Proof of The Galerkin Wick-Square Problem at a Cutoff as a Gaussian-Dressed Problem Relative to the Free-Field Gaussian: the Coupling Condition, the Riccati Coefficients, the Riccati Potential, the Profile and the Running Cost

Each result cited below is universally quantified over the data in its own statement. Fix $N$, and for $i\in[m]$ write $k_{i}=\kappa_{N}(i)$. In Gaussian Dressing of the Wick-Square Cost: the Riccati Coefficients, the Dressed Variances, the Quadratic Profile, the Shifted Pair and the Shifted Operator the temperature is written $a$; here it is $\tfrac{\nu}{2}$ throughout, and a letter $a$ with a mode as subscript is always a renormalised constant of The Riccati Coefficients of the Wick-Square Problem: Existence, the Gaussian Profile and Summable Constants §constants. That lemma is applied with $d=m$, $a=\tfrac{\nu}{2}$, $\lambda_{0}=\gamma$, $\theta=1$, $c=c^{N}$, $\beta=\beta^{N}$ and $g$ the zero running cost on $\mathcal{P}_{2}(\mathbb{R}^{m})$, that is, with the data at cutoff $N$; it applies once clause 1 is proved (Step 1). Finite sums over $[m]$ are those of Properties of Finite Sums, whose claims are cited by number.

Step 1 (clause 1). By The Free-Field Variances of the Fourier Modes §variances, $c^{N}_{i}=c_{k_{i}}=\nu/(2\mu_{k_{i}})$, so $\tfrac{\nu}{2}(c^{N}_{i})^{-1}=\tfrac{\nu}{2}\cdot\tfrac{2\mu_{k_{i}}}{\nu}=\mu_{k_{i}}$. Since $1\le\mu_{k_{i}}$ by The Wick-Square Problem on the Torus: Standing Notation §modes and $\gamma,\beta>0$ by The Wick-Square Problem on the Torus: Standing Notation §parameters, the numbers $\mu_{k_{i}}^{2}$, $\gamma\mu_{k_{i}}$ and $2\beta$ are positive, so $\mu_{k_{i}}^{2}+\gamma\mu_{k_{i}}+2\beta>0$. With the substitutions above this number is $(a/c_{i})^{2}+\lambda_{0}a/c_{i}+2\theta\beta_{i}$, which is the coupling hypothesis of Gaussian Dressing of the Wick-Square Cost: the Riccati Coefficients, the Dressed Variances, the Quadratic Profile, the Shifted Pair and the Shifted Operator.

Step 2 (clause 2). Fix $i$. By The Riccati Coefficients of the Wick-Square Problem: Existence, the Gaussian Profile and Summable Constants §root, $q_{k_{i}}>0$ and $2q_{k_{i}}^{2}+(\gamma+2\mu_{k_{i}})q_{k_{i}}=\beta$. Put $t=2q_{k_{i}}$. Then

$$\tfrac12t^{2}+\bigl(\tfrac{\gamma}{2}+\mu_{k_{i}}\bigr)t=2q_{k_{i}}^{2}+(\gamma+2\mu_{k_{i}})q_{k_{i}}=\beta=\beta^{N}_{i},$$

and $\mu_{k_{i}}+t>0$ because $\mu_{k_{i}}\ge1$ and $q_{k_{i}}>0$. Since $a/c_{i}=\mu_{k_{i}}$ by Step 1, these are the two conditions of Gaussian Dressing of the Wick-Square Cost: the Riccati Coefficients, the Dressed Variances, the Quadratic Profile, the Shifted Pair and the Shifted Operator §riccati with $\theta=1$ and $\lambda_{0}=\gamma$, and the uniqueness asserted there gives $b_{i}=t=2q_{k_{i}}$.

Step 3 (clause 3). By Gaussian Dressing of the Wick-Square Cost: the Riccati Coefficients, the Dressed Variances, the Quadratic Profile, the Shifted Pair and the Shifted Operator §variances and Steps 1 and 2, $c'_{i}=\tfrac{\nu}{2}(\mu_{k_{i}}+2q_{k_{i}})^{-1}$, so $1/c'_{i}=\tfrac{2}{\nu}(\mu_{k_{i}}+2q_{k_{i}})$. By The Diagonal Gaussian Density on Euclidean Space and Its Notation §scaling and the homogeneity of finite sums (claim 3 of Properties of Finite Sums, with $\lambda=\tfrac{\nu}{4}$ and then $\lambda=\tfrac12$), for $z\in\mathbb{R}^{m}$,

$$\tfrac{\nu}{4}|z|_{c'}^{2}=\tfrac{\nu}{4}\sum_{i=1}^{m}\tfrac{2}{\nu}(\mu_{k_{i}}+2q_{k_{i}})z_{i}^{2}=\tfrac12\sum_{i=1}^{m}(\mu_{k_{i}}+2q_{k_{i}})z_{i}^{2},$$

which is $\Phi_{N}(z)$ 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.

Step 4 (clause 4). Let $u_{0}(z)=\tfrac12\sum_{i=1}^{m}b_{i}z_{i}^{2}$ be the function of Gaussian Dressing of the Wick-Square Cost: the Riccati Coefficients, the Dressed Variances, the Quadratic Profile, the Shifted Pair and the Shifted Operator §profile, so that $\Psi_{N}(\rho)=\int_{\mathbb{R}^{m}}u_{0}\,d\rho$. By Step 2 and claim 3 of Properties of Finite Sums, $u_{0}(z)=\sum_{i=1}^{m}q_{k_{i}}z_{i}^{2}$, so 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 the additivity of finite sums (claim 2 of Properties of Finite Sums), $\tilde{u}_{N}(z)=u_{0}(z)+A$ for every $z$, where $A=\sum_{i=1}^{m}a_{k_{i}}$. By the same clause of the dressing lemma, $u_{0}$ is of class $C^{2}$ on $\mathbb{R}^{m}$ with each $\partial_{j}\partial_{i}u_{0}$ equal to $b_{i}$ or to $0$. Every $b_{i}=2q_{k_{i}}$ is positive (Step 2, with $q_{k_{i}}>0$ by The Riccati Coefficients of the Wick-Square Problem: Existence, the Gaussian Profile and Summable Constants §root), so $M=\sum_{i=1}^{m}b_{i}$ satisfies $0\le M$ and $b_{i}\le M$ for every $i\in[m]$, by claims 5 and 6 of Properties of Finite Sums; hence each $\partial_{j}\partial_{i}u_{0}$ is bounded in absolute value by $M$, and so $u_{0}$ is Borel and integrable with respect to every $\rho\in\mathcal{P}_{2}(\mathbb{R}^{m})$ by Integrals of Functions with a Bounded Hessian are Intrinsic Test Functions on the Wasserstein Space §integrable. The constant function $1$ on $\mathbb{R}^{m}$ is the indicator of $\mathbb{R}^{m}$; it is bounded, and Borel (the indicator of the Borel set $\mathbb{R}^{m}$), hence integrable with respect to $\rho$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures, and its integral is $\rho(\mathbb{R}^{m})=1$ by The Integral of an Indicator Function is the Measure of the Set. By Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §integrable and Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §linear, applied with its $m$ equal to $2$ to $u_{0}$ and $1$ with coefficients $1$ and $A$, $\tilde{u}_{N}$ is integrable with respect to $\rho$ and $\int_{\mathbb{R}^{m}}\tilde{u}_{N}\,d\rho=\Psi_{N}(\rho)+A$. Finally, by Gaussian Dressing of the Wick-Square Cost: the Riccati Coefficients, the Dressed Variances, the Quadratic Profile, the Shifted Pair and the Shifted Operator §operator and Step 2,

$$s_{N}=\sum_{i=1}^{m}\bigl(\tfrac{\nu}{2}\cdot2q_{k_{i}}-\beta c_{k_{i}}\bigr)=\sum_{i=1}^{m}\bigl(\nu q_{k_{i}}-\beta c_{k_{i}}\bigr)=\sum_{i=1}^{m}\gamma\,a_{k_{i}}=\gamma A,$$

the third equality because $a_{k}=(\nu q_{k}-\beta c_{k})/\gamma$ for every mode $k$ by The Riccati Coefficients of the Wick-Square Problem: Existence, the Gaussian Profile and Summable Constants §constants, and the last by claim 3 of Properties of Finite Sums. Hence $A=\gamma^{-1}s_{N}$.

Step 5 (clause 5). By The Wick-Square Problem on the Sobolev Triple of Order Two: Standing Notation §triple, $H=H^{-3}(\mathbb{T}^{n})$ with the norm $|\cdot|_{H^{-3}}$ (the triple of The Negative-Order Sobolev Triple of the Torus: a Diagonal Hilbert Triple whose Form Operator Is One Minus the Laplacian §triple with its $s$ equal to $2$). Fix $z\in\mathbb{R}^{m}$ and let $x=\iota_{N}z$; 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, $x\in H^{-1}$, $x(k_{i})=z_{i}$ for $i\in[m]$, and $x$ vanishes outside $\Gamma_{N}$. So the family $k\mapsto x(k)^{2}/\mu_{k}^{3}$ vanishes outside the nonempty finite set $\Gamma_{N}$ (The Wick-Square Problem on the Torus: Standing Notation §cubes); by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §finite-support it is cube-summable with lattice sum $\sum_{k\in\Gamma_{N}}x(k)^{2}/\mu_{k}^{3}$, which equals $\sum_{i=1}^{m}x(k_{i})^{2}/\mu_{k_{i}}^{3}=\sum_{i=1}^{m}z_{i}^{2}/\mu_{k_{i}}^{3}$ by Sum over a Finite Index Set, applied with the bijection $\kappa_{N}:[m]\to\Gamma_{N}$ 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. Hence Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums §sobolev, used with its $m$ equal to $3$, gives $|\iota_{N}z|_{H}^{2}=\sum_{i=1}^{m}z_{i}^{2}/\mu_{k_{i}}^{3}$. Since $\mu_{k_{i}}\ge1$ we have $\mu_{k_{i}}^{3}\ge1$ and $z_{i}^{2}/\mu_{k_{i}}^{3}\le z_{i}^{2}$, so every number $z_{i}^{2}-z_{i}^{2}/\mu_{k_{i}}^{3}$ is nonnegative. By claims 2 and 3 of Properties of Finite Sums (the latter with $\lambda=-1$), $\sum_{i=1}^{m}z_{i}^{2}-\sum_{i=1}^{m}z_{i}^{2}/\mu_{k_{i}}^{3}=\sum_{i=1}^{m}\bigl(z_{i}^{2}-z_{i}^{2}/\mu_{k_{i}}^{3}\bigr)$, which is nonnegative by claim 5 there. Hence $|\iota_{N}z|_{H}^{2}\le\sum_{i=1}^{m}z_{i}^{2}=\lVert z\rVert^{2}$ by Euclidean Norm on $\mathbb{R}^n$; both numbers being nonnegative, $|\iota_{N}z|_{H}\le\lVert z\rVert$.

Now let $C_{g}$, $\ell_{g}$ and $g$ be as in clause 5, and $z,z'\in\mathbb{R}^{m}$. Since $\iota_{N}z\in H^{-1}$, $|g(\iota_{N}z)|\le C_{g}$. The families $\iota_{N}z-\iota_{N}z'$ and $\iota_{N}(z-z')$ coincide, both taking the value $z_{i}-z'_{i}$ at $k_{i}$ and $0$ outside $\Gamma_{N}$ (the operations of $\mathrm{Map}(\mathbb{Z}^{n},\mathbb{R})$ being pointwise), so by the hypothesis on $g$ and the first part,

$$|g(\iota_{N}z)-g(\iota_{N}z')|\le\ell_{g}\,|\iota_{N}(z-z')|_{H}\le\ell_{g}\lVert z-z'\rVert .$$

Thus $g\circ\iota_{N}$ is bounded with bound $C_{g}$ in the sense of Bounded Real-Valued Function on a Set; and since $\lVert z-z'\rVert$ is the Euclidean distance $d_{E}(z,z')$ of Euclidean Distance on $\mathbb{R}^n$, a metric by Euclidean Distance is a Metric on $\mathbb{R}^n$, $g\circ\iota_{N}$ is Lipschitz with constant $\ell_{g}$ in the sense of Lipschitz Map Between Metric Spaces into $\mathbb{R}$ with the metric of The Absolute Value Metric on the Real Line, hence uniformly continuous by A Lipschitz Map is Uniformly Continuous.