Proof of 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

Each result cited is universally quantified over the data in its own statement. Elementary ordered-field arithmetic and order, the properties of the absolute value and of finite sums are used without further comment, as provided by The Real Numbers: Standing Notation and Background §background.

Preliminaries. By The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §pair, $\mathcal{D}_{\Sigma}\subseteq\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$. For every $k\in\mathbb{N}$, $a_{k}>0$, since $a$ is a weight sequence, and $c_{k}>0$, since $c$ is a variance sequence (A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §gaussian). The bound $W$ of the Wick couplings satisfies $0\le|w_{1}|\,c_{1}\le W$ by The Wick-Square Corrector and the Score-Paired Wick-Square Cost Relative to a Diagonal Gaussian Measure on a Hilbert Space §couplings, so $W\ge0$. For $k\in\mathbb{N}$ put

$$s_{k}=\frac{\beta a_{k}}{c_{k}},\qquad t_{k}=\frac{2\theta\,w_{k}c_{k}}{\beta},\qquad E_{k}=s_{k}+\lambda_{0}+t_{k}.$$

Then $s_{k}>0$, $E_{k}\ge\varepsilon_{0}>0$ by hypothesis, $t_{k}\le|t_{k}|=2\theta|w_{k}|c_{k}/\beta\le2\theta W/\beta$, and, dividing $c_{k}\le\kappa a_{k}$ by the positive number $\kappa c_{k}$, $s_{k}\ge\beta/\kappa$.

Clause 1. Fix $k\in\mathbb{N}$. For $b\in\mathbb{R}$ put $z(b)=1+\theta c_{k}b/\beta$. The map $b\mapsto z(b)$ is a bijection of $\mathbb{R}$ onto itself, with inverse $z\mapsto\beta(z-1)/(\theta c_{k})$, and

$$\frac{\beta}{c_{k}}+\theta\,b=\frac{\beta}{c_{k}}\,z(b)\qquad(b\in\mathbb{R}).\tag{1}$$

Let $b\in\mathbb{R}$ and $y=\theta c_{k}b/\beta=z(b)-1$, so that $\theta a_{k}b=s_{k}y$. Multiplying the four quantities $\frac{\theta a_{k}}{2}b^{2}$, $\frac{\lambda_{0}}{2}b$, $\frac{\beta a_{k}}{c_{k}}b$ and $w_{k}$ by $2\theta c_{k}/\beta$ gives $s_{k}y^{2}$, $\lambda_{0}y$, $2\theta a_{k}b=2s_{k}y$ and $t_{k}$ respectively; since $s_{k}(y^{2}+2y)=s_{k}(z(b)^{2}-1)$ and $\lambda_{0}y=\lambda_{0}z(b)-\lambda_{0}$,

$$\frac{2\theta c_{k}}{\beta}\Bigl(\frac{\theta a_{k}}{2}\,b^{2}+\Bigl(\frac{\lambda_{0}}{2}+\frac{\beta a_{k}}{c_{k}}\Bigr)b-w_{k}\Bigr)=s_{k}\,z(b)^{2}+\lambda_{0}\,z(b)-E_{k}.\tag{2}$$

As $2\theta c_{k}/\beta\ne0$ and $\beta/c_{k}>0$, (1) and (2) show that $b$ satisfies the two conditions of clause 1 if and only if $z=z(b)$ satisfies

$$s_{k}z^{2}+\lambda_{0}z=E_{k}\qquad\text{and}\qquad z>0.\tag{3}$$

So, the map $b\mapsto z(b)$ being a bijection, it suffices to show that (3) has exactly one solution $z\in\mathbb{R}$.

Put $\Delta_{k}=\lambda_{0}^{2}+4s_{k}E_{k}$. Since $s_{k}E_{k}>0$, $\Delta_{k}>\lambda_{0}^{2}\ge0$, so $\Delta_{k}$ has a unique nonnegative square root $\sqrt{\Delta_{k}}$; as $0\le\lambda_{0}$, $0\le\sqrt{\Delta_{k}}$ and $\lambda_{0}^{2}<(\sqrt{\Delta_{k}})^{2}$, the monotonicity of squares of nonnegative numbers gives $\lambda_{0}<\sqrt{\Delta_{k}}$. For every $z\in\mathbb{R}$,

$$(2s_{k}z+\lambda_{0})^{2}=4s_{k}\bigl(s_{k}z^{2}+\lambda_{0}z\bigr)+\lambda_{0}^{2},$$

so, $4s_{k}$ being nonzero, $s_{k}z^{2}+\lambda_{0}z=E_{k}$ holds if and only if $(2s_{k}z+\lambda_{0})^{2}=\Delta_{k}$. Existence: $z_{k}=(\sqrt{\Delta_{k}}-\lambda_{0})/(2s_{k})$ is positive and satisfies $2s_{k}z_{k}+\lambda_{0}=\sqrt{\Delta_{k}}$, whose square is $\Delta_{k}$; so $z_{k}$ solves (3). Uniqueness: if $z$ solves (3), then $2s_{k}z+\lambda_{0}>0$ and its square is $\Delta_{k}$, so $2s_{k}z+\lambda_{0}=\sqrt{\Delta_{k}}$ by the uniqueness of the nonnegative square root, that is $z=z_{k}$. Hence $b_{k}=\beta(z_{k}-1)/(\theta c_{k})$ is the unique real number with the two properties of clause 1, and $z(b_{k})=z_{k}$.

The bound. By (3), $s_{k}(z_{k}^{2}-1)+\lambda_{0}(z_{k}-1)=E_{k}-s_{k}-\lambda_{0}=t_{k}$, that is

$$(z_{k}-1)\bigl(s_{k}(z_{k}+1)+\lambda_{0}\bigr)=t_{k}.\tag{4}$$

Since $z_{k}>0$ and $\lambda_{0}>0$, $s_{k}(z_{k}+1)+\lambda_{0}>s_{k}>0$, so (4) gives $|z_{k}-1|=|t_{k}|/(s_{k}(z_{k}+1)+\lambda_{0})\le|t_{k}|/s_{k}$. Therefore

$$|b_{k}|=\frac{\beta\,|z_{k}-1|}{\theta c_{k}}\le\frac{\beta\,|t_{k}|}{\theta c_{k}s_{k}}=\frac{2|w_{k}|\,c_{k}}{\beta a_{k}} .$$

Admissibility. For every $k$, $|b_{k}|\,a_{k}\le2|w_{k}|c_{k}/\beta\le2W/\beta$, that is

$$a_{k}|b_{k}|\le\frac{2W}{\beta},\tag{5}$$

and $0\le2W/\beta$. Moreover $0\le|b_{k}|\,c_{k}\le\frac{2}{\beta}\,|w_{k}|\,c_{k}^{2}/a_{k}$; the series $\sum_{k}|w_{k}|c_{k}^{2}/a_{k}$ converges by The Wick-Square Corrector and the Score-Paired Wick-Square Cost Relative to a Diagonal Gaussian Measure on a Hilbert Space §couplings, so $\sum_{k}\frac{2}{\beta}|w_{k}|c_{k}^{2}/a_{k}$ converges by Elementary Properties of Series of Real Numbers §linearity, and $\sum_{k}|b_{k}|c_{k}$ converges by the comparison test Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §comparison. Thus $b$ is admissible with bound $2W/\beta$.

Clause 2. By Diagonal Quadratic Profiles on the Noise Wasserstein Space: Moment Bounds, a Lipschitz Estimate, the Gradient Field, and Noise Intrinsic Test Functions §scaling, applied with $b$ and the real number $\theta/\beta$, the sequence $d=(\theta b_{k}/\beta)_{k\in\mathbb{N}}$ is admissible. By (1) with $b=b_{k}$ and $z(b_{k})=z_{k}$,

$$c_{k}^{-1}+d_{k}=\frac{1}{\beta}\Bigl(\frac{\beta}{c_{k}}+\theta b_{k}\Bigr)=\frac{z_{k}}{c_{k}}>0,\qquad\text{so}\qquad c'_{k}=\frac{c_{k}}{z_{k}}\qquad(k\in\mathbb{N}).$$

The sequence $c'$ is the sequence of Changing the Variances of a Diagonal Gaussian Reference by a Diagonal Quadratic: the Same Measures and Domains, with Penalty and Score Shifted by a Diagonal Quadratic Profile for this $d$, so it is a variance sequence by Changing the Variances of a Diagonal Gaussian Reference by a Diagonal Quadratic: the Same Measures and Domains, with Penalty and Score Shifted by a Diagonal Quadratic Profile §variances.

The bound $c'_{k}\le\kappa'a_{k}$. Fix $k$ and put $\sigma_{k}=s_{k}z_{k}=\beta a_{k}z_{k}/c_{k}=\beta a_{k}/c'_{k}>0$. Multiplying the equation in (3) by $s_{k}$ gives

$$\sigma_{k}^{2}+\lambda_{0}\sigma_{k}=s_{k}E_{k}.\tag{6}$$

Put $T_{k}=s_{k}+\theta W/\beta>0$. Expanding,

$$T_{k}^{2}+\lambda_{0}T_{k}-s_{k}E_{k}=s_{k}\Bigl(\frac{2\theta W}{\beta}-t_{k}\Bigr)+\frac{\theta^{2}W^{2}}{\beta^{2}}+\frac{\lambda_{0}\theta W}{\beta}\ge0,$$

since $t_{k}\le2\theta W/\beta$ and $W\ge0$. If $\sigma_{k}>T_{k}$, then, as $T_{k}>0$, $\sigma_{k}^{2}>T_{k}^{2}$ and $\lambda_{0}\sigma_{k}>\lambda_{0}T_{k}$, so $\sigma_{k}^{2}+\lambda_{0}\sigma_{k}>T_{k}^{2}+\lambda_{0}T_{k}\ge s_{k}E_{k}$, contradicting (6). Hence $\sigma_{k}\le T_{k}$. Put $K=\lambda_{0}+\theta W/\beta>0$; then $0<\sigma_{k}+\lambda_{0}\le s_{k}+K$, and (6) with $E_{k}\ge\varepsilon_{0}$ gives

$$\sigma_{k}=\frac{s_{k}E_{k}}{\sigma_{k}+\lambda_{0}}\ge\frac{\varepsilon_{0}\,s_{k}}{s_{k}+K}.$$

Since $s_{k}\ge\beta/\kappa$,

$$\frac{s_{k}}{s_{k}+K}-\frac{\beta/\kappa}{\beta/\kappa+K}=\frac{K\,(s_{k}-\beta/\kappa)}{(s_{k}+K)(\beta/\kappa+K)}\ge0,\qquad\frac{\beta/\kappa}{\beta/\kappa+K}=\frac{\beta}{\beta+\kappa\lambda_{0}+\kappa\theta W\beta^{-1}} .$$

Hence $\sigma_{k}\ge\varepsilon_{0}\beta/(\beta+\kappa\lambda_{0}+\kappa\theta W\beta^{-1})$, and

$$c'_{k}=\frac{\beta a_{k}}{\sigma_{k}}\le\frac{\beta+\kappa\lambda_{0}+\kappa\theta W\beta^{-1}}{\varepsilon_{0}}\,a_{k}=\kappa'a_{k}.$$

The number $\kappa'$ is positive, since $\beta>0$, $\kappa\lambda_{0}>0$, $\kappa\theta W\beta^{-1}\ge0$ (as $W\ge0$) and $\varepsilon_{0}>0$.

Clause 3. By clause 1, $b$ is admissible, and $\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$. So Diagonal Quadratic Profiles on the Noise Wasserstein Space: Moment Bounds, a Lipschitz Estimate, the Gradient Field, and Noise Intrinsic Test Functions §test, applied with $Q=\mathcal{D}$, shows that $\Phi_{0}=\Phi_{b}$ is a noise intrinsic test function on $\mathcal{D}$ with $\nabla\Phi_{0}(\nu)=V_{b}(\nu)$ for every $\nu\in\mathcal{D}$.

Clause 4. Multiplying the equation of clause 1 by $c_{k}$ gives $w_{k}c_{k}=\frac{\theta}{2}a_{k}c_{k}b_{k}^{2}+\frac{\lambda_{0}}{2}c_{k}b_{k}+\beta a_{k}b_{k}$, so

$$\beta a_{k}b_{k}-w_{k}c_{k}=-\frac{\theta}{2}\,(a_{k}b_{k})(c_{k}b_{k})-\frac{\lambda_{0}}{2}\,c_{k}b_{k},\tag{7}$$

and by (5)

$$\bigl|\beta a_{k}b_{k}-w_{k}c_{k}\bigr|\le\frac{\theta}{2}\,(a_{k}|b_{k}|)\,c_{k}|b_{k}|+\frac{\lambda_{0}}{2}\,c_{k}|b_{k}|\le\Bigl(\frac{\theta W}{\beta}+\frac{\lambda_{0}}{2}\Bigr)c_{k}|b_{k}| .$$

The series $\sum_{k}|b_{k}|c_{k}$ converges (clause 1), hence so does $\sum_{k}\bigl(\frac{\theta W}{\beta}+\frac{\lambda_{0}}{2}\bigr)c_{k}|b_{k}|$ by Elementary Properties of Series of Real Numbers §linearity, and An Absolutely Convergent Series of Real Numbers Converges §dominated shows that $\sum_{k}(\beta a_{k}b_{k}-w_{k}c_{k})$ converges absolutely. It therefore converges by An Absolutely Convergent Series of Real Numbers Converges §convergence, and its sum $e$ is a real number.

Clause 5. The sequence $d$ of clause 2 is admissible with $c_{k}^{-1}+d_{k}>0$ for every $k$, its sequence $c'$ is the sequence of dressed variances, and $\kappa'$ is positive with $c'_{k}\le\kappa'a_{k}$ (clause 2); so Changing the Variances of a Diagonal Gaussian Reference by a Diagonal Quadratic: the Same Measures and Domains, with Penalty and Score Shifted by a Diagonal Quadratic Profile applies with this $d$ and with this $\kappa'$ as the fixed constant, and its pair $(\mathcal{D}',\mathcal{D}'_{\Sigma},\mathcal{E}',\Sigma')$ is the one of clause 5. By Changing the Variances of a Diagonal Gaussian Reference by a Diagonal Quadratic: the Same Measures and Domains, with Penalty and Score Shifted by a Diagonal Quadratic Profile §space, $\mathcal{P}^{a}_{\gamma_{c'}}=\mathcal{P}^{a}_{\rho}$; by Changing the Variances of a Diagonal Gaussian Reference by a Diagonal Quadratic: the Same Measures and Domains, with Penalty and Score Shifted by a Diagonal Quadratic Profile §domains, $\mathcal{D}'=\mathcal{D}$ and $\mathcal{D}'_{\Sigma}=\mathcal{D}_{\Sigma}$; by Changing the Variances of a Diagonal Gaussian Reference by a Diagonal Quadratic: the Same Measures and Domains, with Penalty and Score Shifted by a Diagonal Quadratic Profile §logsum, $\sum_{k}\log(c'_{k}/c_{k})$ converges absolutely. By Diagonal Quadratic Profiles on the Noise Wasserstein Space: Moment Bounds, a Lipschitz Estimate, the Gradient Field, and Noise Intrinsic Test Functions §scaling (with $b$ and $\theta/\beta$), $\Phi_{d}(\mu)=\frac{\theta}{\beta}\Phi_{b}(\mu)$ and $V_{d}(\mu)=\frac{\theta}{\beta}V_{b}(\mu)$ for $\mu\in\mathcal{P}^{a}_{\rho}$, so $\beta\,\Phi_{d}(\mu)=\theta\,\Phi_{0}(\mu)$ and $\beta\,V_{d}(\mu)=\theta\,V_{b}(\mu)$. Hence Changing the Variances of a Diagonal Gaussian Reference by a Diagonal Quadratic: the Same Measures and Domains, with Penalty and Score Shifted by a Diagonal Quadratic Profile §penalty gives, for $\mu\in\mathcal{D}$,

$$\mathcal{E}'(\mu)=\mathcal{E}(\mu)+\beta\,\Phi_{d}(\mu)+\frac{\beta}{2}\sum_{k=1}^{\infty}\log\frac{c'_{k}}{c_{k}}=\mathcal{E}(\mu)+\theta\,\Phi_{0}(\mu)+\frac{\beta}{2}\sum_{k=1}^{\infty}\log\frac{c'_{k}}{c_{k}},$$

and Changing the Variances of a Diagonal Gaussian Reference by a Diagonal Quadratic: the Same Measures and Domains, with Penalty and Score Shifted by a Diagonal Quadratic Profile §score gives, for $\nu\in\mathcal{D}_{\Sigma}\subseteq\mathcal{D}$, using clause 3,

$$\Sigma'(\nu)=\Sigma(\nu)+\beta\,V_{d}(\nu)=\Sigma(\nu)+\theta\,V_{b}(\nu)=\Sigma(\nu)+\theta\,\nabla\Phi_{0}(\nu).$$

Clause 6. Let $(\nu,q)\in\mathcal{V}^{a}(\mathcal{D}_{\Sigma})$ and $r\in\mathbb{R}$; thus $\nu\in\mathcal{D}_{\Sigma}$ and $q\in L^{2}(\nu;X^{a})$ by The Bundle of Noise Fields over a Set of Measures, First-Order Equation Operators on the Noise Wasserstein Space, and Their Delta-Shifts §bundle. Since $\mathcal{D}'_{\Sigma}=\mathcal{D}_{\Sigma}$ (clause 5) and the bundle over a set depends only on that set, $(\nu,q)\in\mathcal{V}^{a}(\mathcal{D}'_{\Sigma})$, so $F'(\nu,r,q)$ is defined. Write $V=V_{b}(\nu)=\nabla\Phi_{0}(\nu)$ (clause 3, as $\nu\in\mathcal{D}$), $Z=Z^{a}_{\nu}$ for the noise score field of $\nu$ relative to $\gamma_{c}$ and $Z'$ for that relative to $\gamma_{c'}$. By Viscosity Subsolutions, Supersolutions and Solutions on the Noise Wasserstein Space Relative to a Noise Penalty Pair and a Profile §shifted and the second formula of The Hamilton-Jacobi Equation with Gaussian Score Drift and a Score-Paired Wick-Square Cost on a Hilbert Space §operator,

$$F^{\Phi_{0}}(\nu,r,q)=F\bigl(\nu,r+\Phi_{0}(\nu),q+V\bigr)=\lambda_{0}r+\lambda_{0}\Phi_{0}(\nu)+\frac{\theta}{2}\lVert q+V\rVert_{\nu}^{2}+\beta\langle Z,q+V\rangle_{\nu}-G_{w}(\nu)-g(\nu).\tag{8}$$

The space $L^{2}(\nu;X^{a})$ is a real Hilbert space, with $\lVert\eta\rVert_{\nu}^{2}=\langle\eta,\eta\rangle_{\nu}$ for its elements $\eta$, by Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields, The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations and The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §hilbert; by bilinearity and symmetry of the inner product,

$$\frac{\theta}{2}\lVert q+V\rVert_{\nu}^{2}+\beta\langle Z,q+V\rangle_{\nu}=\frac{\theta}{2}\lVert q\rVert_{\nu}^{2}+\beta\langle Z,q\rangle_{\nu}+\theta\langle V,q\rangle_{\nu}+\frac{\theta}{2}\lVert V\rVert_{\nu}^{2}+\beta\langle Z,V\rangle_{\nu}.$$

By Changing the Variances of a Diagonal Gaussian Reference by a Diagonal Quadratic: the Same Measures and Domains, with Penalty and Score Shifted by a Diagonal Quadratic Profile §score, applied with the $d$ of clause 2, $Z'=Z+V_{d}(\nu)=Z+\frac{\theta}{\beta}V$ (clause 5), so $\beta\langle Z',q\rangle_{\nu}=\beta\langle Z,q\rangle_{\nu}+\theta\langle V,q\rangle_{\nu}$. Substituting into (8),

$$F^{\Phi_{0}}(\nu,r,q)=\lambda_{0}r+\frac{\theta}{2}\lVert q\rVert_{\nu}^{2}+\beta\langle Z',q\rangle_{\nu}+R(\nu)-g(\nu),\qquad R(\nu)=\lambda_{0}\Phi_{0}(\nu)+\frac{\theta}{2}\lVert V\rVert_{\nu}^{2}+\beta\langle Z,V\rangle_{\nu}-G_{w}(\nu).\tag{9}$$

We show $R(\nu)=-e$. For $k\in\mathbb{N}$ let $m_{k}=\int_{X}x_{k}^{2}\,\nu(dx)$, a real number by the preamble of Diagonal Quadratic Profiles on the Noise Wasserstein Space: Moment Bounds, a Lipschitz Estimate, the Gradient Field, and Noise Intrinsic Test Functions, as $\nu\in\mathcal{P}^{a}_{\rho}$. We use four convergent series.

(i) By Diagonal Quadratic Profiles on the Noise Wasserstein Space: Moment Bounds, a Lipschitz Estimate, the Gradient Field, and Noise Intrinsic Test Functions §profile, $\sum_{k}b_{k}m_{k}$ converges absolutely, hence converges by An Absolutely Convergent Series of Real Numbers Converges §convergence, and $\Phi_{0}(\nu)=\frac12\sum_{k}b_{k}m_{k}$; so $\lambda_{0}\Phi_{0}(\nu)=\sum_{k}\frac{\lambda_{0}}{2}b_{k}m_{k}$ by Elementary Properties of Series of Real Numbers §linearity.

(ii) By Diagonal Quadratic Profiles on the Noise Wasserstein Space: Moment Bounds, a Lipschitz Estimate, the Gradient Field, and Noise Intrinsic Test Functions §field, $\lVert V\rVert_{\nu}^{2}=\sum_{k}a_{k}b_{k}^{2}m_{k}$. (This series converges: by Diagonal Quadratic Profiles on the Noise Wasserstein Space: Moment Bounds, a Lipschitz Estimate, the Gradient Field, and Noise Intrinsic Test Functions §moments, applied with the nonnegative sequence $(a_{k}b_{k}^{2})_{k\in\mathbb{N}}$ in place of $t$ and $M=(2W/\beta)^{2}$; indeed $a_{k}\cdot a_{k}b_{k}^{2}\le(2W/\beta)^{2}$ and $0\le a_{k}b_{k}^{2}c_{k}\le\frac{2W}{\beta}|b_{k}|c_{k}$ by (5), so $\sum_{k}a_{k}b_{k}^{2}c_{k}$ converges by Elementary Properties of Series of Real Numbers §linearity and Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §comparison.) So $\frac{\theta}{2}\lVert V\rVert_{\nu}^{2}=\sum_{k}\frac{\theta}{2}a_{k}b_{k}^{2}m_{k}$ by Elementary Properties of Series of Real Numbers §linearity.

(iii) The measure $\nu\in\mathcal{D}_{\Sigma}$ lies in $\mathcal{P}^{a}_{\rho}$ and has a relative score with respect to $\gamma_{c}$ and finite Fisher information relative to $\gamma_{c}$ with weights $a$, by The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §score-domain. So The Gaussian Relative Score Paired with the Gradient Field of a Diagonal Quadratic Profile §pairing gives $\langle Z,V\rangle_{\nu}=\sum_{k}a_{k}b_{k}\bigl(\frac{m_{k}}{c_{k}}-1\bigr)$, and $\beta\langle Z,V\rangle_{\nu}=\sum_{k}\bigl(\frac{\beta a_{k}}{c_{k}}b_{k}m_{k}-\beta a_{k}b_{k}\bigr)$ by Elementary Properties of Series of Real Numbers §linearity.

(iv) By The Score-Paired Wick-Square Cost is the Wick-Ordered Series, and Cutoff Counterterms Are Forced up to a Convergent Constant §series, $G_{w}(\nu)=\sum_{k}w_{k}(m_{k}-c_{k})$, so $-G_{w}(\nu)=\sum_{k}(w_{k}c_{k}-w_{k}m_{k})$ by Elementary Properties of Series of Real Numbers §linearity.

By Elementary Properties of Series of Real Numbers §linearity, the termwise sum of the four series in (i) to (iv) converges with sum $R(\nu)$. Its $k$-th term is

$$m_{k}\Bigl(\frac{\theta a_{k}}{2}\,b_{k}^{2}+\Bigl(\frac{\lambda_{0}}{2}+\frac{\beta a_{k}}{c_{k}}\Bigr)b_{k}-w_{k}\Bigr)-\bigl(\beta a_{k}b_{k}-w_{k}c_{k}\bigr)=-\bigl(\beta a_{k}b_{k}-w_{k}c_{k}\bigr),$$

the bracket vanishing by clause 1. On the other hand the series $\sum_{k}(\beta a_{k}b_{k}-w_{k}c_{k})$ converges with sum $e$ (clause 4), so $\sum_{k}\bigl(-(\beta a_{k}b_{k}-w_{k}c_{k})\bigr)=-e$ by Elementary Properties of Series of Real Numbers §linearity. Hence $R(\nu)=-e$, and (9) becomes

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

By The Hamilton-Jacobi Equation with Gaussian Score Drift on a Hilbert Space §operator, read with $c'$ in place of $c$ (a variance sequence with $c'_{k}\le\kappa'a_{k}$, by clause 2), the pair $(\mathcal{D}',\mathcal{D}'_{\Sigma},\mathcal{E}',\Sigma')$, temperature $\beta$, discount $\lambda_{0}$, control cost $\theta$ and running cost $\nu\mapsto g(\nu)+e$ on $\mathcal{D}'=\mathcal{D}$, whose score field at $\nu$ is $Z'$, the right-hand side is $F'(\nu,r,q)$.

Clause 7. By clause 2, $c'$ is a variance sequence and $\kappa'$ is positive with $c'_{k}\le\kappa'a_{k}$ for every $k$, so The Gaussian Entropy Pair is a Noise Penalty Pair, with Nonnegative Penalty and Dense Score Domain, read with $c'$ in place of $c$ and $\kappa'$ in place of $\kappa$, applies to the pair $P'=(\mathcal{D}',\mathcal{D}'_{\Sigma},\mathcal{E}',\Sigma')$ of clause 5. By The Gaussian Entropy Pair is a Noise Penalty Pair, with Nonnegative Penalty and Dense Score Domain §pair, $P'$ is a noise penalty pair on $\mathcal{P}^{a}_{\gamma_{c'}}$, and by The Gaussian Entropy Pair is a Noise Penalty Pair, with Nonnegative Penalty and Dense Score Domain §nonnegative, $0\le\mathcal{E}'(\mu)$ for every $\mu\in\mathcal{D}'$, that is for every $\mu\in\mathcal{D}$, as $\mathcal{D}'=\mathcal{D}$ (clause 5). We check the requirements of Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair for $P'$ on $\mathcal{P}^{a}_{\rho}$, $\rho=\gamma_{c}$. By clause 5 (that is, by Changing the Variances of a Diagonal Gaussian Reference by a Diagonal Quadratic: the Same Measures and Domains, with Penalty and Score Shifted by a Diagonal Quadratic Profile §space), $\mathcal{P}^{a}_{\gamma_{c'}}=\mathcal{P}^{a}_{\rho}$, so $\mathcal{D}'_{\Sigma}\subseteq\mathcal{D}'\subseteq\mathcal{P}^{a}_{\rho}$. The requirement $\Sigma'(\mu)\in T^{a}_{\mu}$ for $\mu\in\mathcal{D}'_{\Sigma}$ and the conditions Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §nonempty, Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §variation and Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §dense involve only the sets $\mathcal{D}'_{\Sigma}\subseteq\mathcal{D}'$, the functions $\mathcal{E}'$ and $\Sigma'$, the spaces $L^{2}(\mu;X^{a})$ of Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields and the noise tangent spaces $T^{a}_{\mu}$ of The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §tangent, which are defined for every $\mu\in\mathcal{P}(X)$, the noise gradients $\nabla_{a}\psi$ of bounded $C^{2}$ cylindrical functions (The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient), which are maps on $X$, push-forwards of members of $\mathcal{D}'_{\Sigma}$, and the values $W_{a}(\nu,\mu)$ of the metric $W_{a}$ for members $\nu,\mu$ of $\mathcal{D}'$, which The Noise Wasserstein Distance §distance defines from $\mu$, $\nu$ and their couplings of finite noise cost; none of them refers to the reference measure except through the set $\mathcal{P}^{a}_{\gamma_{c'}}=\mathcal{P}^{a}_{\rho}$. So these requirements hold for $P'$ on $\mathcal{P}^{a}_{\rho}$, because they hold for $P'$ on $\mathcal{P}^{a}_{\gamma_{c'}}$. Finally, condition Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §bound relative to $\rho$ holds with the nonnegative constant $C=0$: for every $\mu\in\mathcal{D}'$, $W_{a}(\mu,\rho)$ is a real number, as $\mu\in\mathcal{P}^{a}_{\rho}$, and $-0\cdot\bigl(1+W_{a}(\mu,\rho)^{2}\bigr)=0\le\mathcal{E}'(\mu)$. Hence $P'$ is a noise penalty pair on $\mathcal{P}^{a}_{\rho}$ in the setting read with $\rho=\gamma_{c}$.