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

In the settings of The Wick-Square Problem on the Sobolev Triple of Order Two: Standing Notation, The Galerkin Problems of the Wick-Square Problem: Standing Notation for Cubes of Modes, Coordinates, the Free Galerkin Quadratic and the Riccati Potential and The Real Numbers: Standing Notation and Background, let $N\in\mathbb{N}$, with $m$, $\kappa_{N}$, the coordinate map $\iota_{N}$, the free Galerkin quadratic $\tilde{u}_{N}$ and the Riccati potential $\Phi_{N}$ as fixed there. We also work in the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation with its dimension $d$ taken to be $m$; the letters $k$, $m$, $q_{k}$ and $\kappa_{N}$ keep the meanings of The Galerkin Problems of the Wick-Square Problem: Standing Notation for Cubes of Modes, Coordinates, the Free Galerkin Quadratic and the Riccati Potential (not those of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §dimensions), $\gamma$ is the discount rate of The Wick-Square Problem on the Torus: Standing Notation §parameters, a Gaussian measure always carries a subscript, and probability measures on $\mathbb{R}^{m}$ are written $\rho$. Let $c^{N}=(c^{N}_{1},\dots,c^{N}_{m})\in\mathbb{R}^{m}$ with $c^{N}_{i}=c_{\kappa_{N}(i)}$, the free-field variance of the mode $\kappa_{N}(i)$, which is positive, so that $c^{N}$ is a variance vector; let $\beta^{N}=(\beta,\dots,\beta)\in\mathbb{R}^{m}$. The data at cutoff $N$ are the variance vector $c^{N}$, the temperature $\tfrac{\nu}{2}$, the discount $\gamma$, the control cost $1$, the couplings $\beta^{N}$ and the zero running cost on $\mathcal{P}_{2}(\mathbb{R}^{m})$; here $\tfrac{\nu}{2}$ and $\gamma$ are positive by The Wick-Square Problem on the Torus: Standing Notation §parameters. By clause 1 below, Gaussian Dressing of the Wick-Square Cost: the Riccati Coefficients, the Dressed Variances, the Quadratic Profile, the Shifted Pair and the Shifted Operator applies to these data, and we let $b_{1},\dots,b_{m}$ be its Riccati coefficients, $c'$ its dressed variances, $\Psi_{N}$ its quadratic profile (written $\Phi_{0}$ there) and $s_{N}$ the real number written $e$ in Gaussian Dressing of the Wick-Square Cost: the Riccati Coefficients, the Dressed Variances, the Quadratic Profile, the Shifted Pair and the Shifted Operator §operator, all for the data at cutoff $N$.

1. (The coupling condition)¶ For every $i\in[m]$, $\tfrac{\nu}{2}\,(c^{N}_{i})^{-1}=\mu_{\kappa_{N}(i)}$ and

$$\mu_{\kappa_{N}(i)}^{2}+\gamma\,\mu_{\kappa_{N}(i)}+2\beta>0 ,$$

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 for the data at cutoff $N$.

2. (Relation to the Riccati coefficients $q_{k}$)¶ $b_{i}=2q_{\kappa_{N}(i)}$ for every $i\in[m]$.

3. (The Riccati potential)¶ For every $z\in\mathbb{R}^{m}$, $\Phi_{N}(z)=\tfrac{\nu}{4}|z|_{c'}^{2}$, with the weighted square of The Diagonal Gaussian Density on Euclidean Space and Its Notation §scaling.

4. (The profile)¶ For every $\rho\in\mathcal{P}_{2}(\mathbb{R}^{m})$ the function $\tilde{u}_{N}$ is integrable with respect to $\rho$ and

$$\int_{\mathbb{R}^{m}}\tilde{u}_{N}\,d\rho=\Psi_{N}(\rho)+\gamma^{-1}s_{N}.$$

5. (The running cost)¶ $|\iota_{N}z|_{H}\le\lVert z\rVert$ for every $z\in\mathbb{R}^{m}$. Consequently, if $C_{g},\ell_{g}\in\mathbb{R}$ are nonnegative and $g:H^{-1}\to\mathbb{R}$ satisfies $|g(x)|\le C_{g}$ and $|g(x)-g(y)|\le\ell_{g}\,|x-y|_{H}$ for all $x,y\in H^{-1}$, then $|g(\iota_{N}z)|\le C_{g}$ and $|g(\iota_{N}z)-g(\iota_{N}z')|\le\ell_{g}\lVert z-z'\rVert$ for all $z,z'\in\mathbb{R}^{m}$; in particular the function $g\circ\iota_{N}:\mathbb{R}^{m}\to\mathbb{R}$ is bounded and uniformly continuous for the Euclidean distance and the metric of The Absolute Value Metric on the Real Line.