Proof of The Counterterm Is Forced at the Level of Galerkin Solutions: Other Counterterms Shift the Solutions by a Constant, and the Bare Solutions Diverge

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

Notation. The numbers $\beta$, $\gamma$ and $\nu$ are positive by The Wick-Square Problem on the Torus: Standing Notation §parameters. Fix $N\in\mathbb{N}$ until Step 4, write $\kappa=\kappa_{N}$ and $m$ as in The Galerkin Problems of the Wick-Square Problem: Standing Notation for Cubes of Modes, Coordinates, the Free Galerkin Quadratic and the Riccati Potential §cutoff, and put $s=s_{N}(b)$. The operator $\mathcal{F}_{N}$ of The Galerkin Solutions of the Wick-Square Problem Converge to the Renormalised Viscosity Solution for Every Bounded Lipschitz Running Cost §galerkin is the Wick-ordered Galerkin operator for $g$, which by The Galerkin Hamilton-Jacobi-Bellman Equations of the Wick-Square Problem in the Coordinates of a Cube of Modes §wick is the operator $\mathcal{F}^{c}_{N}$ of The Galerkin Hamilton-Jacobi-Bellman Equations of the Wick-Square Problem in the Coordinates of a Cube of Modes §operator with counterterm $k\mapsto c_{k}$. Viscosity sub- and supersolutions of $\mathcal{F}_{N}$ and $\mathcal{F}^{b}_{N}$ are those of Viscosity Subsolution and Supersolution of a Second-Order Equation on the open set $\mathbb{R}^{m}$, by The Galerkin Hamilton-Jacobi-Bellman Equations of the Wick-Square Problem in the Coordinates of a Cube of Modes §equation, and a viscosity solution is a function that is both.

Step 1 ($s_{N}(b)$ in coordinates). The map $[m]\to\mathbb{R}$, $i\mapsto c_{\kappa(i)}-b(\kappa(i))$, is the composite of $k\mapsto c_{k}-b(k)$ on $\Gamma_{N}$ with the bijection $\kappa:[m]\to\Gamma_{N}$ of The Galerkin Problems of the Wick-Square Problem: Standing Notation for Cubes of Modes, Coordinates, the Free Galerkin Quadratic and the Riccati Potential §cutoff. By claim 2 (reindexing along a bijection, with $G=[m]$, $F=\Gamma_{N}$ and $\theta=\kappa$) and claim 1 of Properties of a Sum over a Finite Index Set,

$$\sum_{k\in\Gamma_{N}}\bigl(c_{k}-b(k)\bigr)=\sum_{y\in[m]}\bigl(c_{\kappa(y)}-b(\kappa(y))\bigr)=\sum_{i=1}^{m}\bigl(c_{\kappa(i)}-b(\kappa(i))\bigr), \qquad\text{so}\qquad s=\beta\sum_{i=1}^{m}\bigl(c_{\kappa(i)}-b(\kappa(i))\bigr).$$

Step 2 (The operators differ by a shift of $r$). By The Galerkin Hamilton-Jacobi-Bellman Equations of the Wick-Square Problem in the Coordinates of a Cube of Modes §operator, the operators $\mathcal{F}^{b}_{N}$ and $\mathcal{F}_{N}=\mathcal{F}^{c}_{N}$ have the same terms except the counterterm sums, so for all $z,p\in\mathbb{R}^{m}$, $r\in\mathbb{R}$ and $X\in\mathcal{S}(m)$, by additivity of finite sums and Step 1,

$$\mathcal{F}^{b}_{N}(z,r,p,X)-\mathcal{F}_{N}(z,r,p,X)=-\beta\sum_{i=1}^{m}\bigl(z_{i}^{2}-b(\kappa(i))\bigr)+\beta\sum_{i=1}^{m}\bigl(z_{i}^{2}-c_{\kappa(i)}\bigr)=-\beta\sum_{i=1}^{m}\bigl(c_{\kappa(i)}-b(\kappa(i))\bigr)=-s .$$

Since $\mathcal{F}_{N}$ depends on $r$ only through the term $\gamma r$, we have $\mathcal{F}_{N}(z,r-s/\gamma,p,X)=\mathcal{F}_{N}(z,r,p,X)-s$, and therefore

$$\mathcal{F}^{b}_{N}(z,r,p,X)=\mathcal{F}_{N}(z,r-s/\gamma,p,X)\qquad(z,p\in\mathbb{R}^{m},\ r\in\mathbb{R},\ X\in\mathcal{S}(m)).\qquad(1)$$

Step 3 (Solutions shift by the constant $s/\gamma$). Let $\varepsilon_{0}:\mathbb{R}^{m}\to\mathbb{R}$ be the constant function with value $s/\gamma$. We claim: a function $U:\mathbb{R}^{m}\to\mathbb{R}$ is a viscosity subsolution (respectively supersolution) of $\mathcal{F}^{b}_{N}$ on $\mathbb{R}^{m}$ if and only if $\tilde{W}=U-\varepsilon_{0}$ is a viscosity subsolution (respectively supersolution) of $\mathcal{F}_{N}$ on $\mathbb{R}^{m}$.

Semicontinuity: $\varepsilon_{0}$ is continuous, since $|\varepsilon_{0}(z)-\varepsilon_{0}(y)|=0$ for all $z,y$, hence upper and lower semicontinuous by Semicontinuity Under Negation and Characterization of Continuity. If $U$ is upper (lower) semicontinuous, so is $\tilde{W}=U-\varepsilon_{0}$ by claim 3 of Negation, Restriction, and Separated Differences of Semicontinuous Functions; if $\tilde{W}$ is upper (lower) semicontinuous, so is $U=\tilde{W}+\varepsilon_{0}$ by claim 1 (claim 3) of Sums and Nonnegative Multiples of Semicontinuous Functions.

Test functions: $\varepsilon_{0}$ is smooth on $\mathbb{R}^{m}$ by claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, hence of class $C^{2}$ by Smooth Map on a Euclidean Open Set; so for $\varphi:\mathbb{R}^{m}\to\mathbb{R}$ the function $\psi=\varphi-\varepsilon_{0}$ is of class $C^{2}$ if and only if $\varphi=\psi+\varepsilon_{0}$ is, by claim 3 of that lemma. The difference quotient of $\varepsilon_{0}$ in Partial Derivative on a Euclidean Open Set is $0$ for every nonzero increment, so every partial derivative of $\varepsilon_{0}$ exists and vanishes at every point; by the sum and scalar rules of claim 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, applied once to $\psi=\varphi+(-1)\varepsilon_{0}$ and once more to the resulting equality $\partial_{i}\psi=\partial_{i}\varphi$ of functions, $D\psi(z)=D\varphi(z)$ and $D^{2}\psi(z)=D^{2}\varphi(z)$ for every $z$. Moreover $U(z)-\varphi(z)=(U(z)-s/\gamma)-(\varphi(z)-s/\gamma)=\tilde{W}(z)-\psi(z)$ for every $z$, so $U-\varphi=\tilde{W}-\psi$ as functions, with the same local maxima and minima. By (1) with $r=U(z)$, $p=D\varphi(z)$ and $X=D^{2}\varphi(z)$,

$$\mathcal{F}^{b}_{N}\bigl(z,U(z),D\varphi(z),D^{2}\varphi(z)\bigr)=\mathcal{F}_{N}\bigl(z,\tilde{W}(z),D\psi(z),D^{2}\psi(z)\bigr)\qquad(z\in\mathbb{R}^{m}).\qquad(2)$$

Now suppose $U$ is a viscosity subsolution of $\mathcal{F}^{b}_{N}$. Then $\tilde{W}$ is upper semicontinuous. If $\psi$ is of class $C^{2}$ and $\tilde{W}-\psi$ has a local maximum at $z$, then $\varphi=\psi+\varepsilon_{0}$ is of class $C^{2}$, $U-\varphi=\tilde{W}-\psi$ has a local maximum at $z$, so the left side of (2) is at most $0$, and hence so is the right side; thus $\tilde{W}$ is a viscosity subsolution of $\mathcal{F}_{N}$. Conversely, if $\tilde{W}$ is a viscosity subsolution of $\mathcal{F}_{N}$, then $U$ is upper semicontinuous, and for $\varphi$ of class $C^{2}$ with $U-\varphi$ having a local maximum at $z$, the function $\psi=\varphi-\varepsilon_{0}$ is of class $C^{2}$ and $\tilde{W}-\psi=U-\varphi$ has a local maximum at $z$, so the right side of (2) is at most $0$, and so is the left side. For supersolutions: if $U$ is a viscosity supersolution of $\mathcal{F}^{b}_{N}$, then $\tilde{W}$ is lower semicontinuous, and if $\psi$ is of class $C^{2}$ and $\tilde{W}-\psi$ has a local minimum at $z$, then $\varphi=\psi+\varepsilon_{0}$ is of class $C^{2}$ and $U-\varphi=\tilde{W}-\psi$ has a local minimum at $z$, so the left side of (2) is at least $0$, and hence so is the right side. Conversely, if $\tilde{W}$ is a viscosity supersolution of $\mathcal{F}_{N}$, then $U$ is lower semicontinuous, and for $\varphi$ of class $C^{2}$ with $U-\varphi$ having a local minimum at $z$, the function $\psi=\varphi-\varepsilon_{0}$ is of class $C^{2}$ and $\tilde{W}-\psi=U-\varphi$ has a local minimum at $z$, so the right side of (2) is at least $0$, and so is the left side. This proves the claim.

Step 4 (Clause 1). By The Galerkin Solutions of the Wick-Square Problem Converge to the Renormalised Viscosity Solution for Every Bounded Lipschitz Running Cost §galerkin, $U_{N}$ is a viscosity solution of $\mathcal{F}_{N}$ on $\mathbb{R}^{m}$ with $U_{N}-\tilde u_{N}$ bounded, and it is the only such function. Since $U^{b}_{N}-\varepsilon_{0}=U_{N}$, Step 3 shows that $U^{b}_{N}$ is a viscosity solution of $\mathcal{F}^{b}_{N}$; and $U^{b}_{N}-\tilde u_{N}=(U_{N}-\tilde u_{N})+s/\gamma$ is bounded, since adding the constant $s/\gamma$ shifts upper and lower bounds by $s/\gamma$. Let $U$ be any viscosity solution of $\mathcal{F}^{b}_{N}$ on $\mathbb{R}^{m}$ with $U-\tilde u_{N}$ bounded. By Step 3, $U-\varepsilon_{0}$ is a viscosity solution of $\mathcal{F}_{N}$, and $(U-\varepsilon_{0})-\tilde u_{N}=(U-\tilde u_{N})-s/\gamma$ is bounded. By the uniqueness in The Galerkin Solutions of the Wick-Square Problem Converge to the Renormalised Viscosity Solution for Every Bounded Lipschitz Running Cost §galerkin, $U-\varepsilon_{0}=U_{N}$, that is $U=U_{N}+s/\gamma=U^{b}_{N}$. As $N$ was arbitrary, clause 1 holds for every $N$.

Step 5 (Clause 2). Let $x\in H^{-1}$. For every $N$, $u^{b}_{N}(x)=U^{b}_{N}(\mathrm{pr}_{N}x)=U_{N}(\mathrm{pr}_{N}x)+s_{N}(b)/\gamma=u_{N}(x)+s_{N}(b)/\gamma$, with $u_{N}$ as in The Galerkin Solutions of the Wick-Square Problem Converge to the Renormalised Viscosity Solution for Every Bounded Lipschitz Running Cost §bounds, and $(u_{N}(x))_{N}$ converges to $u(x)$ by The Galerkin Solutions of the Wick-Square Problem Converge to the Renormalised Viscosity Solution for Every Bounded Lipschitz Running Cost §pointwise. Let $S_{N}=\sum_{k\in\Gamma_{N}}(c_{k}-b(k))$ be the $N$th cube sum of the family $k\mapsto c_{k}-b(k)$ (Cube Sums of Families on the Integer Lattice §cube-sums); by the definition of $s_{N}(b)$, $s_{N}(b)/\gamma=\tfrac{\beta}{\gamma}S_{N}$, so

$$u^{b}_{N}(x)=u_{N}(x)+\tfrac{\beta}{\gamma}S_{N}\qquad(N\in\mathbb{N}).$$

Suppose first that the family is cube-summable, with lattice sum $\Sigma=\sum_{k\in\mathbb{Z}^{n}}(c_{k}-b(k))$; by Cube Sums of Families on the Integer Lattice §lattice-sum, $(S_{N})$ converges to $\Sigma$. By Arithmetic of Limits of Real Sequences §scalar, $(\tfrac{\beta}{\gamma}S_{N})$ converges to $\tfrac{\beta}{\gamma}\Sigma$, and by Arithmetic of Limits of Real Sequences §sums, $(u^{b}_{N}(x))$ converges to $u(x)+\tfrac{\beta}{\gamma}\Sigma$. Conversely, suppose $(u^{b}_{N}(x))$ converges to some $L$. By the difference assertion of Arithmetic of Limits of Real Sequences §scalar, the sequence $\bigl(u^{b}_{N}(x)-u_{N}(x)\bigr)=(\tfrac{\beta}{\gamma}S_{N})$ converges to $L-u(x)$, and by the same clause, multiplying by $\tfrac{\gamma}{\beta}$, $(S_{N})$ converges to $\tfrac{\gamma}{\beta}(L-u(x))$. So the family $k\mapsto c_{k}-b(k)$ is cube-summable by Cube Sums of Families on the Integer Lattice §lattice-sum, with lattice sum $\Sigma=\tfrac{\gamma}{\beta}(L-u(x))$, and hence $L=u(x)+\tfrac{\beta}{\gamma}\Sigma$. This proves clause 2.

Step 6 (Clause 3). For the zero counterterm, $s_{N}(0)=\beta\sum_{k\in\Gamma_{N}}(c_{k}-0)=\beta\sum_{k\in\Gamma_{N}}c_{k}=C_{N}$, so $U^{0}_{N}=U_{N}+C_{N}/\gamma$ is the function $U^{b}_{N}$ of clause 1 (shift) for $b=0$, and by clause 1, proved in Step 4, it is the unique viscosity solution of the bare Galerkin operator $\mathcal{F}^{0}_{N}$ (The Galerkin Hamilton-Jacobi-Bellman Equations of the Wick-Square Problem in the Coordinates of a Cube of Modes §wick) on $\mathbb{R}^{m}$ for which $U^{0}_{N}-\tilde u_{N}$ is bounded, as asserted in clause 3. By Step 5 with $b=0$, $u^{0}_{N}(x)-C_{N}/\gamma=u_{N}(x)$ for every $x\in H^{-1}$ and $N\in\mathbb{N}$. By The Galerkin Solutions of the Wick-Square Problem Converge to the Renormalised Viscosity Solution for Every Bounded Lipschitz Running Cost §pointwise, $(u^{0}_{N}(x)-C_{N}/\gamma)_{N}$ converges to $u(x)$.

Now let $2\le n$ and $x\in H^{-1}$, and suppose, to reach a contradiction, that $(u^{0}_{N}(x))_{N}$ is bounded above: there is $B\in\mathbb{R}$ with $u^{0}_{N}(x)\le B$ for every $N$. The sequence $(u_{N}(x))_{N}$ converges, so it is bounded by claim 2 of Uniqueness of Limits and Boundedness of Convergent Real Sequences: there is a positive $A\in\mathbb{R}$ with $|u_{N}(x)|\le A$ for every $N$. Hence

$$C_{N}/\gamma=u^{0}_{N}(x)-u_{N}(x)\le B+A\qquad(N\in\mathbb{N}).$$

By The Free-Field Variances of the Fourier Modes §variances, $c_{k}=\tfrac{\nu}{2\mu_{k}}=\tfrac{\nu}{2}\cdot\tfrac{1}{\mu_{k}}$, so by claim 4 (homogeneity) of Properties of a Sum over a Finite Index Set, $C_{N}=\tfrac{\beta\nu}{2}\sum_{k\in\Gamma_{N}}\tfrac{1}{\mu_{k}}$. Since $\tfrac{2\gamma}{\beta\nu}$ is positive,

$$\sum_{k\in\Gamma_{N}}\frac{1}{\mu_{k}}=\frac{2\gamma}{\beta\nu}\cdot\frac{C_{N}}{\gamma}\le\frac{2\gamma}{\beta\nu}(B+A)\qquad(N\in\mathbb{N}),$$

so the set of cube sums $\bigl\{\sum_{k\in\Gamma_{N}}\tfrac{1}{\mu_{k}}:N\in\mathbb{N}\bigr\}$ is bounded above. Here $\mu_{k}$ and $\Gamma_{N}$ are the Fourier weights and cubes of The Wick-Square Problem on the Torus: Standing Notation §modes and The Wick-Square Problem on the Torus: Standing Notation §cubes, which are those of Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums, and $1\le n$. This contradicts Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums §divergent, which applies because $2\le n$. Hence $(u^{0}_{N}(x))_{N}$ is not bounded above, and clause 3 is proved.