At Each Cutoff the Lifted Wick-Square Solution Relative to the Free-Field Gaussian Is the Integral of the Galerkin Solution, and Its Pointwise Recovery Converges to the Renormalised Solution

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 $C_{g},\ell_{g}\in\mathbb{R}$ be nonnegative and let $g:H^{-1}\to\mathbb{R}$ satisfy

$$|g(x)|\le C_{g}\quad\text{and}\quad|g(x)-g(y)|\le\ell_{g}\,|x-y|_{H}\qquad\text{for all }x,y\in H^{-1}.$$

Let $u:H^{-1}\to\mathbb{R}$ be the renormalised viscosity solution for the running cost $g$ with $u-u_{0}$ bounded and Lipschitz, as in The Galerkin Solutions of the Wick-Square Problem Converge to the Renormalised Viscosity Solution for Every Bounded Lipschitz Running Cost, and for $N\in\mathbb{N}$ let $U_{N}:\mathbb{R}^{m}\to\mathbb{R}$ be the Galerkin solution at cutoff $N$ for $g$, with $m$, $\iota_{N}$ and $\mathrm{pr}_{N}$ 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. For each $N\in\mathbb{N}$ the following data are fixed. Letters and data are read as in 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 for this $N$: we work in the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation with $d=m$, probability measures on $\mathbb{R}^{m}$ are written $\rho$, and $c^{N}$, $\beta^{N}$, the dressed variances $c'$, the quadratic profile $\Psi_{N}$ and the real number $s_{N}$ are those of that lemma. Let $(\mathcal{D}_{N},\mathcal{D}_{N,\Sigma},\mathcal{E}_{N},\Sigma_{N})$ be the Gaussian free-energy pair with variances $c^{N}$ and temperature $\tfrac{\nu}{2}$. By 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 §cost and Running Costs for N-Particle Systems: Integrals of Bounded Uniformly Continuous Functions and Costs of the Empirical Measure §integral, the function $\mathcal{G}_{N}:\mathcal{P}_{2}(\mathbb{R}^{m})\to\mathbb{R}$, $\mathcal{G}_{N}(\rho)=\int_{\mathbb{R}^{m}}g\circ\iota_{N}\,d\rho$, is defined, bounded by $C_{g}$ and uniformly continuous from $(\mathcal{P}_{2}(\mathbb{R}^{m}),W_{2})$ to $\mathbb{R}$ with the metric of The Absolute Value Metric on the Real Line. By 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 §condition and clauses Well-Posedness of the Lifted Hamilton-Jacobi Equation with a Wick-Square Cost Relative to a Diagonal Gaussian Measure, by Gaussian Dressing §existence and Well-Posedness of the Lifted Hamilton-Jacobi Equation with a Wick-Square Cost Relative to a Diagonal Gaussian Measure, by Gaussian Dressing §uniqueness of that theorem, applied with variances $c^{N}$, temperature $\tfrac{\nu}{2}$, discount $\gamma>0$, control cost $1$ (so $0<1\le1$), couplings $\beta^{N}$ and running cost $\mathcal{G}_{N}$, there is exactly one viscosity solution $\mathcal{U}_{N}:\mathcal{D}_{N}\to\mathbb{R}$ of the lifted Wick-square Hamilton-Jacobi equation with these data relative to the profile $\Psi_{N}$ for which $\mathcal{U}_{N}-\Psi_{N}$ is bounded. For $z\in\mathbb{R}^{m}$ and positive $\varepsilon\in\mathbb{R}$, $\eta_{z,\varepsilon}$ is the translated diagonal Gaussian measure of The Lifted Ornstein-Uhlenbeck Hamilton-Jacobi Equation with a Linear Running Cost: Its Viscosity Solution Is the Integral of the Pointwise Solution, Which It Determines §recovery (with $d=m$); by that clause, applied with $d=m$, variances $c'$, temperature $\tfrac{\nu}{2}$, discount $\gamma$, control cost $1$ and running cost $g\circ\iota_{N}$, it lies in the penalty domain of the Gaussian free-energy pair with variances $c'$ and temperature $\tfrac{\nu}{2}$, which is $\mathcal{D}_{N}$ by Gaussian Dressing of the Wick-Square Cost: the Riccati Coefficients, the Dressed Variances, the Quadratic Profile, the Shifted Pair and the Shifted Operator §pairs.

1. (Identification)¶ Let $N\in\mathbb{N}$. The function $U_{N}$ is integrable with respect to every $\rho\in\mathcal{P}_{2}(\mathbb{R}^{m})$, and $\mathcal{U}_{N}(\rho)=\int_{\mathbb{R}^{m}}U_{N}\,d\rho$ for every $\rho\in\mathcal{D}_{N}$.

2. (Pointwise recovery)¶ Let $N\in\mathbb{N}$. For every $z\in\mathbb{R}^{m}$ and every sequence $(\varepsilon_{j})_{j\in\mathbb{N}}$ of positive real numbers converging to $0$, the sequence $(\mathcal{U}_{N}(\eta_{z,\varepsilon_{j}}))_{j\in\mathbb{N}}$ converges to $U_{N}(z)$.

3. (The renormalised limit)¶ Let $x\in H^{-1}$ and let $(\varepsilon_{j})_{j\in\mathbb{N}}$ be a sequence of positive real numbers converging to $0$. For every $N\in\mathbb{N}$ the sequence $(\mathcal{U}_{N}(\eta_{\mathrm{pr}_{N}x,\varepsilon_{j}}))_{j\in\mathbb{N}}$ converges to $U_{N}(\mathrm{pr}_{N}x)$, and the sequence $(U_{N}(\mathrm{pr}_{N}x))_{N\in\mathbb{N}}$ converges to $u(x)$.