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

Each result cited is universally quantified over the data in its own statement. Elementary ordered-field arithmetic and order, and the elementary properties of finite sums and squares, are used without further comment, as provided by The Real Numbers: Standing Notation and Background §background. For $i\in[d]$ put

$$p_{i}=\frac{\lambda_{0}}{2}+\frac{a}{c_{i}},\qquad D_{i}=p_{i}^{2}+2\theta\beta_{i}.$$

Clause 1. Fix $i\in[d]$. Expanding the square,

$$D_{i}-\Bigl(\frac{\lambda_{0}}{2}\Bigr)^{2}=\Bigl(\frac{a}{c_{i}}\Bigr)^{2}+\lambda_{0}\,\frac{a}{c_{i}}+2\theta\beta_{i},$$

which is positive by hypothesis. Hence $D_{i}>(\lambda_{0}/2)^{2}\ge0$; so the radicand $D_{i}=(\lambda_{0}/2+a/c_{i})^{2}+2\theta\beta_{i}$ of clause 1 is positive, and $\sqrt{D_{i}}$ is the unique nonnegative real number whose square is $D_{i}$, by Existence and Uniqueness of the Nonnegative Square Root. Since $0\le\lambda_{0}/2$, $0\le\sqrt{D_{i}}$ and $(\lambda_{0}/2)^{2}<D_{i}=(\sqrt{D_{i}})^{2}$, claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives

$$\frac{\lambda_{0}}{2}<\sqrt{D_{i}}.\tag{A}$$

For $s\in\mathbb{R}$ put $t_{s}=\theta s+p_{i}$. Since $\theta\ne0$, multiplying by $2\theta$ and then adding $p_{i}^{2}$ shows

$$\frac{\theta}{2}\,s^{2}+p_{i}\,s=\beta_{i}\iff\theta^{2}s^{2}+2\theta p_{i}\,s=2\theta\beta_{i}\iff t_{s}^{2}=D_{i},$$

and moreover $a/c_{i}+\theta s=t_{s}-\lambda_{0}/2$.

Existence: for $b_{i}=\theta^{-1}(\sqrt{D_{i}}-p_{i})$ we have $t_{b_{i}}=\sqrt{D_{i}}$, so $t_{b_{i}}^{2}=D_{i}$ and the equation of clause 1 holds, and $a/c_{i}+\theta b_{i}=\sqrt{D_{i}}-\lambda_{0}/2>0$ by (A). Uniqueness: if $s\in\mathbb{R}$ satisfies the equation and $a/c_{i}+\theta s>0$, then $t_{s}=(a/c_{i}+\theta s)+\lambda_{0}/2>0$ and $t_{s}^{2}=D_{i}$, so $t_{s}=\sqrt{D_{i}}$ by the uniqueness part of Existence and Uniqueness of the Nonnegative Square Root, that is $s=\theta^{-1}(\sqrt{D_{i}}-p_{i})=b_{i}$. This is the formula of clause 1.

Clause 2. By clause 1, $a/c_{i}+\theta b_{i}>0$ for every $i\in[d]$, and $a>0$; so $c'_{i}=a\,(a/c_{i}+\theta b_{i})^{-1}$ is positive for every $i$, and $c'$ is a variance vector by The Diagonal Gaussian Density on Euclidean Space and Its Notation §variances. Moreover

$$\frac{1}{c'_{i}}=\frac{1}{a}\Bigl(\frac{a}{c_{i}}+\theta b_{i}\Bigr)=\frac{1}{c_{i}}+\frac{\theta}{a}\,b_{i}\qquad(i\in[d]).$$

Let $x\in\mathbb{R}^{d}$. By The Diagonal Gaussian Density on Euclidean Space and Its Notation §scaling, the $i$th component of $a\,S_{c'}(x)$ is $a\,x_{i}/c'_{i}=a\,x_{i}/c_{i}+\theta b_{i}x_{i}$, which is the $i$th component of $a\,S_{c}(x)+\theta(b_{1}x_{1},\dots,b_{d}x_{d})$; and by the same clause

$$|x|_{c'}^{2}-|x|_{c}^{2}=\sum_{i=1}^{d}x_{i}^{2}\Bigl(\frac{1}{c'_{i}}-\frac{1}{c_{i}}\Bigr)=\frac{\theta}{a}\sum_{i=1}^{d}b_{i}x_{i}^{2}.$$

Clause 3. Let $B$ be the diagonal real $d\times d$ matrix with diagonal entries $b_{1},\dots,b_{d}$; it is symmetric, so $B\in\mathcal{S}(d)$. For $z\in\mathbb{R}^{d}$, $Bz=(b_{1}z_{1},\dots,b_{d}z_{d})$ by the definition of the matrix-vector product, and $\tfrac12\,z\cdot(Bz)=\tfrac12\sum_{i=1}^{d}b_{i}z_{i}^{2}=u_{0}(z)$ by the definition of the dot product. Hence Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity §quadratic, applied with $n=d$ (note $1\le d$ by The Wasserstein Space and Its Lift to Square-Integrable Random Vectors: Standing Notation §data), the matrix $B$, linear coefficient $0$ and constant term $0$, shows that $u_{0}$ is of class $C^{2}$ on $\mathbb{R}^{d}$ with $Du_{0}(x)=Bx=(b_{1}x_{1},\dots,b_{d}x_{d})$ and Hessian matrix $D^{2}u_{0}(x)=B$ for every $x$; the entries of the Hessian matrix being the iterated partial derivatives (Hessian Matrix of a C^2 Function), $\partial_{j}\partial_{i}u_{0}(x)$ is $b_{i}$ if $i=j$ and $0$ otherwise ($B$ being diagonal, the order of the two indices is immaterial).

Put $M_{b}=\sum_{k=1}^{d}|b_{k}|$. Then $0\le M_{b}$, and $|\partial_{j}\partial_{i}u_{0}(x)|\le M_{b}$ for all $x\in\mathbb{R}^{d}$ and $i,j\in[d]$, since $|b_{i}|$ is one of the nonnegative summands of $M_{b}$. So Integrals of Functions with a Bounded Hessian are Intrinsic Test Functions on the Wasserstein Space §test, applied with $f=u_{0}$ and $M=M_{b}$, shows that $\Phi_{0}=\varphi_{u_{0}}$ is an intrinsic test function on $\mathcal{P}_{2}(\mathbb{R}^{d})$, that $\nabla\Phi_{0}(\mu)$ is the class $Du_{0}$ for every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, and that $H_{\Phi_{0}}(\mu)=\int_{\mathbb{R}^{d}}D^{2}u_{0}\,d\mu$. The entries of $D^{2}u_{0}$ are the constant functions with values $b_{i}$ (on the diagonal) and $0$ (off it), whose $\mu$-integrals are $b_{i}$ and $0$ because $\mu$ is a probability measure; so $H_{\Phi_{0}}(\mu)=B$.

Clause 4. By The Gaussian Free-Energy Pair: Relative Entropy and Relative Score with Respect to a Diagonal Gaussian Measure §pair, $\mathcal{D}$ (respectively $\mathcal{D}'$) is the set of $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ of finite relative entropy with respect to $\gamma_{c}$ (respectively $\gamma_{c'}$), and $\mathcal{D}_{\Sigma}$ (respectively $\mathcal{D}'_{\Sigma}$) is the set of $\mu\in\mathcal{D}$ (respectively $\mu\in\mathcal{D}'$) of finite Fisher information relative to $\gamma_{c}$ (respectively $\gamma_{c'}$). By Changing the Variances of a Diagonal Gaussian Reference Measure: Equal Domains, the Shift of the Relative Entropy and the Shift of the Relative Score §domains the two entropy conditions agree and the two Fisher-information conditions agree; hence $\mathcal{D}'=\mathcal{D}$ and $\mathcal{D}'_{\Sigma}=\mathcal{D}_{\Sigma}$.

Let $\mu\in\mathcal{D}$. By clauses 2 and 3, $|x|_{c'}^{2}-|x|_{c}^{2}=\frac{2\theta}{a}\,u_{0}(x)$ for every $x\in\mathbb{R}^{d}$. The function $u_{0}$ is $\mu$-integrable by Integrals of Functions with a Bounded Hessian are Intrinsic Test Functions on the Wasserstein Space §integrable (with $M=M_{b}$), so Linearity and Monotonicity of the Lebesgue Integral §integrable gives $\int_{\mathbb{R}^{d}}(|x|_{c'}^{2}-|x|_{c}^{2})\,\mu(dx)=\frac{2\theta}{a}\,\Phi_{0}(\mu)$. Therefore, by The Gaussian Free-Energy Pair: Relative Entropy and Relative Score with Respect to a Diagonal Gaussian Measure §pair and Changing the Variances of a Diagonal Gaussian Reference Measure: Equal Domains, the Shift of the Relative Entropy and the Shift of the Relative Score §entropy,

$$\mathcal{E}'(\mu)=a\,H(\mu\,|\,\gamma_{c'})=a\,H(\mu\,|\,\gamma_{c})+\frac{a}{2}\cdot\frac{2\theta}{a}\,\Phi_{0}(\mu)+a\,(Z_{c'}-Z_{c})=\mathcal{E}(\mu)+\theta\,\Phi_{0}(\mu)+a\,(Z_{c'}-Z_{c}).$$

Let $\nu\in\mathcal{D}_{\Sigma}$. By Changing the Variances of a Diagonal Gaussian Reference Measure: Equal Domains, the Shift of the Relative Entropy and the Shift of the Relative Score §score, $\zeta^{c'}_{\nu}=\zeta^{c}_{\nu}+S_{c'}-S_{c}$ in $L^{2}(\nu;\mathbb{R}^{d})$. By clause 2, $a\,S_{c'}(x)-a\,S_{c}(x)=\theta\,Du_{0}(x)$ for every $x$, so $a\,S_{c'}-a\,S_{c}=\theta\,Du_{0}$ as classes in $L^{2}(\nu;\mathbb{R}^{d})$. Hence, by The Gaussian Free-Energy Pair: Relative Entropy and Relative Score with Respect to a Diagonal Gaussian Measure §pair and clause 3,

$$\Sigma'(\nu)=a\,\zeta^{c'}_{\nu}=a\,\zeta^{c}_{\nu}+\theta\,Du_{0}=\Sigma(\nu)+\theta\,\nabla\Phi_{0}(\nu).$$

Clause 5. Let $(\nu,q)\in\mathcal{V}(\mathcal{D}_{\Sigma})$, $r\in\mathbb{R}$ and $Y\in\mathcal{S}(d)$; thus $\nu\in\mathcal{D}_{\Sigma}=\mathcal{D}'_{\Sigma}$ (clause 4) and $q\in L^{2}(\nu;\mathbb{R}^{d})$, so both sides of the asserted identity are defined. By clause 3 and Restrictions, Sums, Real Multiples and Differences of Intrinsic Test Functions on the Wasserstein Space §restriction, regarding $\Phi_{0}$ as an intrinsic test function on $\mathcal{D}$ changes neither $\nabla\Phi_{0}(\nu)=Du_{0}$ nor $H_{\Phi_{0}}(\nu)=B$. By Viscosity Subsolutions, Supersolutions and Solutions Relative to a Penalty Pair and a Profile §shifted and the formula of The Lifted Hamilton-Jacobi Equation with a Wick-Square Cost Relative to a Diagonal Gaussian Measure §operator,

$$F^{\Phi_{0}}(\nu,r,q,Y)=F\bigl(\nu,r+\Phi_{0}(\nu),q+Du_{0},Y+B\bigr)=\lambda_{0}r+\lambda_{0}\Phi_{0}(\nu)+\frac{\theta}{2}\lVert q+Du_{0}\rVert_{\nu}^{2}+a\,\langle\zeta^{c}_{\nu},q+Du_{0}\rangle_{\nu}-\int_{\mathbb{R}^{d}}V_{c,\beta}\,d\nu-g(\nu).\tag{B}$$

The inner product of $L^{2}(\nu;\mathbb{R}^{d})$ is bilinear and symmetric and $\lVert\eta\rVert_{\nu}^{2}=\langle\eta,\eta\rangle_{\nu}$, by Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields; hence

$$\frac{\theta}{2}\lVert q+Du_{0}\rVert_{\nu}^{2}+a\,\langle\zeta^{c}_{\nu},q+Du_{0}\rangle_{\nu}=\frac{\theta}{2}\lVert q\rVert_{\nu}^{2}+\langle a\,\zeta^{c}_{\nu}+\theta\,Du_{0},q\rangle_{\nu}+\frac{\theta}{2}\lVert Du_{0}\rVert_{\nu}^{2}+a\,\langle\zeta^{c}_{\nu},Du_{0}\rangle_{\nu}.$$

By clause 4, $a\,\zeta^{c}_{\nu}+\theta\,Du_{0}=a\,\zeta^{c'}_{\nu}$, so $\langle a\,\zeta^{c}_{\nu}+\theta\,Du_{0},q\rangle_{\nu}=a\,\langle\zeta^{c'}_{\nu},q\rangle_{\nu}$. It remains to evaluate

$$R(\nu)=\lambda_{0}\Phi_{0}(\nu)+\frac{\theta}{2}\lVert Du_{0}\rVert_{\nu}^{2}+a\,\langle\zeta^{c}_{\nu},Du_{0}\rangle_{\nu}-\int_{\mathbb{R}^{d}}V_{c,\beta}\,d\nu .$$

We express each term as an integral. First, $\Phi_{0}(\nu)=\int u_{0}\,d\nu$. Second, by Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields, $\lVert Du_{0}\rVert_{\nu}^{2}=\int\lVert Du_{0}(x)\rVert^{2}\,\nu(dx)$, where $\lVert Du_{0}(x)\rVert^{2}=\sum_{i}b_{i}^{2}x_{i}^{2}$; the functions $u_{0}$ and $x\mapsto\lVert Du_{0}(x)\rVert^{2}$ are $\nu$-integrable by Integrals of Functions with a Bounded Hessian are Intrinsic Test Functions on the Wasserstein Space §integrable. Third, $\nu$ has finite Fisher information relative to $\gamma_{c}$ by The Gaussian Free-Energy Pair: Relative Entropy and Relative Score with Respect to a Diagonal Gaussian Measure §pair, so Gaussian Integration by Parts Relative to a Diagonal Gaussian Measure for Functions with a Bounded Hessian §ibp, applied with $f=u_{0}$ and $M=M_{b}$, gives $\langle\zeta^{c}_{\nu},Du_{0}\rangle_{\nu}=-\int L_{c}u_{0}\,d\nu$, the function $L_{c}u_{0}$ being $\nu$-integrable by Gaussian Integration by Parts Relative to a Diagonal Gaussian Measure for Functions with a Bounded Hessian §generator; by The Laplacian of a Twice Continuously Differentiable Function §laplacian, clause 3, The Diagonal Gaussian Density on Euclidean Space and Its Notation §scaling and the definition of the dot product,

$$L_{c}u_{0}(x)=\Delta u_{0}(x)-S_{c}(x)\cdot Du_{0}(x)=\sum_{i=1}^{d}b_{i}-\sum_{i=1}^{d}\frac{b_{i}}{c_{i}}\,x_{i}^{2}.$$

Fourth, $V_{c,\beta}(x)=\sum_{i}\beta_{i}(x_{i}^{2}-c_{i})$ by The Wick-Square Potential Relative to a Diagonal Gaussian Measure §potential, so that $V_{c,\beta}(x)=\tfrac12\,x\cdot(M_{\beta}x)+0_{\mathbb{R}^{d}}\cdot x+\bigl(-\sum_{i=1}^{d}\beta_{i}c_{i}\bigr)$ for every $x\in\mathbb{R}^{d}$, where $M_{\beta}\in\mathcal{S}(d)$ is the diagonal matrix with diagonal entries $2\beta_{1},\dots,2\beta_{d}$ (as $M_{\beta}x=(2\beta_{1}x_{1},\dots,2\beta_{d}x_{d})$ by the definition of the matrix-vector product). Hence Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity §quadratic, applied with $n=d$, the matrix $M_{\beta}$, linear coefficient $0_{\mathbb{R}^{d}}$ and constant term $-\sum_{i=1}^{d}\beta_{i}c_{i}$, shows that $V_{c,\beta}$ is of class $C^{2}$ on $\mathbb{R}^{d}$ with Hessian matrix $M_{\beta}$, whose entries are bounded in absolute value by $2\sum_{i=1}^{d}|\beta_{i}|$; so $V_{c,\beta}$ is $\nu$-integrable by Integrals of Functions with a Bounded Hessian are Intrinsic Test Functions on the Wasserstein Space §integrable, applied with $f=V_{c,\beta}$ and $M=2\sum_{i=1}^{d}|\beta_{i}|$. By Linearity and Monotonicity of the Lebesgue Integral §integrable, $R(\nu)=\int h\,d\nu$ with $h=\lambda_{0}u_{0}+\frac{\theta}{2}\lVert Du_{0}\rVert^{2}-a\,L_{c}u_{0}-V_{c,\beta}$, and for every $x\in\mathbb{R}^{d}$

$$h(x)=\sum_{i=1}^{d}\Bigl(\frac{\lambda_{0}}{2}b_{i}+\frac{\theta}{2}b_{i}^{2}+\frac{a}{c_{i}}b_{i}-\beta_{i}\Bigr)x_{i}^{2}-\sum_{i=1}^{d}\bigl(a\,b_{i}-\beta_{i}c_{i}\bigr).$$

Each coefficient $\frac{\theta}{2}b_{i}^{2}+\bigl(\frac{\lambda_{0}}{2}+\frac{a}{c_{i}}\bigr)b_{i}-\beta_{i}$ vanishes by clause 1, so $h(x)=-e$ for every $x$, and $R(\nu)=-e$ since $\nu$ is a probability measure. Substituting into (B),

$$F^{\Phi_{0}}(\nu,r,q,Y)=\lambda_{0}r+\frac{\theta}{2}\lVert q\rVert_{\nu}^{2}+a\,\langle\zeta^{c'}_{\nu},q\rangle_{\nu}-\bigl(g(\nu)+e\bigr).$$

By The Lifted Ornstein-Uhlenbeck Hamilton-Jacobi Equation Relative to a Diagonal Gaussian Measure §operator, applied to the Gaussian free-energy pair with variances $c'$ and temperature $a$ (whose score domain is $\mathcal{D}_{\Sigma}$ by clause 4, and whose score is $a\,\zeta^{c'}_{\nu}$), with discount $\lambda_{0}$, control cost $\theta$ and running cost $\nu\mapsto g(\nu)+e$, the right-hand side is $F'(\nu,r,q,Y)$. The argument $Y$ enters neither operator.