Proof of 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

Each result cited below is universally quantified over the data in its own statement. Fix $N\in\mathbb{N}$. Probability measures on $\mathbb{R}^{m}$ are written $\rho$ or $\eta$; the temperature is $\tfrac{\nu}{2}$ throughout, and a letter $a$ with a mode as subscript is a renormalised constant, not a temperature. We use 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 (the representation theorem) with $d=m$, variances $c'$, temperature $\tfrac{\nu}{2}$, $\lambda_{0}=\gamma$ and $\theta=1$; its pair is the Gaussian free-energy pair with variances $c'$ and temperature $\tfrac{\nu}{2}$, whose penalty domain 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, that lemma applying 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. For a bounded uniformly continuous $\tilde{g}:\mathbb{R}^{m}\to\mathbb{R}$ write $F_{\tilde{g}}$ for its operator with running cost $\tilde{g}$; by The Penalty-Drift Hamilton-Jacobi Equation on an Open Subset of Euclidean Space §operator, with noise intensity $2\cdot\tfrac{\nu}{2}=\nu$,

$$F_{\tilde{g}}(z,r,p,X)=\gamma r+\tfrac12\lVert p\rVert^{2}+DV_{c'}(z)\cdot p-\tfrac{\nu}{2}\operatorname{tr}(X)-\tilde{g}(z).$$

Step 1 (the shifted Galerkin solution). Let $w=U_{N}-\tilde{u}_{N}$. 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 subsolution and supersolution of $\mathcal{F}_{N}$ on $\mathbb{R}^{m}$ and $w$ is bounded. By 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 §solutions (its hypotheses hold as $C_{g}\ge0$ and $|g|\le C_{g}$), $w$ is a viscosity subsolution and supersolution of the shifted Galerkin operator $\mathcal{F}^{\sharp}_{N}$ of The Galerkin Hamilton-Jacobi-Bellman Equations of the Wick-Square Problem in the Coordinates of a Cube of Modes §shifted.

Step 2 ($w$ is the pointwise solution). 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 §potential, $V_{c'}(z)=\tfrac{\nu}{4}|z|_{c'}^{2}=\Phi_{N}(z)$ for every $z$, so $DV_{c'}=D\Phi_{N}$, and comparing the display above with the formula of The Galerkin Hamilton-Jacobi-Bellman Equations of the Wick-Square Problem in the Coordinates of a Cube of Modes §shifted (the two versions of the penalty-drift equation item, The Penalty-Drift Hamilton-Jacobi Equation on an Open Subset of Euclidean Space §operator and The Penalty-Drift Hamilton-Jacobi Equation on an Open Subset of Euclidean Space §operator, give the same formula) gives $F_{g\circ\iota_{N}}=\mathcal{F}^{\sharp}_{N}$; for both operators viscosity sub- and supersolutions on $\mathbb{R}^{m}$ are those of Viscosity Subsolution and Supersolution of a Second-Order Equation (The Penalty-Drift Hamilton-Jacobi Equation on an Open Subset of Euclidean Space §equation, The Galerkin Hamilton-Jacobi-Bellman Equations of the Wick-Square Problem in the Coordinates of a Cube of Modes §equation). 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, $g\circ\iota_{N}$ is bounded by $C_{g}$ and uniformly continuous. Hence, by Step 1 and the uniqueness in 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 §pointwise, $w$ is the pointwise solution of the representation theorem for the running cost $g\circ\iota_{N}$; in particular $w$ is Borel and integrable with respect to every $\rho\in\mathcal{P}_{2}(\mathbb{R}^{m})$.

Step 3 (adding a constant). The constant function $1$ on $\mathbb{R}^{m}$ is the indicator of $\mathbb{R}^{m}$, bounded and Borel, hence integrable with respect to every $\rho\in\mathcal{P}_{2}(\mathbb{R}^{m})$ (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures) with integral $\rho(\mathbb{R}^{m})=1$ (The Integral of an Indicator Function is the Measure of the Set). Let $\tilde{g}'=g\circ\iota_{N}+s_{N}$. It is bounded by $C_{g}+|s_{N}|$, and $\tilde{g}'(z)-\tilde{g}'(z')=g(\iota_{N}z)-g(\iota_{N}z')$, so 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 it is Lipschitz with constant $\ell_{g}$ for the Euclidean distance, hence uniformly continuous by A Lipschitz Map is Uniformly Continuous. By Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §linear (with the functions $g\circ\iota_{N}$ and $1$), the cost of the representation theorem for $\tilde{g}'$ is $\rho\mapsto\int_{\mathbb{R}^{m}}\tilde{g}'\,d\rho=\mathcal{G}_{N}(\rho)+s_{N}$.

Let $w'=w+\gamma^{-1}s_{N}$, a bounded function. For all $(z,r,p,X)$, $F_{\tilde{g}'}(z,r+\gamma^{-1}s_{N},p,X)=F_{g\circ\iota_{N}}(z,r,p,X)$, the terms $\gamma\cdot\gamma^{-1}s_{N}$ and $-s_{N}$ cancelling. Since $w$ is upper and lower semicontinuous (Step 1 and Viscosity Subsolution and Supersolution of a Second-Order Equation), so is $w'$, the inequalities of Upper Semicontinuous Function on a Subset of a Metric Space and Lower Semicontinuous Function on a Subset of a Metric Space being unchanged when one constant is added to both sides. If $\varphi$ is of class $C^{2}$ on $\mathbb{R}^{m}$ and $w'-\varphi$ has a local maximum at $z$, then so has $w-\varphi=(w'-\varphi)-\gamma^{-1}s_{N}$ (Local Maximum of a Function Relative to a Subset of a Metric Space), so by Steps 1 and 2

$$F_{\tilde{g}'}\bigl(z,w'(z),D\varphi(z),D^{2}\varphi(z)\bigr)=\mathcal{F}^{\sharp}_{N}\bigl(z,w(z),D\varphi(z),D^{2}\varphi(z)\bigr)\le0 ;$$

likewise, if $w'-\varphi$ has a local minimum at $z$, then so has $w-\varphi=(w'-\varphi)-\gamma^{-1}s_{N}$ (Local Minimum of a Function Relative to a Subset of a Metric Space), and Steps 1 and 2 give $F_{\tilde{g}'}\bigl(z,w'(z),D\varphi(z),D^{2}\varphi(z)\bigr)=\mathcal{F}^{\sharp}_{N}\bigl(z,w(z),D\varphi(z),D^{2}\varphi(z)\bigr)\ge0$. Thus $w'$ is a bounded viscosity subsolution and supersolution of $F_{\tilde{g}'}$, hence by 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 §pointwise the pointwise solution for $\tilde{g}'$. By Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §linear, $\int_{\mathbb{R}^{m}}w'\,d\rho=\int_{\mathbb{R}^{m}}w\,d\rho+\gamma^{-1}s_{N}$ for $\rho\in\mathcal{P}_{2}(\mathbb{R}^{m})$.

Step 4 (clause 1). By 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 §representation for $\tilde{g}'$, every bounded viscosity solution on $\mathcal{D}_{N}$ of the lifted Ornstein-Uhlenbeck equation with variances $c'$, temperature $\tfrac{\nu}{2}$, discount $\gamma$, control cost $1$ and running cost $\rho\mapsto\mathcal{G}_{N}(\rho)+s_{N}$ equals $\rho\mapsto\int_{\mathbb{R}^{m}}w'\,d\rho$. By Well-Posedness of the Lifted Hamilton-Jacobi Equation with a Wick-Square Cost Relative to a Diagonal Gaussian Measure, by Gaussian Dressing §representation, applied with the data of the statement (its $c'$, $\Phi_{0}$ and $e$ being $c'$, $\Psi_{N}$ and $s_{N}$), $\mathcal{U}_{N}-\Psi_{N}$, which is bounded, is a viscosity solution of exactly this equation. Hence $\mathcal{U}_{N}(\rho)-\Psi_{N}(\rho)=\int_{\mathbb{R}^{m}}w'\,d\rho$ for $\rho\in\mathcal{D}_{N}$.

Now $U_{N}=\tilde{u}_{N}+w$, where $\tilde{u}_{N}$ is integrable with respect to every $\rho\in\mathcal{P}_{2}(\mathbb{R}^{m})$ 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 §profile and $w$ by Step 2; so $U_{N}$ is integrable by Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §integrable. For $\rho\in\mathcal{D}_{N}\subseteq\mathcal{P}_{2}(\mathbb{R}^{m})$ (The Gaussian Free-Energy Pair: Relative Entropy and Relative Score with Respect to a Diagonal Gaussian Measure §pair), by Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §linear, the profile clause and Step 3,

$$\int_{\mathbb{R}^{m}}U_{N}\,d\rho=\Psi_{N}(\rho)+\gamma^{-1}s_{N}+\int_{\mathbb{R}^{m}}w\,d\rho=\Psi_{N}(\rho)+\int_{\mathbb{R}^{m}}w'\,d\rho=\mathcal{U}_{N}(\rho).$$

Step 5 (clause 2). Fix $z$ and $(\varepsilon_{j})$, and write $\eta_{j}=\eta_{z,\varepsilon_{j}}\in\mathcal{D}_{N}$. By clause 1 and Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §linear, $\mathcal{U}_{N}(\eta_{j})=\int_{\mathbb{R}^{m}}\tilde{u}_{N}\,d\eta_{j}+\int_{\mathbb{R}^{m}}w\,d\eta_{j}$. By Step 2 and 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 for the cost $g\circ\iota_{N}$, $\int_{\mathbb{R}^{m}}w\,d\eta_{j}\to w(z)$. Since $U_{N}(z)=\tilde{u}_{N}(z)+w(z)$, it remains to show $\int_{\mathbb{R}^{m}}\tilde{u}_{N}\,d\eta_{j}\to\tilde{u}_{N}(z)$.

Let $T:\mathbb{R}^{m}\to\mathbb{R}^{m}$ be the constant map with value $z$, Borel since each preimage is $\emptyset$ or $\mathbb{R}^{m}$; then $T_{\#}\eta_{j}(B)=\eta_{j}(T^{-1}(B))$ is $1$ or $0$ according as $z\in B$ or not, so $T_{\#}\eta_{j}=\delta_{z}$ (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward, The Dirac Measure at a Point of Euclidean Space §dirac), and $\delta_{z}\in\mathcal{P}_{2}(\mathbb{R}^{m})$ by Dirac Measures on Euclidean Space: Probability Measure, Integrals, Push-Forwards and the Coupling of Two Dirac Measures §measure. The identity map $\mathrm{id}$ of $\mathbb{R}^{m}$ is Borel and $\mathrm{id}_{\#}\eta_{j}=\eta_{j}$, since $\mathrm{id}^{-1}(B)=B$ (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward), and $T_{\#}\eta_{j}=\delta_{z}$; so Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §pushforward (with $S=\mathrm{id}$ and $T$ the constant map) shows that $(\mathrm{id},T)_{\#}\eta_{j}\in\Pi(\eta_{j},\delta_{z})$ has quadratic cost $\int_{\mathbb{R}^{m}}\lVert x-z\rVert^{2}\,\eta_{j}(dx)$, the integrand $x\mapsto\lVert x-z\rVert^{2}$ being Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions (applied to the Borel maps $\mathrm{id}$ and $T$). As $\eta_{j}$ is the image of the diagonal Gaussian measure $\gamma_{v_{j}}$ with variance vector $v_{j}=(\varepsilon_{j},\dots,\varepsilon_{j})$ under $y\mapsto z+y$, the change of variables of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward, then Diagonal Gaussian Measures on Euclidean Space §measure with claim 3 of Image Measures, Measures with Densities, and Change of Variables, and The Diagonal Gaussian Density on Euclidean Space: Regularity, Gradient, Normalization, Second Moments and Exponential Moments of Diagonal Quadratic Forms §moments give that this cost is $\int_{\mathbb{R}^{m}}\lVert y\rVert^{2}\rho_{v_{j}}(y)\,\lambda_{m}(dy)=\sum_{i=1}^{m}\varepsilon_{j}=m\varepsilon_{j}$, the last step by claim 3 of Properties of Finite Sums (with $\lambda=\varepsilon_{j}$, the sum of $m$ ones being $m$). So $W_{2}(\eta_{j},\delta_{z})^{2}\le m\varepsilon_{j}$ by The Quadratic Wasserstein Distance on Euclidean Space §distance; given a positive $t$, $m\varepsilon_{j}<t^{2}$ for all large $j$, whence $W_{2}(\eta_{j},\delta_{z})<t$; thus $W_{2}(\eta_{j},\delta_{z})\to0$.

The function $\tilde{u}_{N}$ is of class $C^{2}$ (The Galerkin Problems of the Wick-Square Problem: Standing Notation for Cubes of Modes, Coordinates, the Free Galerkin Quadratic and the Riccati Potential §functions), hence continuous by claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous and Borel (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps); and since $q_{\kappa_{N}(i)}>0$ (The Riccati Coefficients of the Wick-Square Problem: Existence, the Gaussian Profile and Summable Constants §root) and $x_{i}^{2}\le\lVert x\rVert^{2}$ (Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §coordinate, squaring nonnegative numbers), each summand of $\tilde{u}_{N}(x)$ satisfies $|q_{\kappa_{N}(i)}x_{i}^{2}+a_{\kappa_{N}(i)}|\le q_{\kappa_{N}(i)}\lVert x\rVert^{2}+|a_{\kappa_{N}(i)}|\le(q_{\kappa_{N}(i)}+|a_{\kappa_{N}(i)}|)(1+\lVert x\rVert^{2})$; by claims 2 and 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers and claim 3 of Properties of Finite Sums, $|\tilde{u}_{N}(x)|\le B(1+\lVert x\rVert^{2})$ with $B=\sum_{i=1}^{m}(q_{\kappa_{N}(i)}+|a_{\kappa_{N}(i)}|)$. By Convergence in the Wasserstein Distance Implies Weak Convergence and Convergence of Integrals of Continuous Functions of Quadratic Growth §quadratic and Dirac Measures on Euclidean Space: Probability Measure, Integrals, Push-Forwards and the Coupling of Two Dirac Measures §integral, $\int_{\mathbb{R}^{m}}\tilde{u}_{N}\,d\eta_{j}\to\int_{\mathbb{R}^{m}}\tilde{u}_{N}\,d\delta_{z}=\tilde{u}_{N}(z)$. Adding the two limits by Arithmetic of Limits of Real Sequences §sums, $\mathcal{U}_{N}(\eta_{j})\to U_{N}(z)$.

Step 6 (clause 3). Let $x\in H^{-1}$ and $N\in\mathbb{N}$. Since $\mathrm{pr}_{N}x\in\mathbb{R}^{m}$ (The Galerkin Problems of the Wick-Square Problem: Standing Notation for Cubes of Modes, Coordinates, the Free Galerkin Quadratic and the Riccati Potential §coordinates), the first assertion is clause 2 with $z=\mathrm{pr}_{N}x$. With $u_{N}(x)=U_{N}(\mathrm{pr}_{N}x)$ as in The Galerkin Solutions of the Wick-Square Problem Converge to the Renormalised Viscosity Solution for Every Bounded Lipschitz Running Cost §bounds, The Galerkin Solutions of the Wick-Square Problem Converge to the Renormalised Viscosity Solution for Every Bounded Lipschitz Running Cost §pointwise gives $U_{N}(\mathrm{pr}_{N}x)\to u(x)$ as $N\to\infty$.