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

In the setting of The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Gaussian entropy pair with temperature $1$, whose hypothesis holds with the constant $1$ 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 §ratio. Measures on $X$ are written $\nu$, and $\mu_{k}$ with a subscript is always a Fourier weight; $\langle\cdot,\cdot\rangle_{\nu}$ is the inner product of $L^{2}(\nu;X^{a})$. Let $N\in\mathbb{N}$, let $y_{N}:X\to\mathbb{R}^{N}$ be the map $y_{N}(\Phi)=\bigl(\Phi(\kappa(1)),\dots,\Phi(\kappa(N))\bigr)$, let $h\in C^{2}_{b}(\mathbb{R}^{N})$, the set of Bounded Twice Continuously Differentiable Functions with Bounded First and Second Partial Derivatives on Euclidean Space §bounded with $q=N$, with partial derivatives $\partial_{i}h$ and $\partial_{i}\partial_{i}h$ as in Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives, and let $\varphi=h\circ y_{N}:X\to\mathbb{R}$. Let $\mathcal{L}h:X\to\mathbb{R}$ be the function

$$(\mathcal{L}h)(\Phi)=\sum_{i=1}^{N}\Bigl(\partial_{i}\partial_{i}h\bigl(y_{N}(\Phi)\bigr)-\mu_{\kappa(i)}\,\Phi(\kappa(i))\,\partial_{i}h\bigl(y_{N}(\Phi)\bigr)\Bigr).$$

1. (Functions of finitely many modes)¶ $\varphi=f\circ p_{N}$ for the function $f\in C^{2}_{b}(\mathbb{R}^{N})$ given by $f(x)=h\bigl(x_{1}/\rho_{\kappa(1)}^{m},\dots,x_{N}/\rho_{\kappa(N)}^{m}\bigr)$. Consequently, with $U_{\varphi}$ and $\nabla_{a}\varphi$ as in Integrals of Bounded C^2 Cylindrical Functions are Noise Intrinsic Test Functions with the Noise Gradient as Gradient, applied with $N$ in place of the natural number written $n$ there, $U_{\varphi}$ is a noise intrinsic test function on $\mathcal{D}$ with $\nabla U_{\varphi}(\nu)=\nabla_{a}\varphi$ for every $\nu\in\mathcal{D}$, 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 §test.

2. (The drift is minus the Ornstein-Uhlenbeck generator)¶ For every $\nu\in\mathcal{D}_{\Sigma}$ the function $\mathcal{L}h$ is integrable with respect to $\nu$, and

$$\langle\Sigma(\nu),\nabla U_{\varphi}(\nu)\rangle_{\nu}=-\int_{X}\mathcal{L}h\,d\nu .$$

3. (The operator on functions of finitely many modes)¶ Let $\lambda_{0},\theta\in\mathbb{R}$ be positive, let $g:\mathcal{D}\to\mathbb{R}$, and let $F$ be the Hamilton-Jacobi operator with Gaussian score drift relative to $\gamma_{c}$ with temperature $1$, discount $\lambda_{0}$, control cost $\theta$ and running cost $g$. Then for every $\nu\in\mathcal{D}_{\Sigma}$ and $r\in\mathbb{R}$,

$$F\bigl(\nu,r,\nabla U_{\varphi}(\nu)\bigr)=\lambda_{0}\,r+\frac{\theta}{2}\sum_{i=1}^{N}\int_{X}\partial_{i}h\bigl(y_{N}(\Phi)\bigr)^{2}\,\nu(d\Phi)-\int_{X}\mathcal{L}h\,d\nu-g(\nu).$$

4. (The drift coefficients are those of the Laplacian minus one)¶ Let $u:\mathbb{R}^{n}\to\mathbb{R}$ belong to $C^{2}_{\mathrm{per}}$, and let $U=[\,u|_{Q}\,]$ and $V=[\,(\Delta u-u)|_{Q}\,]$ in $L^{2}(\mathbb{T}^{n})$. Then $\hat{V}(k)=-\mu_{k}\,\hat{U}(k)$ for every $k\in\mathbb{Z}^{n}$.