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

In the setting of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation, so that the reference measure is $\rho=\gamma_{c}$, let $\beta$ and $\kappa$ be positive real numbers with $c_{k}\le\kappa\,a_{k}$ for every $k\in\mathbb{N}$, and let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be the Gaussian entropy pair with temperature $\beta$, whose hypothesis holds with this $\kappa$. Let $n\in\mathbb{N}$, let $f\in C^{2}_{b}(\mathbb{R}^{n})$, 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 $\partial_{k}f$ and $\partial_{k}\partial_{k}f$ as in Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives, let $\varphi=f\circ p_{n}$, and let $U_{\varphi}$, $\nabla_{a}\varphi$ and $\nabla U_{\varphi}$ be as in Integrals of Bounded C^2 Cylindrical Functions are Noise Intrinsic Test Functions with the Noise Gradient as Gradient. Here $\mathcal{D}_{\Sigma}\subseteq\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$ and $\mathcal{D}\subseteq\mathcal{P}_{2}(X)$ by The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §penalty-domain and The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §score-domain, so that for $\mu\in\mathcal{D}$ the noise Ornstein-Uhlenbeck functional $L^{a}_{\mu}(f)$ of $\mu$ at $f$ is defined. $\langle\cdot,\cdot\rangle_{\mu}$ is the inner product of $L^{2}(\mu;X^{a})$.

1. (Test function)¶ $U_{\varphi}$ is a noise intrinsic test function on $\mathcal{D}$, and $\nabla U_{\varphi}(\mu)=\nabla_{a}\varphi$ for every $\mu\in\mathcal{D}$, by Integrals of Bounded C^2 Cylindrical Functions are Noise Intrinsic Test Functions with the Noise Gradient as Gradient §test and Integrals of Bounded C^2 Cylindrical Functions are Noise Intrinsic Test Functions with the Noise Gradient as Gradient §gradient.

2. (The score drift is the Ornstein-Uhlenbeck generator)¶ For every $\mu\in\mathcal{D}_{\Sigma}$,

$$\langle\Sigma(\mu),\nabla U_{\varphi}(\mu)\rangle_{\mu}=\beta\,L^{a}_{\mu}(f)=\beta\sum_{k=1}^{n}a_{k}\int_{X}\Bigl(\frac{x_{k}}{c_{k}}\,\partial_{k}f(p_{n}(x))-\partial_{k}\partial_{k}f(p_{n}(x))\Bigr)\mu(dx).$$

3. (The Gaussian-score-drift operator on integrals)¶ 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 $\beta$, discount $\lambda_{0}$, control cost $\theta$ and running cost $g$. Then for every $\mu\in\mathcal{D}_{\Sigma}$ and $r\in\mathbb{R}$ the pair $(\mu,\nabla U_{\varphi}(\mu))$ belongs to the bundle $\mathcal{V}^{a}(\mathcal{D}_{\Sigma})$ of noise fields over $\mathcal{D}_{\Sigma}$, on which $F$ is defined, and

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