E_{i}\rangle_{H}.$$

On the other side, $\hat{x}\in H^{-1}$ and $p_{\sigma}\in H$, 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 §pairings $\langle A\hat{x},p_{\sigma}\rangle_{H}$ is the lattice sum of $k\mapsto\hat{x}(k)p_{\sigma}(k)/\mu_{k}^{2}$. By (7) this family vanishes off $\Gamma_{N}$; for $M\ge N$ one has $\Gamma_{N}\subseteq\Gamma_{M}$ (The Wick-Square Problem on the Torus: Standing Notation §cubes), so the $M$th cube sum equals $\sum_{k\in\Gamma_{N}}\hat{x}(k)p_{\sigma}(k)/\mu_{k}^{2}$, and by Cube Sums of Families on the Integer Lattice §lattice-sum so does the lattice sum. With (8),

$$\langle A\hat{x},p_{\sigma}\rangle_{H}=\sum_{i=1}^{m}\frac{\hat{x}(k_{i})p_{\sigma}(k_{i})}{\mu_{k_{i}}^{2}}=\sum_{i=1}^{m}\mu_{k_{i}}\hat{x}(k_{i})\langle p_{\sigma},E_{i}\rangle_{H}.$$

By The Shifted Hamilton-Jacobi-Bellman Equation of the Wick-Square Problem with Noise, Control and Cost Cut Off to a Cube of Modes §data and Step 1(a), $B^{N}_{q}\hat{x}=\Pi_{N}(B_{q}\hat{x})=\sum_{i=1}^{m}2q_{k_{i}}\hat{x}(k_{i})E_{i}$, so by Inner Products Against Finite Sums, and Orthonormal Families, in a Real Inner Product Space §combinations $\langle B^{N}_{q}\hat{x},p_{\sigma}\rangle_{H}=\sum_{i=1}^{m}2q_{k_{i}}\hat{x}(k_{i})\langle E_{i},p_{\sigma}\rangle_{H}$. Adding, by Elementary Identities in a Real Inner Product Space §bilinear,

$$\langle A\hat{x}+B^{N}_{q}\hat{x},p_{\sigma}\rangle_{H}=D\Phi_{N}(\hat{z})\cdot D\Psi(\hat{z}).$$

Trace. The Hessian $D^{2}\Psi(\hat{z})$ is the matrix with entries $\partial_{i}\partial_{l}\Psi(\hat{z})$ (Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives, Hessian Matrix of a C^2 Function), so by Trace of a Real Square Matrix its trace is $\sum_{i=1}^{m}\partial_{i}\partial_{i}\Psi(\hat{z})$, and by Step 7, $\operatorname{tr}D^{2}\Psi(\hat{z})=\sum_{i=1}^{m}\bigl(D^{2}\varphi(\hat{x})(E_{i},E_{i})+\sigma\delta\mu_{k_{i}}^{-2}\bigr)=\sum_{i=1}^{m}X_{\sigma}(E_{i},E_{i})$, since $X_{\sigma}(E_{i},E_{i})=D^{2}\varphi(\hat{x})(E_{i},E_{i})+\sigma\delta\langle E_{i},E_{i}\rangle_{V}$ by Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction §identity and Bounded Symmetric Bilinear Forms on a Real Inner Product Space: Norm, Order, Identity Form and Restriction §restriction, and $\langle E_{i},E_{i}\rangle_{V}=\mu_{k_{i}}^{-2}$ by Step 1. On the other side, with $\kappa$ the enumeration of The Wick-Square Problem on the Sobolev Triple of Order Two: Standing Notation §noise, a bijection of $\mathbb{N}$ onto $\mathbb{Z}^{n}$ (an enumeration being by definition such a bijection, Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families), the set $J=\{j\in\mathbb{N}:\kappa(j)\in\Gamma_{N}\}$ is finite and nonempty and $\kappa$ maps it bijectively onto $\Gamma_{N}$; the terms $X_{\sigma}(f^{N}_{j},f^{N}_{j})$ vanish for $j\notin J$, since then $f^{N}_{j}=0_{H}$ and $X_{\sigma}$ is bilinear. So every partial sum of index at least $\max J$ of the series defining $\mathrm{Tr}_{f^{N}}X_{\sigma}$ in Square-Summable Sequences in the Form Space of a Hilbert Triple and the Trace of a Form along Them §trace equals $\sum_{j\in J}X_{\sigma}(e_{\kappa(j)},e_{\kappa(j)})=\sum_{k\in\Gamma_{N}}X_{\sigma}(e_{k},e_{k})=\sum_{i=1}^{m}X_{\sigma}(E_{i},E_{i})$, and the series converges to this number. Hence $\operatorname{tr}D^{2}\Psi(\hat{z})=\mathrm{Tr}_{f^{N}}X_{\sigma}$.

Running cost. $\iota_{N}\hat{z}=\Pi_{N}\hat{x}\in H^{-1}$, so the running cost $g|_{H^{-1}}$ of $\mathcal{F}^{\sharp}_{N}$ takes the value $g(\Pi_{N}\hat{x})$ at $\iota_{N}\hat{z}$.

Combining these identities,

$$F^{\sigma}\bigl(\hat{x},s,D\varphi(\hat{x}),D^{2}\varphi(\hat{x})\bigr)=\gamma Z(\hat{z})-\tfrac{\nu}{2}\mathrm{Tr}_{f^{N}}X_{\sigma}+\tfrac12G_{N}(p_{\sigma},p_{\sigma})+\langle A\hat{x}+B^{N}_{q}\hat{x},p_{\sigma}\rangle_{H}-g(\Pi_{N}\hat{x})=\mathcal{F}^{\sharp}_{N}\bigl(\hat{z},Z(\hat{z}),D\Psi(\hat{z}),D^{2}\Psi(\hat{z})\bigr),$$

and therefore, by (9), $\sigma F^{\sigma}(\hat{x},s,D\varphi(\hat{x}),D^{2}\varphi(\hat{x}))\le0$.

Step 9 (Conclusion). Take $y=\hat{x}$, which lies in $D(A)=W$ by Step 6, the number $s=w^{\sigma}_{\delta}(\hat{x})$ of Step 8, the vector $D\varphi(\hat{x})\in H$ in the gradient slot of Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple and the form $Y=D^{2}\varphi(\hat{x})\in\mathrm{Sym}(H)$. Then $|y-\hat{x}|_{H}=0$, $|w^{\sigma}_{\delta}(y)-w^{\sigma}_{\delta}(\hat{x})|=0$, $|s-w^{\sigma}_{\delta}(\hat{x})|=0$, $|D\varphi(\hat{x})-D\varphi(\hat{x})|_{H}=0$ and $\lVert Y-D^{2}\varphi(\hat{x})\rVert=\lVert0_{\mathrm{Sym}}\rVert=0$ (Elementary Properties of Bounded Symmetric Bilinear Forms: Norm, Quadratic Form, Order and Continuity §norm-axioms), all less than $\varepsilon$. If $\sigma=1$, Step 8 gives $(F^{\sharp}_{N})^{-}_{\delta}(y,s,D\varphi(\hat{x}),Y)\le0<\varepsilon$; if $\sigma=-1$, it gives $-\varepsilon<0\le(F^{\sharp}_{N})^{+}_{\delta}(y,s,D\varphi(\hat{x}),Y)$. Since $\delta$, $\varphi$, $\hat{x}$ and $\varepsilon$ were arbitrary, $w$ is a viscosity subsolution of $F^{\sharp}_{N}$ on $H$ when $Z$ is a viscosity subsolution of $\mathcal{F}^{\sharp}_{N}$ on $\mathbb{R}^{m}$, by Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple §subsolution, which is clause 2; and $w$ is a viscosity supersolution of $F^{\sharp}_{N}$ on $H$ when $Z$ is a viscosity supersolution of $\mathcal{F}^{\sharp}_{N}$ on $\mathbb{R}^{m}$, by Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on a Hilbert Triple §supersolution, which is clause 3.