The Gaussian Relative Score Paired with the Gradient Field of a Diagonal Quadratic Profile

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 $\nu\in\mathcal{P}^{a}_{\rho}$ have a relative score $(\zeta_{k})_{k\in\mathbb{N}}$ with respect to $\gamma_{c}$ and finite Fisher information relative to $\gamma_{c}$ with weights $a$, these notions being applicable since $\nu\in\mathcal{P}_{2}(X)$ by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §moments. Let $Z^{a}_{\nu}\in L^{2}(\nu;X^{a})$ be its noise score field, let $\langle\cdot,\cdot\rangle_{\nu}$ be the inner product of $L^{2}(\nu;X^{a})$, and let $\langle\cdot,\cdot\rangle_{L^{2}(\nu)}$ be that of $L^{2}(\nu)$ as in The Relative Score with Respect to a Diagonal Gaussian Measure on a Hilbert Space. Admissible sequences $b$ and the gradient fields $V_{b}(\nu)\in L^{2}(\nu;X^{a})$ are those of Diagonal Quadratic Profiles on the Noise Wasserstein Space: Moment Bounds, a Lipschitz Estimate, the Gradient Field, and Noise Intrinsic Test Functions §field.

1. (Coordinates)¶ For every $k\in\mathbb{N}$ the coordinate function $x\mapsto x_{k}$ is square-integrable with respect to $\nu$, its class again being written $x_{k}$, and

$$\langle\zeta_{k},x_{k}\rangle_{L^{2}(\nu)}=\frac{1}{c_{k}}\int_{X}x_{k}^{2}\,\nu(dx)-1 .$$

2. (Pairing)¶ Let $b$ be admissible. The series

$$\sum_{k=1}^{\infty}a_{k}\,b_{k}\Bigl(\frac{1}{c_{k}}\int_{X}x_{k}^{2}\,\nu(dx)-1\Bigr)$$

converges, and its sum is $\langle Z^{a}_{\nu},V_{b}(\nu)\rangle_{\nu}$.