Proof of On Functions of Finitely Many Fourier Modes the Free-Field Score Drift is Minus the Ornstein-Uhlenbeck Generator with Drift the Laplacian Minus One

Each result cited is universally quantified over the data in its own statement.

Preliminaries. By The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation §gaussian the setting of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation is in force with the space $X$, the basis $(e_{j})_{j\in\mathbb{N}}$, the weights $a$ and the variances $c$ of The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation §data, and items stated there with a natural number written $n$ are read with $N$ in its place. In particular $x_{k}=\langle x,e_{k}\rangle_{H^{-m}}$ and $p_{N}(x)=(x_{1},\dots,x_{N})$ are the coordinates and the coordinate map of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates. Since $c_{k}\le 1\cdot a_{k}$ for every $k\in\mathbb{N}$ by White Noise in the Square-Integrable Space and the Free Field on the Torus as Noise Weights and a Variance Sequence on a Negative Sobolev Space §ratio, the hypotheses of On Integrals of Cylindrical Functions the Gaussian Score Drift is the Noise Ornstein-Uhlenbeck Functional, and the Score-Drift Hamilton-Jacobi Operator Takes Its Pointwise Form hold with temperature $1$ and its constant equal to $1$, and the Gaussian entropy pair there is the pair $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ of the statement.

For $i\in[N]$ write $s_{i}=\rho_{\kappa(i)}^{m}$. The numbers $\rho_{k}$ are positive, as recorded in Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families (whose numbers $\rho_{k}$ are those of The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation §torus), so $s_{i}$ is positive and has a multiplicative inverse $s_{i}^{-1}$. By White Noise in the Square-Integrable Space and the Free Field on the Torus as Noise Weights and a Variance Sequence on a Negative Sobolev Space §basis, for every $\Phi\in X$ and $i\in[N]$,

$$\Phi_{i}=s_{i}\,\Phi(\kappa(i)). \tag{1}$$

By the definition of $a_{i}$ and $c_{i}$ in White Noise in the Square-Integrable Space and the Free Field on the Torus as Noise Weights and a Variance Sequence on a Negative Sobolev Space, $a_{i}=s_{i}^{2}$ and $c_{i}=(\rho_{\kappa(i)}^{m+1})^{2}$ is positive, so the identity $a_{i}=\mu_{\kappa(i)}c_{i}$ gives

$$a_{i}=s_{i}^{2},\qquad a_{i}\,c_{i}^{-1}=\mu_{\kappa(i)}\qquad(i\in[N]). \tag{2}$$

Step 1: calculus for $f$. Let $A:\mathbb{R}^{N}\to\mathbb{R}^{N}$ be the map $A(x)=(s_{1}^{-1}x_{1},\dots,s_{N}^{-1}x_{N})$, so that $f=h\circ A$. For $l\in[N]$ the coordinate function $A_{l}(x)=s_{l}^{-1}x_{l}$ is the function $Q$ of Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity §quadratic with the zero matrix $0_{N}$ for $M$, for $q$ the point of $\mathbb{R}^{N}$ whose $l$th entry is $s_{l}^{-1}$ and whose other entries are $0$, and with constant $0$; hence $A_{l}$ is of class $C^{2}$ on $\mathbb{R}^{N}$, so $A$ is of class $C^{2}$ by claim 1 of Coordinate Functions, the $C^k$ Hierarchy, and Partial Derivatives of a Smooth Map, and of class $C^{1}$ by claim 2 there. For $x\in\mathbb{R}^{N}$, $i\in[N]$ and real $t\ne0$, the difference quotient of $A_{l}$ at $x$ in the $i$th variable with increment $t$ equals $s_{l}^{-1}$ if $i=l$ and $0$ otherwise, so by Partial Derivative on a Euclidean Open Set the partial derivative $\partial_{i}A_{l}(x)$ exists and equals $s_{l}^{-1}$ if $i=l$ and $0$ if $i\ne l$. The function $h$ is of class $C^{2}$, hence of class $C^{1}$ by claim 2 of Coordinate Functions, the $C^k$ Hierarchy, and Partial Derivatives of a Smooth Map, and each $\partial_{i}h$ is of class $C^{1}$ by clause 2 of C^k Maps on a Euclidean Open Set. Claim 1 of A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$, applied with $F=A$ and $G=h$, and then with $F=A$ and $G=\partial_{i}h$, gives for all $i,j\in[N]$ and $x\in\mathbb{R}^{N}$, only the term $l=i$, respectively $l=j$, of its sum being nonzero,

$$\partial_{i}f(x)=s_{i}^{-1}\,\partial_{i}h(A(x)),\qquad \partial_{j}(\partial_{i}h\circ A)(x)=s_{j}^{-1}\,\partial_{j}\partial_{i}h(A(x)). \tag{3}$$

Thus $\partial_{i}f=s_{i}^{-1}\,(\partial_{i}h\circ A)$ on $\mathbb{R}^{N}$, and claim 2 of Partial Derivatives, Continuity and $C^k$ Regularity under a Scaling Substitution, used with both open sets equal to $\mathbb{R}^{N}$, the point written $c$ there equal to the origin, the scalar written $\lambda$ there equal to $1$ (so that $c+\lambda x=x$) and the scalar written $\mu$ there equal to $s_{i}^{-1}$, together with (3), gives that $\partial_{j}\partial_{i}f(x)$ exists for every $x$ and

$$\partial_{j}\partial_{i}f(x)=s_{i}^{-1}s_{j}^{-1}\,\partial_{j}\partial_{i}h(A(x)). \tag{4}$$

The function $f$ is of class $C^{2}$ on $\mathbb{R}^{N}$ by claim 2 of A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$ with $k=2$. Since $h$, $\partial_{i}h$ and $\partial_{j}\partial_{i}h$ are bounded by Bounded Twice Continuously Differentiable Functions with Bounded First and Second Partial Derivatives on Euclidean Space §bounded, a bound $B$ for one of them yields the bound $B$, $s_{i}^{-1}B$ or $s_{i}^{-1}s_{j}^{-1}B$ for $f$, $\partial_{i}f$ or $\partial_{j}\partial_{i}f$ respectively, by (3) and (4), the factors $s_{i}^{-1}>0$ and $s_{j}^{-1}>0$ being pulled out of the absolute value; hence $f\in C^{2}_{b}(\mathbb{R}^{N})$ by Bounded Twice Continuously Differentiable Functions with Bounded First and Second Partial Derivatives on Euclidean Space §bounded. Finally, by (1), for every $\Phi\in X$ the $i$th entry of $A(p_{N}(\Phi))$ is $s_{i}^{-1}s_{i}\Phi(\kappa(i))=\Phi(\kappa(i))$, so

$$A\circ p_{N}=y_{N}. \tag{5}$$

Claim 1. By (5), $\varphi=h\circ y_{N}=h\circ A\circ p_{N}=f\circ p_{N}$, with $f\in C^{2}_{b}(\mathbb{R}^{N})$ by Step 1. Hence On Integrals of Cylindrical Functions the Gaussian Score Drift is the Noise Ornstein-Uhlenbeck Functional, and the Score-Drift Hamilton-Jacobi Operator Takes Its Pointwise Form §test, applied with temperature $1$, its constant equal to $1$, $N$ in place of $n$ and this $f$, shows that $U_{\varphi}$ is a noise intrinsic test function on $\mathcal{D}$ with $\nabla U_{\varphi}(\nu)=\nabla_{a}\varphi$ for every $\nu\in\mathcal{D}$.

Claim 2. Let $\nu\in\mathcal{D}_{\Sigma}$. Then $\nu\in\mathcal{D}\subseteq\mathcal{P}_{2}(X)$ by The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §score-domain and The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §penalty-domain. For $i\in[N]$ let $\psi_{i}:X\to\mathbb{R}$ be $\psi_{i}(x)=c_{i}^{-1}x_{i}\,\partial_{i}f(p_{N}(x))-\partial_{i}\partial_{i}f(p_{N}(x))$; it is integrable with respect to $\nu$, as recorded in The Noise Ornstein-Uhlenbeck Functional of a Probability Measure at a Bounded C^2 Function of Finitely Many Coordinates §functional applied to $\nu$ and $f$, and $L^{a}_{\nu}(f)=\sum_{i=1}^{N}a_{i}\int_{X}\psi_{i}\,d\nu$ by that clause. For $\Phi\in X$, by (1), (2), (3), (4) with $j=i$, and (5),

$$a_{i}\psi_{i}(\Phi)=a_{i}c_{i}^{-1}\,s_{i}\Phi(\kappa(i))\,s_{i}^{-1}\partial_{i}h(y_{N}(\Phi))-s_{i}^{2}s_{i}^{-2}\,\partial_{i}\partial_{i}h(y_{N}(\Phi))=\mu_{\kappa(i)}\Phi(\kappa(i))\,\partial_{i}h(y_{N}(\Phi))-\partial_{i}\partial_{i}h(y_{N}(\Phi)),$$

so $\mathcal{L}h=\sum_{i=1}^{N}(-a_{i})\psi_{i}$ pointwise on $X$. By Linearity and Monotonicity of the Lebesgue Integral §integrable, applied to the $N$ summands one at a time (induction on the number of summands), $\mathcal{L}h$ is integrable with respect to $\nu$ and $\int_{X}\mathcal{L}h\,d\nu=-\sum_{i=1}^{N}a_{i}\int_{X}\psi_{i}\,d\nu=-L^{a}_{\nu}(f)$. By On Integrals of Cylindrical Functions the Gaussian Score Drift is the Noise Ornstein-Uhlenbeck Functional, and the Score-Drift Hamilton-Jacobi Operator Takes Its Pointwise Form §drift with temperature $1$, applied to $\nu$,

$$\langle\Sigma(\nu),\nabla U_{\varphi}(\nu)\rangle_{\nu}=1\cdot L^{a}_{\nu}(f)=-\int_{X}\mathcal{L}h\,d\nu .$$

Claim 3. Let $\nu\in\mathcal{D}_{\Sigma}$ and $r\in\mathbb{R}$. By On Integrals of Cylindrical Functions the Gaussian Score Drift is the Noise Ornstein-Uhlenbeck Functional, and the Score-Drift Hamilton-Jacobi Operator Takes Its Pointwise Form §operator with temperature $1$, applied to $\nu$,

$$F\bigl(\nu,r,\nabla U_{\varphi}(\nu)\bigr)=\lambda_{0}\,r+\frac{\theta}{2}\sum_{i=1}^{N}a_{i}\int_{X}\partial_{i}f(p_{N}(x))^{2}\,\nu(dx)+1\cdot L^{a}_{\nu}(f)-g(\nu).$$

For $i\in[N]$ the function $(\partial_{i}f)\circ p_{N}$ is a bounded $C^{1}$ cylindrical function, as recorded in The Noise Ornstein-Uhlenbeck Functional of a Probability Measure at a Bounded C^2 Function of Finitely Many Coordinates for $f$, so by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §bounded-borel it is Borel and bounded, with a bound $B$ say. Its square is then Borel: the map $t\mapsto t^{2}$ of $\mathbb{R}$ is continuous by claim 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space, applied with both functions equal to the identity map of $\mathbb{R}$ (which is continuous, with $\delta=\varepsilon$), hence Borel by claims 3 and 2 of Borel Measurability and Bounded Integration on a Metric Space, and the composite is Borel by claim 4 there. It is also bounded, with bound $B^{2}$, since $|t|\le B$ gives $t^{2}=|t|\,|t|\le B^{2}$. Hence the square is integrable with respect to the probability measure $\nu$, a Borel measure on $X$ with $\nu(X)=1$, by claim 6(b) of Borel Measurability and Bounded Integration on a Metric Space. By (3), (5) and (2), $a_{i}\,\partial_{i}f(p_{N}(\Phi))^{2}=s_{i}^{2}s_{i}^{-2}\,\partial_{i}h(y_{N}(\Phi))^{2}=\partial_{i}h(y_{N}(\Phi))^{2}$ for every $\Phi\in X$, so Linearity and Monotonicity of the Lebesgue Integral §integrable gives $a_{i}\int_{X}\partial_{i}f(p_{N}(x))^{2}\,\nu(dx)=\int_{X}\partial_{i}h(y_{N}(\Phi))^{2}\,\nu(d\Phi)$. Substituting this and $L^{a}_{\nu}(f)=-\int_{X}\mathcal{L}h\,d\nu$, obtained in the proof of claim 2, yields the asserted formula.

Claim 4. Let $u\in C^{2}_{\mathrm{per}}$. By The Laplacian of the Trigonometric System, and the Fourier Coefficients of a Laplacian on the Torus §coefficients, $u$ and $\Delta u$ belong to $C_{\mathrm{per}}$, and for every $k\in\mathbb{Z}^{n}$

$$\bigl\langle[\,(\Delta u)|_{Q}\,],E_{k}\bigr\rangle_{L^{2}}=-4\pi^{2}\lVert k\rVert^{2}\bigl\langle[\,u|_{Q}\,],E_{k}\bigr\rangle_{L^{2}} .$$

The map $\Delta u-u$ has value $1\cdot\Delta u(x)+(-1)\cdot u(x)$ at $x\in\mathbb{R}^{n}$, so Finite Linear Combinations of Continuous Periodic Functions and Their Classes on the Torus §class, used with two summands, coefficients $1$ and $-1$ and the functions $\Delta u$ and $u$, gives $V=[\,(\Delta u)|_{Q}\,]-U$ in $L^{2}(\mathbb{T}^{n})$. By Properties of the Fourier Coefficients on the Torus, and the Realisation of Weighted Coefficient Families §linear and The Fourier Coefficients of a Square-Integrable Class on the Torus §coefficients, for every $k\in\mathbb{Z}^{n}$,

$$\hat{V}(k)=\bigl\langle[\,(\Delta u)|_{Q}\,],E_{k}\bigr\rangle_{L^{2}}-\hat{U}(k)=-4\pi^{2}\lVert k\rVert^{2}\,\hat{U}(k)-\hat{U}(k)=-\bigl(1+4\pi^{2}\lVert k\rVert^{2}\bigr)\hat{U}(k)=-\mu_{k}\,\hat{U}(k),$$

the last step by the definition of the Fourier weights $\mu_{k}$ in The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation §torus.