Gaussian Dressing of the Score-Paired Wick-Square Cost on a Hilbert Space: Riccati Coefficients, Dressed Variances, the Quadratic Profile, the Constant, the Dressed Pair and the Shifted Operator

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 $\kappa$. Let $w$ be a sequence of Wick couplings with bound $W$, let $\lambda_{0},\theta\in\mathbb{R}$ be positive, let $\varepsilon_{0}\in\mathbb{R}$ be positive with

$$\frac{\beta a_{k}}{c_{k}}+\lambda_{0}+\frac{2\theta\,w_{k}c_{k}}{\beta}\ge\varepsilon_{0}\qquad\text{for every }k\in\mathbb{N},$$

and let $g:\mathcal{D}\to\mathbb{R}$. admissible sequences, diagonal quadratic profiles $\Phi_{b}$ and their gradient fields $V_{b}$ are those of Diagonal Quadratic Profiles on the Noise Wasserstein Space: Moment Bounds, a Lipschitz Estimate, the Gradient Field, and Noise Intrinsic Test Functions, $\log$ is the natural logarithm, and convergent series are those of that definition. In this item the letter $q$ denotes a noise field and the letter $r$ a real number.

1. (The Riccati coefficients)¶ For every $k\in\mathbb{N}$ there is exactly one $b_{k}\in\mathbb{R}$ with

$$\frac{\theta a_{k}}{2}\,b_{k}^{2}+\Bigl(\frac{\lambda_{0}}{2}+\frac{\beta a_{k}}{c_{k}}\Bigr)b_{k}=w_{k}\qquad\text{and}\qquad\frac{\beta}{c_{k}}+\theta\,b_{k}>0,$$

and it satisfies $|b_{k}|\le2|w_{k}|\,c_{k}/(\beta a_{k})$. The sequence $b=(b_{k})_{k\in\mathbb{N}}$ is admissible with bound $2W/\beta$.

2. (The dressed variances)¶ The sequence $d=(\theta b_{k}/\beta)_{k\in\mathbb{N}}$ is admissible and satisfies $c_{k}^{-1}+d_{k}>0$ for every $k\in\mathbb{N}$. The sequence $c'$ with $c'_{k}=(c_{k}^{-1}+\theta b_{k}/\beta)^{-1}$, called the dressed variances, is a variance sequence, and $c'_{k}\le\kappa'a_{k}$ for every $k\in\mathbb{N}$, where $\kappa'=\bigl(\beta+\kappa\lambda_{0}+\kappa\theta W\beta^{-1}\bigr)\varepsilon_{0}^{-1}$.

3. (The quadratic profile)¶ The diagonal quadratic profile $\Phi_{0}=\Phi_{b}$ is a noise intrinsic test function on $\mathcal{D}$, with $\nabla\Phi_{0}(\nu)=V_{b}(\nu)$ for $\nu\in\mathcal{D}$.

4. (The constant)¶ The series $e=\sum_{k=1}^{\infty}\bigl(\beta a_{k}b_{k}-w_{k}c_{k}\bigr)$ converges absolutely.

5. (The dressed pair)¶ Read A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation with $c'$ in place of $c$, and let $(\mathcal{D}',\mathcal{D}'_{\Sigma},\mathcal{E}',\Sigma')$ be the Gaussian entropy pair relative to $\gamma_{c'}$ with temperature $\beta$, whose hypothesis holds with $\kappa'$. Then $\mathcal{P}^{a}_{\gamma_{c'}}=\mathcal{P}^{a}_{\rho}$, $\mathcal{D}'=\mathcal{D}$, $\mathcal{D}'_{\Sigma}=\mathcal{D}_{\Sigma}$, and

$$\mathcal{E}'(\mu)=\mathcal{E}(\mu)+\theta\,\Phi_{0}(\mu)+\frac{\beta}{2}\sum_{k=1}^{\infty}\log\frac{c'_{k}}{c_{k}}\quad(\mu\in\mathcal{D}),\qquad\Sigma'(\nu)=\Sigma(\nu)+\theta\,\nabla\Phi_{0}(\nu)\quad(\nu\in\mathcal{D}_{\Sigma}),$$

the series converging absolutely.

6. (The shifted operator)¶ Let $F$ be the Hamilton-Jacobi operator with Gaussian score drift and the Wick-square cost relative to $\gamma_{c}$ with temperature $\beta$, discount $\lambda_{0}$, control cost $\theta$, Wick couplings $w$ and running cost $g$, and, $\Phi_{0}$ being a noise intrinsic test function on $\mathcal{D}$ by clause 3 and the pair a noise penalty pair by The Gaussian Entropy Pair is a Noise Penalty Pair, with Nonnegative Penalty and Dense Score Domain §pair, let $F^{\Phi_{0}}$ be the operator $F$ shifted by $\Phi_{0}$. 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 $\nu\mapsto g(\nu)+e$ on $\mathcal{D}'=\mathcal{D}$. Then

$$F^{\Phi_{0}}(\nu,r,q)=F'(\nu,r,q)\qquad\text{for all }(\nu,q)\in\mathcal{V}^{a}(\mathcal{D}_{\Sigma})\text{ and }r\in\mathbb{R}.$$

7. (The dressed pair on the original space)¶ The dressed pair $(\mathcal{D}',\mathcal{D}'_{\Sigma},\mathcal{E}',\Sigma')$ of clause 5 is a noise penalty pair on $\mathcal{P}^{a}_{\rho}$ in the setting of A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation read with the reference measure $\rho=\gamma_{c}$, and $0\le\mathcal{E}'(\mu)$ for every $\mu\in\mathcal{D}$.