The Wick-Square Corrector and the Score-Paired Wick-Square Cost Relative to a Diagonal Gaussian Measure on a Hilbert Space

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 $\beta,\kappa\in\mathbb{R}$ be positive 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 $\kappa$; thus $\Sigma(\nu)=\beta\,Z^{a}_{\nu}$ for $\nu\in\mathcal{D}_{\Sigma}$, where $Z^{a}_{\nu}$ is the noise score field. $\langle\cdot,\cdot\rangle_{\nu}$ is the inner product of $L^{2}(\nu;X^{a})$, and convergent series are those of that definition.

1. (Hypothesis on the couplings)¶ The corrector and the cost below are defined for a sequence of Wick couplings: a real sequence $w=(w_{k})_{k\in\mathbb{N}}$ such that the series $\sum_{k=1}^{\infty}|w_{k}|\,c_{k}^{2}/a_{k}$ converges and there is $W\in\mathbb{R}$ with $|w_{k}|\,c_{k}\le W$ for every $k\in\mathbb{N}$; such a $W$ is called a bound for $w$.

2. (The corrector)¶ Let $w$ be a sequence of Wick couplings with bound $W$, and let $\varpi=(\varpi_{k})_{k\in\mathbb{N}}$ with $\varpi_{k}=w_{k}c_{k}/(\beta a_{k})$. Then $\sum_{k=1}^{\infty}|\varpi_{k}|\,c_{k}$ converges, being $\beta^{-1}$ times the series of clause 1, and $|\varpi_{k}|\,a_{k}=\beta^{-1}|w_{k}|\,c_{k}\le\beta^{-1}W$, so $\varpi$ is admissible with bound $\beta^{-1}W$. The Wick-square corrector with couplings $w$ is the diagonal quadratic profile $\Pi_{w}=\Phi_{\varpi}:\mathcal{P}^{a}_{\rho}\to\mathbb{R}$, which is a noise intrinsic test function on $\mathcal{D}$, with gradient along noise couplings $\nabla\Pi_{w}(\nu)=V_{\varpi}(\nu)$ at $\nu\in\mathcal{D}$, by Diagonal Quadratic Profiles on the Noise Wasserstein Space: Moment Bounds, a Lipschitz Estimate, the Gradient Field, and Noise Intrinsic Test Functions §test, since $\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$ by The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §penalty-domain.

3. (The score-paired Wick-square cost)¶ Let $w$ be a sequence of Wick couplings. For $\nu\in\mathcal{D}_{\Sigma}$ both $\Sigma(\nu)$ and $\nabla\Pi_{w}(\nu)$ lie in $L^{2}(\nu;X^{a})$, by The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §pair and by Diagonal Quadratic Profiles on the Noise Wasserstein Space: Moment Bounds, a Lipschitz Estimate, the Gradient Field, and Noise Intrinsic Test Functions §field as used in clause 2. The score-paired Wick-square cost with couplings $w$ is the function

$$G_{w}:\mathcal{D}_{\Sigma}\to\mathbb{R},\qquad G_{w}(\nu)=\langle\Sigma(\nu),\nabla\Pi_{w}(\nu)\rangle_{\nu}=\beta\,\langle Z^{a}_{\nu},\nabla\Pi_{w}(\nu)\rangle_{\nu}.$$