Gaussian Dressing of the Wick-Square Cost: the Riccati Coefficients, the Dressed Variances, the Quadratic Profile, the Shifted Pair and the Shifted Operator

In the settings of The Real Numbers: Standing Notation and Background and The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $c$ be a variance vector, with scaling map $S_{c}$, weighted square $|x|_{c}^{2}$ ($x\in\mathbb{R}^{d}$) and normalizing constant $Z_{c}$, let $a,\lambda_{0},\theta\in\mathbb{R}$ be positive, let $\beta=(\beta_{1},\dots,\beta_{d})\in\mathbb{R}^{d}$ satisfy

$$\Bigl(\frac{a}{c_{i}}\Bigr)^{2}+\lambda_{0}\,\frac{a}{c_{i}}+2\theta\beta_{i}>0\qquad\text{for every }i\in[d],$$

and let $g:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$; $\sqrt{s}$ is the nonnegative square root of $s\ge0$. Being of class $C^{2}$ on $\mathbb{R}^{d}$, the gradient $Du(x)$ and the partial derivatives are those of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives, $\mathbb{R}^{d}$ being open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous; intrinsic test functions, their gradients along couplings and their translation Hessians are those of that definition. In this item the letter $q$ denotes a vector field and the letter $r$ a real number.

1. (The Riccati coefficients)¶ For every $i\in[d]$ there is exactly one $b_{i}\in\mathbb{R}$ with

$$\frac{\theta}{2}\,b_{i}^{2}+\Bigl(\frac{\lambda_{0}}{2}+\frac{a}{c_{i}}\Bigr)b_{i}=\beta_{i}\qquad\text{and}\qquad\frac{a}{c_{i}}+\theta\,b_{i}>0,$$

namely $b_{i}=\theta^{-1}\bigl(\sqrt{(\lambda_{0}/2+a/c_{i})^{2}+2\theta\beta_{i}}-\lambda_{0}/2-a/c_{i}\bigr)$, the radicand being positive. We write $b=(b_{1},\dots,b_{d})$.

2. (The dressed variances)¶ The vector $c'=(c'_{1},\dots,c'_{d})$ with $c'_{i}=a\,(a/c_{i}+\theta b_{i})^{-1}$ is a variance vector, called the dressed variances, and for every $x\in\mathbb{R}^{d}$

$$a\,S_{c'}(x)=a\,S_{c}(x)+\theta\,(b_{1}x_{1},\dots,b_{d}x_{d}),\qquad|x|_{c'}^{2}-|x|_{c}^{2}=\frac{\theta}{a}\sum_{i=1}^{d}b_{i}x_{i}^{2},$$

where $S_{c'}$, $|x|_{c'}^{2}$ and, below, $Z_{c'}$ are the scaling map, weighted square and normalizing constant of $c'$.

3. (The quadratic profile)¶ The function $u_{0}:\mathbb{R}^{d}\to\mathbb{R}$, $u_{0}(x)=\tfrac12\sum_{i=1}^{d}b_{i}x_{i}^{2}$, is of class $C^{2}$ on $\mathbb{R}^{d}$ with $Du_{0}(x)=(b_{1}x_{1},\dots,b_{d}x_{d})$ and $\partial_{j}\partial_{i}u_{0}(x)$ equal to $b_{i}$ if $i=j$ and to $0$ otherwise. The function

$$\Phi_{0}:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R},\qquad\Phi_{0}(\mu)=\int_{\mathbb{R}^{d}}u_{0}\,d\mu,$$

called the quadratic profile, is an intrinsic test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$; for every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, $\nabla\Phi_{0}(\mu)$ is the class $Du_{0}$ in $L^{2}(\mu;\mathbb{R}^{d})$ of the gradient map of $u_{0}$, and $H_{\Phi_{0}}(\mu)$ is the diagonal matrix with diagonal entries $b_{1},\dots,b_{d}$.

4. (The dressed pair)¶ Let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ and $(\mathcal{D}',\mathcal{D}'_{\Sigma},\mathcal{E}',\Sigma')$ be the Gaussian free-energy pairs with temperature $a$ and variances $c$ and $c'$ respectively. Then $\mathcal{D}'=\mathcal{D}$, $\mathcal{D}'_{\Sigma}=\mathcal{D}_{\Sigma}$,

$$\mathcal{E}'(\mu)=\mathcal{E}(\mu)+\theta\,\Phi_{0}(\mu)+a\,(Z_{c'}-Z_{c})\quad(\mu\in\mathcal{D}),\qquad\Sigma'(\nu)=\Sigma(\nu)+\theta\,\nabla\Phi_{0}(\nu)\quad(\nu\in\mathcal{D}_{\Sigma}).$$

5. (The shifted operator)¶ Let $e=\sum_{i=1}^{d}(a\,b_{i}-\beta_{i}c_{i})$, let $F$ be the lifted Wick-square Hamilton-Jacobi operator with variances $c$, temperature $a$, discount $\lambda_{0}$, control cost $\theta$, couplings $\beta$ and running cost $g$, and let $F'$ be the lifted Ornstein-Uhlenbeck Hamilton-Jacobi operator with variances $c'$, temperature $a$, discount $\lambda_{0}$, control cost $\theta$ and running cost $\nu\mapsto g(\nu)+e$. The function $\Phi_{0}$ is an intrinsic test function on $\mathcal{D}$ by clause 3 and Restrictions, Sums, Real Multiples and Differences of Intrinsic Test Functions on the Wasserstein Space §restriction, and the operator $F$ shifted by $\Phi_{0}$ satisfies

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