Proof of The Riccati Shift of the Galerkin Equations of the Wick-Square Problem: a Penalty-Drift Equation with Bounded Cost, and Well-Posedness at Each Cutoff

Each result cited below is universally quantified over the data in its own statement. Elementary facts about real numbers, finite sums, squares and absolute values are used through The Real Numbers: Standing Notation and Background without further mention.

Notation. Write $\kappa=\kappa_{N}$ and, for $i\in[m]$,

$$\hat q_{i}=q_{\kappa(i)},\qquad \hat a_{i}=a_{\kappa(i)},\qquad \hat c_{i}=c_{\kappa(i)},\qquad \hat\mu_{i}=\mu_{\kappa(i)},\qquad w_{i}=\hat\mu_{i}+2\hat q_{i}.$$

Each $\hat q_{i}$ is positive by The Riccati Coefficients of the Wick-Square Problem: Existence, the Gaussian Profile and Summable Constants §root, applied at the mode $\kappa(i)$, and $1\le\hat\mu_{i}$ by The Wick-Square Problem on the Torus: Standing Notation §modes; hence $1\le w_{i}$. The numbers $\gamma$, $\nu$ and $\beta$ are positive by The Wick-Square Problem on the Torus: Standing Notation §parameters. Since $\kappa$ is a bijection from $[m]$ onto the nonempty set $\Gamma_{N}$ (The Galerkin Problems of the Wick-Square Problem: Standing Notation for Cubes of Modes, Coordinates, the Free Galerkin Quadratic and the Riccati Potential §cutoff), the set $[m]$ is nonempty, so $1\le m$. For $v,v'\in\mathbb{R}^{m}$ we use $\lVert v\rVert^{2}=\sum_{i=1}^{m}v_{i}^{2}$ (the Euclidean norm being the square root of this sum) and $v\cdot v'=\sum_{i=1}^{m}v_{i}v'_{i}$. Finally $\delta_{li}$ is $1$ if $l=i$ and $0$ otherwise.

Step 1 (Partial derivatives of the building blocks). Let $z\in\mathbb{R}^{m}$ and $i,j\in[m]$. For the constant function with value $b\in\mathbb{R}$, the difference quotient in Partial Derivative on a Euclidean Open Set equals $(b-b)/h=0$ for every nonzero $h$ (the perturbed point lies in the domain $\mathbb{R}^{m}$), so its partial derivative with respect to the $i$th variable exists at $z$ with value $0$. For the coordinate function $\pi_{j}(z)=z_{j}$ the quotient equals $\bigl((z_{j}+h\delta_{ij})-z_{j}\bigr)/h=\delta_{ij}$, so $\partial_{i}\pi_{j}(z)=\delta_{ij}$. By the product and scalar rules of claim 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, the function $z\mapsto\alpha z_{j}^{2}=\alpha\,\pi_{j}(z)\pi_{j}(z)$ has $i$th partial derivative $2\alpha\delta_{ij}z_{j}$ at $z$, for every $\alpha\in\mathbb{R}$. By the sum rule of the same claim, applied by induction on the number of summands, for reals $\alpha_{1},\dots,\alpha_{m},b_{1},\dots,b_{m}$ the function $f(z)=\sum_{j=1}^{m}(\alpha_{j}z_{j}^{2}+b_{j})$ satisfies

$$\partial_{i}f(z)=\sum_{j=1}^{m}2\alpha_{j}\delta_{ij}z_{j}=2\alpha_{i}z_{i},$$

and, the function $z\mapsto2\alpha_{i}z_{i}=2\alpha_{i}\pi_{i}(z)$ having $l$th partial derivative $2\alpha_{i}\delta_{li}$ by claim 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, also $\partial_{l}\partial_{i}f(z)=2\alpha_{i}\delta_{li}$ for every $l\in[m]$.

Step 2 (Gradients and Hessians of $\tilde u_{N}$ and $\Phi_{N}$). Both functions are of class $C^{2}$ 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, so their gradients and Hessians are those of Second-Order Equations on Euclidean Open Sets §test-functions, namely the vector of first partial derivatives (Gradient of a Real-Valued Function on a Euclidean Open Set) and the matrix of iterated partial derivatives (Hessian Matrix of a C^2 Function). Step 1 with $\alpha_{j}=\hat q_{j}$ and $b_{j}=\hat a_{j}$ gives, for every $z$,

$$\partial_{i}\tilde u_{N}(z)=2\hat q_{i}z_{i},\qquad \partial_{l}\partial_{i}\tilde u_{N}(z)=2\hat q_{i}\delta_{li},$$

so $D\tilde u_{N}(z)$ has components $2\hat q_{i}z_{i}$ and $D^{2}\tilde u_{N}(z)$ is the diagonal matrix with diagonal entries $2\hat q_{i}$; this is the first assertion of clause 1. Step 1 with $\alpha_{j}=\tfrac12w_{j}$ and $b_{j}=0$ gives likewise that $D\Phi_{N}(z)$ has components $w_{i}z_{i}$ and that $D^{2}\Phi_{N}(z)$ is the diagonal matrix with diagonal entries $w_{i}$.

Step 3 (The identity of clause 1). Fix $z,p\in\mathbb{R}^{m}$, $r\in\mathbb{R}$ and $X\in\mathcal{S}(m)$, and put $r'=r+\tilde u_{N}(z)$, $p'=p+D\tilde u_{N}(z)$ and $X'=X+D^{2}\tilde u_{N}(z)$, which lies in $\mathcal{S}(m)$ because $\mathcal{S}(m)$ is closed under sums (Second-Order Equations on Euclidean Open Sets §matrices). By Step 2, the definition of the trace (Trace of a Real Square Matrix) and additivity of finite sums,

$$\operatorname{tr}(X')=\operatorname{tr}(X)+\sum_{i=1}^{m}2\hat q_{i},\qquad \tfrac12\lVert p'\rVert^{2}=\tfrac12\sum_{i=1}^{m}(p_{i}+2\hat q_{i}z_{i})^{2}=\tfrac12\lVert p\rVert^{2}+\sum_{i=1}^{m}2\hat q_{i}z_{i}p_{i}+\sum_{i=1}^{m}2\hat q_{i}^{2}z_{i}^{2},$$ $$\sum_{i=1}^{m}\hat\mu_{i}z_{i}p'_{i}=\sum_{i=1}^{m}\hat\mu_{i}z_{i}p_{i}+\sum_{i=1}^{m}2\hat\mu_{i}\hat q_{i}z_{i}^{2},\qquad \gamma r'=\gamma r+\sum_{i=1}^{m}\gamma\hat q_{i}z_{i}^{2}+\sum_{i=1}^{m}\gamma\hat a_{i}.$$

The Wick-ordered Galerkin operator is the operator of The Galerkin Hamilton-Jacobi-Bellman Equations of the Wick-Square Problem in the Coordinates of a Cube of Modes §operator with counterterm $k\mapsto c_{k}$, by The Galerkin Hamilton-Jacobi-Bellman Equations of the Wick-Square Problem in the Coordinates of a Cube of Modes §wick, so that the counterterm contributes $\hat c_{i}$ at the coordinate $i$. Substituting the four displays into

$$\mathcal{F}_{N}(z,r',p',X')=\gamma r'-\tfrac{\nu}{2}\operatorname{tr}(X')+\tfrac12\lVert p'\rVert^{2}+\sum_{i=1}^{m}\hat\mu_{i}z_{i}p'_{i}-\beta\sum_{i=1}^{m}(z_{i}^{2}-\hat c_{i})-g(\iota_{N}z)$$

and collecting terms (using $\tfrac{\nu}{2}\cdot2\hat q_{i}=\nu\hat q_{i}$) gives

$$\mathcal{F}_{N}(z,r',p',X')=\gamma r-\tfrac{\nu}{2}\operatorname{tr}(X)+\tfrac12\lVert p\rVert^{2}+\sum_{i=1}^{m}w_{i}z_{i}p_{i}-g(\iota_{N}z)+\sum_{i=1}^{m}\bigl(2\hat q_{i}^{2}+(\gamma+2\hat\mu_{i})\hat q_{i}-\beta\bigr)z_{i}^{2}+\sum_{i=1}^{m}\bigl(\gamma\hat a_{i}-\nu\hat q_{i}+\beta\hat c_{i}\bigr).$$

For each $i$, the coefficient $2\hat q_{i}^{2}+(\gamma+2\hat\mu_{i})\hat q_{i}-\beta$ vanishes by The Riccati Coefficients of the Wick-Square Problem: Existence, the Gaussian Profile and Summable Constants §root at the mode $\kappa(i)$, and $\gamma\hat a_{i}=\nu\hat q_{i}-\beta\hat c_{i}$ by The Riccati Coefficients of the Wick-Square Problem: Existence, the Gaussian Profile and Summable Constants §constants at the mode $\kappa(i)$ (multiply its defining formula by $\gamma$), so the last two sums are sums of zeros. By Step 2, $\sum_{i=1}^{m}w_{i}z_{i}p_{i}=D\Phi_{N}(z)\cdot p$. Hence

$$\mathcal{F}_{N}(z,r',p',X')=\gamma r+\tfrac12\lVert p\rVert^{2}+D\Phi_{N}(z)\cdot p-\tfrac{\nu}{2}\operatorname{tr}(X)-g(\iota_{N}z),$$

which is $\mathcal{F}^{\sharp}_{N}(z,r,p,X)$ by The Galerkin Hamilton-Jacobi-Bellman Equations of the Wick-Square Problem in the Coordinates of a Cube of Modes §shifted. This proves clause 1.

Step 4 ($\Phi_{N}$ is a penalty). Regularity, Penalty on an Open Subset of Euclidean Space §regularity, holds because $\Phi_{N}$ is of class $C^{2}$ 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. Since $1\le w_{i}$ and $0\le z_{i}^{2}$, comparison of finite sums gives

$$\Phi_{N}(z)=\tfrac12\sum_{i=1}^{m}w_{i}z_{i}^{2}\ \ge\ \tfrac12\sum_{i=1}^{m}z_{i}^{2}=\tfrac12\lVert z\rVert^{2}\ \ge\ 0\qquad(z\in\mathbb{R}^{m}).$$

Let $t\in\mathbb{R}$ and $K_{t}=\{z\in\mathbb{R}^{m}:\Phi_{N}(z)\le t\}$. Being of class $C^{2}$, and so of class $C^{1}$, $\Phi_{N}$ is continuous at every point of $\mathbb{R}^{m}$ by Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives, hence lower semicontinuous on $\mathbb{R}^{m}$ by the final assertion of Semicontinuity Under Negation and Characterization of Continuity; by claim 3 of Semicontinuity via Sublevel and Superlevel Sets, used with both its space and its subset (written $X$ and $A$ there) equal to $\mathbb{R}^{m}$ and the metric $d_{E}$, the set $K_{t}$ is closed in $(\mathbb{R}^{m},\mathcal{T}_{d_{E}})$. For $z\in K_{t}$ the display gives $\lVert z\rVert^{2}\le2\Phi_{N}(z)\le2|t|$; if $1\le\lVert z\rVert$ then $\lVert z\rVert\le\lVert z\rVert^{2}\le2|t|$, so in every case $d_{E}(z,0_{\mathbb{R}^{m}})=\lVert z\rVert\le1+2|t|$, and $K_{t}$ is bounded in $(\mathbb{R}^{m},d_{E})$. By Heine-Borel Theorem in $\mathbb{R}^n$, $K_{t}$ is compact. Thus Penalty on an Open Subset of Euclidean Space §sublevel holds, and $\Phi_{N}$ is a penalty on the open set $\mathbb{R}^{m}$.

Step 5 (The two conditions, clause 2). Let $x,y\in\mathbb{R}^{m}$. By Step 2, $D\Phi_{N}(x)-D\Phi_{N}(y)$ has components $w_{i}(x_{i}-y_{i})$, so

$$\bigl(D\Phi_{N}(x)-D\Phi_{N}(y)\bigr)\cdot(x-y)=\sum_{i=1}^{m}w_{i}(x_{i}-y_{i})^{2}\ \ge\ 0=-0\cdot\lVert x-y\rVert^{2}.$$

This holds for all $x,y\in\mathbb{R}^{m}$, in particular on every sublevel set $\{x:\Phi_{N}(x)<R\}$, so Well-Posedness of the Penalty-Drift Hamilton-Jacobi Equation under a Dissipation Inequality §one-sided holds with $c_{R}=0$ for every $R$. By Step 2 and Trace of a Real Square Matrix, $\operatorname{tr}(D^{2}\Phi_{N}(x))=\sum_{i=1}^{m}w_{i}$, so $\tfrac{\nu}{2}\operatorname{tr}(D^{2}\Phi_{N}(x))=C$ with $C=\tfrac{\nu}{2}\sum_{i=1}^{m}w_{i}$. Since $\gamma$ is positive and $\Phi_{N}(x)\ge0$ by Step 4,

$$\tfrac{\nu}{2}\operatorname{tr}\bigl(D^{2}\Phi_{N}(x)\bigr)=C\le(1-1)\lVert D\Phi_{N}(x)\rVert^{2}+\gamma\Phi_{N}(x)+C ,$$

which is Well-Posedness of the Penalty-Drift Hamilton-Jacobi Equation under a Dissipation Inequality §dissipation for the potential $\Phi_{N}$, discount $\gamma$ and noise intensity $\nu$, with $\varepsilon=1$ (and $0<1\le1$). With Step 4 this proves clause 2.

Step 6 (Semicontinuity is unchanged by the shift). As in Step 4, $\tilde u_{N}$ is continuous on $\mathbb{R}^{m}$, hence both upper and lower semicontinuous there by Semicontinuity Under Negation and Characterization of Continuity. Let $U:\mathbb{R}^{m}\to\mathbb{R}$ and $V=U-\tilde u_{N}$. If $U$ is upper semicontinuous, so is $V$ by claim 3 of Negation, Restriction, and Separated Differences of Semicontinuous Functions (first sentence, with $w=\tilde u_{N}$); if $V$ is upper semicontinuous, so is $U=V+\tilde u_{N}$ by claim 1 of Sums and Nonnegative Multiples of Semicontinuous Functions. In the same way, by the second sentence of claim 3 of Negation, Restriction, and Separated Differences of Semicontinuous Functions and by claim 3 of Sums and Nonnegative Multiples of Semicontinuous Functions, $U$ is lower semicontinuous if and only if $V$ is.

Step 7 (Test functions correspond). Let $\varphi,\psi:\mathbb{R}^{m}\to\mathbb{R}$ with $\psi=\varphi-\tilde u_{N}$, that is $\varphi=\psi+\tilde u_{N}$. Since $\tilde u_{N}$ is of class $C^{2}$, claim 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set (with $\varphi-\tilde u_{N}=\varphi+(-1)\tilde u_{N}$) shows that $\varphi$ is of class $C^{2}$ if and only if $\psi$ is. Suppose they are. By the sum rule of claim 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, $\partial_{i}\varphi(z)=\partial_{i}\psi(z)+\partial_{i}\tilde u_{N}(z)$ at every $z$, so $\partial_{i}\varphi=\partial_{i}\psi+\partial_{i}\tilde u_{N}$ as functions. Since $\psi$ and $\tilde u_{N}$ are of class $C^{2}$, the functions $\partial_{i}\psi$ and $\partial_{i}\tilde u_{N}$ have partial derivatives with respect to every variable at every point, so claim 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set applies to their sum, and applying its sum rule again, $\partial_{l}\partial_{i}\varphi(z)=\partial_{l}\partial_{i}\psi(z)+\partial_{l}\partial_{i}\tilde u_{N}(z)$. Hence $D\varphi(z)=D\psi(z)+D\tilde u_{N}(z)$ and $D^{2}\varphi(z)=D^{2}\psi(z)+D^{2}\tilde u_{N}(z)$. Moreover, for $U$ and $V=U-\tilde u_{N}$ as in Step 6, $U(z)-\varphi(z)=\bigl(U(z)-\tilde u_{N}(z)\bigr)-\bigl(\varphi(z)-\tilde u_{N}(z)\bigr)=V(z)-\psi(z)$ for every $z$, so $U-\varphi=V-\psi$ as functions and they have the same local maxima and the same local minima. Applying Step 3 with $r=V(z)$, $p=D\psi(z)$ and $X=D^{2}\psi(z)\in\mathcal{S}(m)$ gives

$$\mathcal{F}_{N}\bigl(z,U(z),D\varphi(z),D^{2}\varphi(z)\bigr)=\mathcal{F}^{\sharp}_{N}\bigl(z,V(z),D\psi(z),D^{2}\psi(z)\bigr)\qquad(z\in\mathbb{R}^{m})\qquad(*)$$

Step 8 (Clause 3). By The Galerkin Hamilton-Jacobi-Bellman Equations of the Wick-Square Problem in the Coordinates of a Cube of Modes §equation, viscosity sub- and supersolutions are those of Viscosity Subsolution and Supersolution of a Second-Order Equation for the operators $\mathcal{F}_{N}$ and $\mathcal{F}^{\sharp}_{N}$ on the open set $\mathbb{R}^{m}$. Let $U:\mathbb{R}^{m}\to\mathbb{R}$ and $V=U-\tilde u_{N}$. Suppose $U$ is a viscosity subsolution of $\mathcal{F}_{N}$. Then $V$ is upper semicontinuous by Step 6. Let $\psi$ be of class $C^{2}$ and let $V-\psi$ have a local maximum at $z$. Then $\varphi=\psi+\tilde u_{N}$ is of class $C^{2}$ and $U-\varphi=V-\psi$ has a local maximum at $z$ (Step 7), so $\mathcal{F}_{N}(z,U(z),D\varphi(z),D^{2}\varphi(z))\le0$, and by $(*)$ $\mathcal{F}^{\sharp}_{N}(z,V(z),D\psi(z),D^{2}\psi(z))\le0$. So $V$ is a viscosity subsolution of $\mathcal{F}^{\sharp}_{N}$. Conversely, suppose $V$ is a viscosity subsolution of $\mathcal{F}^{\sharp}_{N}$. Then $U$ is upper semicontinuous by Step 6. Let $\varphi$ be of class $C^{2}$ and let $U-\varphi$ have a local maximum at $z$. Then $\psi=\varphi-\tilde u_{N}$ is of class $C^{2}$ and $V-\psi=U-\varphi$ has a local maximum at $z$, so $\mathcal{F}^{\sharp}_{N}(z,V(z),D\psi(z),D^{2}\psi(z))\le0$, and by $(*)$ $\mathcal{F}_{N}(z,U(z),D\varphi(z),D^{2}\varphi(z))\le0$. So $U$ is a viscosity subsolution of $\mathcal{F}_{N}$. Now suppose $U$ is a viscosity supersolution of $\mathcal{F}_{N}$. Then $V$ is lower semicontinuous by Step 6 (second sentence of claim 3 of Negation, Restriction, and Separated Differences of Semicontinuous Functions). Let $\psi$ be of class $C^{2}$ and let $V-\psi$ have a local minimum at $z$. Then $\varphi=\psi+\tilde u_{N}$ is of class $C^{2}$ and $U-\varphi=V-\psi$ has a local minimum at $z$ (Step 7), so $0\le\mathcal{F}_{N}(z,U(z),D\varphi(z),D^{2}\varphi(z))$, and by $(*)$ $0\le\mathcal{F}^{\sharp}_{N}(z,V(z),D\psi(z),D^{2}\psi(z))$; so $V$ is a viscosity supersolution of $\mathcal{F}^{\sharp}_{N}$. Conversely, suppose $V$ is a viscosity supersolution of $\mathcal{F}^{\sharp}_{N}$. Then $U=V+\tilde u_{N}$ is lower semicontinuous by Step 6 (claim 3 of Sums and Nonnegative Multiples of Semicontinuous Functions). Let $\varphi$ be of class $C^{2}$ and let $U-\varphi$ have a local minimum at $z$. Then $\psi=\varphi-\tilde u_{N}$ is of class $C^{2}$ and $V-\psi=U-\varphi$ has a local minimum at $z$, so $0\le\mathcal{F}^{\sharp}_{N}(z,V(z),D\psi(z),D^{2}\psi(z))$, and by $(*)$ $0\le\mathcal{F}_{N}(z,U(z),D\varphi(z),D^{2}\varphi(z))$; so $U$ is a viscosity supersolution of $\mathcal{F}_{N}$. This proves clause 3.

Step 9 (Existence in clause 4). We apply Well-Posedness of the Penalty-Drift Hamilton-Jacobi Equation under a Dissipation Inequality; since its letters $n$, $D$, $P$, $\lambda$, $\theta$, $\kappa$ and $M$ have other meanings here, we name its data by their roles. The dimension of that theorem (written $n$ there) is $m\ge1$; its open nonempty set (written $D$ there, not to be confused with the gradient $D$) is $\mathbb{R}^{m}$; its penalty (written $P$ there) is $\Phi_{N}$ (Step 4); its discount (written $\lambda$ there) is $\gamma$, which is positive; its control cost (written $\theta$ there) is $1$ and its noise intensity (written $\kappa$ there, not to be confused with $\kappa=\kappa_{N}$) is $\nu$, both nonnegative; its running cost is $g\circ\iota_{N}:\mathbb{R}^{m}\to\mathbb{R}$, which is continuous by hypothesis; and its bound on the running cost (written $M$ there) is $C_{g}$, which bounds $g\circ\iota_{N}$ because $\iota_{N}z\in H^{-1}$ (The Galerkin Problems of the Wick-Square Problem: Standing Notation for Cubes of Modes, Coordinates, the Free Galerkin Quadratic and the Riccati Potential §coordinates) and $|g|\le C_{g}$ on $H^{-1}$. The operator $F$ of that theorem is then $\mathcal{F}^{\sharp}_{N}$ by The Galerkin Hamilton-Jacobi-Bellman Equations of the Wick-Square Problem in the Coordinates of a Cube of Modes §shifted, and its two conditions hold by Step 5. By Well-Posedness of the Penalty-Drift Hamilton-Jacobi Equation under a Dissipation Inequality §well-posed there is exactly one $W:\mathbb{R}^{m}\to\mathbb{R}$ that is a viscosity subsolution and a viscosity supersolution of $\mathcal{F}^{\sharp}_{N}$ on $\mathbb{R}^{m}$ and has $\Phi_{N}$-subordinate growth from above and from below; it is continuous on $\mathbb{R}^{m}$ and $-C_{g}/\gamma\le W(z)\le C_{g}/\gamma$ for every $z$. Put $U_{N}=\tilde u_{N}+W$. Since $U_{N}-\tilde u_{N}=W$, clause 3 shows that $U_{N}$ is a viscosity subsolution and a viscosity supersolution of $\mathcal{F}_{N}$ on $\mathbb{R}^{m}$; moreover $U_{N}-\tilde u_{N}=W$ is bounded above by $C_{g}/\gamma$ and below by $-C_{g}/\gamma$, it is continuous, and $|U_{N}(z)-\tilde u_{N}(z)|\le C_{g}/\gamma$ for every $z$.

Step 10 (Uniqueness in clause 4). Let $U:\mathbb{R}^{m}\to\mathbb{R}$ be a viscosity subsolution and a viscosity supersolution of $\mathcal{F}_{N}$ on $\mathbb{R}^{m}$ with $V=U-\tilde u_{N}$ bounded: there are reals $s,t$ with $t\le V(z)\le s$ for every $z$, and with $B=|s|+|t|$ we have $-B\le V(z)\le B$. By clause 3, $V$ is a viscosity subsolution and a viscosity supersolution of $\mathcal{F}^{\sharp}_{N}$ on $\mathbb{R}^{m}$. Let $\eta$ be positive. Since $\Phi_{N}\ge0$ (Step 4), $\eta\Phi_{N}(z)\ge0$, so

$$V(z)\le B\le B+\eta\Phi_{N}(z)\qquad\text{and}\qquad-B-\eta\Phi_{N}(z)\le-B\le V(z)\qquad(z\in\mathbb{R}^{m}).$$

Taking the growth parameter of Penalty-Subordinate Growth of a Function on an Open Subset of Euclidean Space (written $\delta$ there) to be $\eta$ and its constant (written $C$ there, a different number from the constant $C$ of Step 5) to be $B$, this shows that $V$ has $\Phi_{N}$-subordinate growth from above (Penalty-Subordinate Growth of a Function on an Open Subset of Euclidean Space §above) and from below (Penalty-Subordinate Growth of a Function on an Open Subset of Euclidean Space §below). By the uniqueness assertion of Well-Posedness of the Penalty-Drift Hamilton-Jacobi Equation under a Dissipation Inequality §well-posed, $V=W$, so $U=\tilde u_{N}+W=U_{N}$. This proves clause 4.