Proof 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

Each result cited is universally quantified over the data in its own statement.

Throughout, $B\ge0$ is a bound for the sequence $d$, which is admissible: the series $\sum_{k=1}^{\infty}|d_{k}|\,c_{k}$ converges and $|d_{k}|\,a_{k}\le B$ for every $k\in\mathbb{N}$. Since $c_{k}^{-1}+d_{k}$ is positive, $c'_{k}$ is a positive real number and

$$\frac{1}{c'_{k}}=\frac{1}{c_{k}}+d_{k}\qquad(k\in\mathbb{N}).\tag{0}$$

Put $r_{k}=c'_{k}/c_{k}$, a positive real number. For $\mu\in\mathcal{P}(X)$ and $n\in\mathbb{N}$ write $\mu_{n}=(p_{n})_{\#}\mu$, a Borel probability measure on $\mathbb{R}^{n}$ by Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §pushforward.

Step 0 (Coordinate second moments). Let $\mu\in\mathcal{P}_{2}(X)$ and $k\in\mathbb{N}$. By Borel Sets of a Hilbert Space with an Orthonormal Basis: Coordinates, Determination by Finite-Dimensional Projections, and Pairs §continuity the function $x\mapsto x_{k}$ is Borel, hence so is $x\mapsto x_{k}^{2}$ by the arithmetic of measurable maps; and by the same claim, applied with $n=k$,

$$x_{k}^{2}\le\sum_{j=1}^{k}x_{j}^{2}=\lVert p_{k}(x)\rVert^{2}=|P_{k}x|^{2}\le|P_{k}x|^{2}+|Q_{k}x|^{2}=|x|^{2}\qquad(x\in X).$$

By monotonicity of the integral, Linearity and Monotonicity of the Lebesgue Integral §nonnegative, the integral of $x\mapsto x_{k}^{2}$ against $\mu$ is at most the second moment $M_{2}(\mu)$, which is finite by The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §space. So $x\mapsto x_{k}^{2}$ is integrable with respect to $\mu$ by the criterion recorded in Measure Spaces and the Lebesgue Integral: Standing Notation §integral, and we write

$$m_{k}(\mu)=\int_{X}x_{k}^{2}\,\mu(dx),$$

a nonnegative real number. Consequently $x\mapsto x_{k}$ is $2$-integrable with respect to $\mu$ in the sense of Power-Integrable Functions and the p-Seminorm §space; its class in $L^{2}(\mu)$ is again written $x_{k}$, and by The Lebesgue Space of Square-Integrable Functions is a Real Hilbert Space §inner-product

$$\lVert x_{k}\rVert_{L^{2}(\mu)}^{2}=m_{k}(\mu),\qquad\langle x_{k},\varphi\rangle_{L^{2}(\mu)}=\int_{X}x_{k}\,\varphi(x)\,\mu(dx)\quad\text{for every }\varphi\in\mathcal{F}C^{1}_{b}(X),$$

the classes of bounded $C^{1}$ cylindrical functions lying in $L^{2}(\mu)$ by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §square-integrable.

Step 1 (Claim 1). The series $\sum_{k}|d_{k}|c_{k}$ converges, so its terms tend to $0$ by Elementary Properties of Series of Real Numbers §terms-vanish; by the definition of convergence of a real sequence there is $K\in\mathbb{N}$ with $|d_{k}|\,c_{k}\le\frac12$ for every $k\ge K$. For such $k$, by (0),

$$\frac{1}{c'_{k}}\ge\frac{1}{c_{k}}-|d_{k}|=\frac{1-|d_{k}|c_{k}}{c_{k}}\ge\frac{1}{2c_{k}},\qquad\text{so}\qquad c'_{k}\le2c_{k}\qquad(k\ge K).\tag{1}$$

Let $n\in\mathbb{N}$. Splitting the partial sums at $K$ and using (1), the nonnegativity of the terms, and Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates for the convergent series $\sum_{k}c_{k}$ (convergent by Variance Sequences and Their Truncations §variances) and $\sum_{k}|d_{k}|c_{k}$,

$$\sum_{k=1}^{n}c'_{k}\le\sum_{k=1}^{K-1}c'_{k}+2\sum_{k=1}^{\infty}c_{k},\qquad\sum_{k=1}^{n}|d_{k}|\,c'_{k}\le\sum_{k=1}^{K-1}|d_{k}|\,c'_{k}+2\sum_{k=1}^{\infty}|d_{k}|\,c_{k}$$

(an empty sum being $0$). The partial sums of these two series of nonnegative terms are therefore bounded above, and both series converge by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §criterion. In particular $c'$, a sequence of positive real numbers with $\sum_{k}c'_{k}$ convergent, is a variance sequence, which proves the first assertion of claim 1, and

$$\text{the series }\sum_{k=1}^{\infty}|d_{k}|\,c'_{k}\text{ converges.}\tag{2}$$

Put $\kappa'=2\kappa+\sum_{j=1}^{K-1}c'_{j}/a_{j}$, a positive real number since $\kappa$ is positive and the added terms are nonnegative. If $k\ge K$, then $c'_{k}\le2c_{k}\le2\kappa a_{k}\le\kappa'a_{k}$ by (1) and the hypothesis on $\kappa$. If $k<K$, then $c'_{k}=(c'_{k}/a_{k})\,a_{k}\le\bigl(\sum_{j=1}^{K-1}c'_{j}/a_{j}\bigr)a_{k}\le\kappa'a_{k}$, all terms being nonnegative. This proves the second assertion of claim 1.

For the third, let $k\in\mathbb{N}$. By (0), $r_{k}-1=c'_{k}\bigl(c_{k}^{-1}-c'^{-1}_{k}\bigr)=-d_{k}c'_{k}$ and $1-r_{k}^{-1}=c_{k}\bigl(c_{k}^{-1}-c'^{-1}_{k}\bigr)=-d_{k}c_{k}$. The bounds $1-t^{-1}\le\log t\le t-1$ of The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §log, applied with $t=r_{k}$, give $-d_{k}c_{k}\le\log r_{k}\le-d_{k}c'_{k}$, hence

$$|\log r_{k}|\le|d_{k}|\,c_{k}+|d_{k}|\,c'_{k}\qquad(k\in\mathbb{N}).\tag{3}$$

The series $\sum_{k}\bigl(|d_{k}|c_{k}+|d_{k}|c'_{k}\bigr)$ converges by admissibility, (2) and Elementary Properties of Series of Real Numbers §linearity, so by (3) and An Absolutely Convergent Series of Real Numbers Converges §dominated the series $\sum_{k}\log(c'_{k}/c_{k})=\sum_{k}\log r_{k}$ converges absolutely. This proves claim 1. We fix $\kappa'$ as above (the sets $\mathcal{D}'$ and $\mathcal{D}'_{\Sigma}$ do not depend on this choice, as the statement records), and note that $\sum_{k}\log r_{k}$ converges by An Absolutely Convergent Series of Real Numbers Converges §convergence.

Since $c$ and $c'$ are variance sequences, for every $n\in\mathbb{N}$ the truncations $c^{(n)}$ and $c'^{(n)}$ are variance vectors by Variance Sequences and Their Truncations §truncations, and by Diagonal Gaussian Measures on a Hilbert Space §measure

$$(p_{n})_{\#}\gamma_{c}=\gamma_{c^{(n)}},\qquad(p_{n})_{\#}\gamma_{c'}=\gamma_{c'^{(n)}}\qquad(n\in\mathbb{N}).\tag{4}$$

Step 2 (The finite-dimensional comparison). For $\mu\in\mathcal{P}_{2}(X)$ and $n\in\mathbb{N}$ put

$$A_{n}(\mu)=\frac12\sum_{k=1}^{n}d_{k}\,m_{k}(\mu)+\frac12\sum_{k=1}^{n}\log r_{k}.$$

We show: $\mu_{n}$ has finite relative entropy with respect to $\gamma_{c^{(n)}}$ if and only if it has finite relative entropy with respect to $\gamma_{c'^{(n)}}$, and in that case

$$H(\mu_{n}\,|\,\gamma_{c'^{(n)}})=H(\mu_{n}\,|\,\gamma_{c^{(n)}})+A_{n}(\mu).\tag{5}$$

By The Wasserstein Distance on a Hilbert Space as the Increasing Limit of the Euclidean Wasserstein Distances of the Coordinate Projections §projected, $\mu_{n}\in\mathcal{P}_{2}(\mathbb{R}^{n})$, so Changing the Variances of a Diagonal Gaussian Reference Measure: Equal Domains, the Shift of the Relative Entropy and the Shift of the Relative Score applies with $n$ in place of the dimension $d$ there (as in A Diagonal Gaussian Reference Measure on the Noise Wasserstein Space, Rescaled Heads and Gaussian Tails: Standing Notation §euclidean), the variance vectors $c^{(n)}$ and $c'^{(n)}$, and the measure $\mu_{n}$. The equivalence is 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. Suppose now that $\mu_{n}$ has finite relative entropy with respect to one, hence both, of the two measures. 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 §entropy, the function $g(u)=|u|^{2}_{c'^{(n)}}-|u|^{2}_{c^{(n)}}$ is integrable with respect to $\mu_{n}$ and

$$H(\mu_{n}\,|\,\gamma_{c'^{(n)}})=H(\mu_{n}\,|\,\gamma_{c^{(n)}})+\frac12\int_{\mathbb{R}^{n}}g\,d\mu_{n}+Z_{c'^{(n)}}-Z_{c^{(n)}},$$

with the weighted squares and normalizing constants of The Diagonal Gaussian Density on Euclidean Space and Its Notation §scaling and The Diagonal Gaussian Density on Euclidean Space and Its Notation §density. We evaluate the last two terms.

By the definition of the weighted square and (0), $g(u)=\sum_{k=1}^{n}u_{k}^{2}\bigl(c'^{-1}_{k}-c^{-1}_{k}\bigr)=\sum_{k=1}^{n}d_{k}u_{k}^{2}$, so $(g\circ p_{n})(x)=\sum_{k=1}^{n}d_{k}x_{k}^{2}$ for $x\in X$. By Step 0 and the linearity of the integral, Linearity and Monotonicity of the Lebesgue Integral §integrable, $g\circ p_{n}$ is integrable with respect to $\mu$ with integral $\sum_{k=1}^{n}d_{k}m_{k}(\mu)$; and by the change of variables formula (claim 2 of Image Measures, Measures with Densities, and Change of Variables), $\int_{\mathbb{R}^{n}}g\,d\mu_{n}=\int_{X}g\circ p_{n}\,d\mu=\sum_{k=1}^{n}d_{k}m_{k}(\mu)$.

Write $\kappa_{0}$ for the positive constant written $\kappa$ in The Diagonal Gaussian Density on Euclidean Space and Its Notation, so that $Z_{c'^{(n)}}-Z_{c^{(n)}}=\sum_{k=1}^{n}\bigl(\log(\kappa_{0}\sqrt{c'_{k}})-\log(\kappa_{0}\sqrt{c_{k}})\bigr)$. Fix $k\le n$ and let $s_{k}=\sqrt{r_{k}}$, the nonnegative square root of Existence and Uniqueness of the Nonnegative Square Root; $s_{k}$ is positive since $s_{k}^{2}=r_{k}>0$. The number $\sqrt{c_{k}}\,s_{k}$ is nonnegative with square $c_{k}r_{k}=c'_{k}$, so it equals $\sqrt{c'_{k}}$ by the uniqueness in Existence and Uniqueness of the Nonnegative Square Root. By the product rule $\log(st)=\log s+\log t$ of The Natural Logarithm,

$$\log(\kappa_{0}\sqrt{c'_{k}})-\log(\kappa_{0}\sqrt{c_{k}})=\log s_{k}=\frac12\bigl(\log s_{k}+\log s_{k}\bigr)=\frac12\log(s_{k}s_{k})=\frac12\log r_{k}.$$

Summing over $k\le n$, $Z_{c'^{(n)}}-Z_{c^{(n)}}=\frac12\sum_{k=1}^{n}\log r_{k}$. Inserting both evaluations gives (5).

Step 3 (Transfer to $X$). Let $\mu\in\mathcal{P}_{2}(X)$ be such that the series $\sum_{k}d_{k}m_{k}(\mu)$ converges absolutely. We show: $\mu$ has finite relative entropy with respect to $\gamma_{c}$ if and only if it has finite relative entropy with respect to $\gamma_{c'}$, and in that case

$$H(\mu\,|\,\gamma_{c'})=H(\mu\,|\,\gamma_{c})+\frac12\sum_{k=1}^{\infty}d_{k}m_{k}(\mu)+\frac12\sum_{k=1}^{\infty}\log r_{k}.\tag{6}$$

Both series converge by An Absolutely Convergent Series of Real Numbers Converges §convergence (the second by Step 1); let $T(\mu)$ denote the right-hand correction $\frac12\sum_{k}d_{k}m_{k}(\mu)+\frac12\sum_{k}\log r_{k}$ and put $L(\mu)=\frac12\sum_{k}|d_{k}|m_{k}(\mu)+\frac12\sum_{k}|\log r_{k}|$, a real number. For every $n$, the triangle inequality for finite sums and Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates give $|A_{n}(\mu)|\le L(\mu)$, and $A_{n}(\mu)\to T(\mu)$ as $n\to\infty$, being half the sum of the partial sums of two convergent series (Series of Real Numbers §convergent).

We apply Relative Entropy of the Finite-Dimensional Projections of Borel Probability Measures on a Hilbert Space: Monotonicity and Approximation with this $\mu$ and with $\gamma=\gamma_{c}$, respectively $\gamma=\gamma_{c'}$; by (4) the projected reference measures there are $\gamma_{c^{(n)}}$, respectively $\gamma_{c'^{(n)}}$.

Suppose $\mu$ has finite relative entropy with respect to $\gamma_{c}$. By Relative Entropy of the Finite-Dimensional Projections of Borel Probability Measures on a Hilbert Space: Monotonicity and Approximation §projections, for every $n$ the measure $\mu_{n}$ has finite relative entropy with respect to $\gamma_{c^{(n)}}$, with $H(\mu_{n}\,|\,\gamma_{c^{(n)}})\le H(\mu\,|\,\gamma_{c})$. By Step 2, $\mu_{n}$ has finite relative entropy with respect to $\gamma_{c'^{(n)}}$ and $H(\mu_{n}\,|\,\gamma_{c'^{(n)}})=H(\mu_{n}\,|\,\gamma_{c^{(n)}})+A_{n}(\mu)\le H(\mu\,|\,\gamma_{c})+L(\mu)$. By Relative Entropy of the Finite-Dimensional Projections of Borel Probability Measures on a Hilbert Space: Monotonicity and Approximation §bounded, $\mu$ has finite relative entropy with respect to $\gamma_{c'}$.

Conversely, suppose $\mu$ has finite relative entropy with respect to $\gamma_{c'}$. By Relative Entropy of the Finite-Dimensional Projections of Borel Probability Measures on a Hilbert Space: Monotonicity and Approximation §projections and Step 2, for every $n$ the measure $\mu_{n}$ has finite relative entropy with respect to $\gamma_{c^{(n)}}$, and $H(\mu_{n}\,|\,\gamma_{c^{(n)}})=H(\mu_{n}\,|\,\gamma_{c'^{(n)}})-A_{n}(\mu)\le H(\mu\,|\,\gamma_{c'})+L(\mu)$. By Relative Entropy of the Finite-Dimensional Projections of Borel Probability Measures on a Hilbert Space: Monotonicity and Approximation §bounded, $\mu$ has finite relative entropy with respect to $\gamma_{c}$.

Finally, when $\mu$ has finite relative entropy with respect to both, Relative Entropy of the Finite-Dimensional Projections of Borel Probability Measures on a Hilbert Space: Monotonicity and Approximation §limit gives $H(\mu_{n}\,|\,\gamma_{c^{(n)}})\to H(\mu\,|\,\gamma_{c})$ and $H(\mu_{n}\,|\,\gamma_{c'^{(n)}})\to H(\mu\,|\,\gamma_{c'})$; letting $n\to\infty$ in (5), with $A_{n}(\mu)\to T(\mu)$, gives (6) by uniqueness of limits.

Step 4 (Claim 2). First, every probability measure $\gamma$ on a measurable space $(S,\mathcal{S})$ has finite relative entropy with respect to itself, with $H(\gamma\,|\,\gamma)=0$. Indeed the constant function $1$ is measurable and nonnegative, and $\int_{S}\mathbf{1}_{A}\cdot1\,d\gamma=\int_{S}\mathbf{1}_{A}\,d\gamma=\gamma(A)$ for $A\in\mathcal{S}$ by The Integral of an Indicator Function is the Measure of the Set, so $1$ is a density of $\gamma$ with respect to $\gamma$ in the sense of The Radon-Nikodym Theorem for a Finite Measure and a Sigma-Finite Measure, and Uniqueness of Densities. With $\phi$ as in Relative Entropy of Probability Measures, $\phi(1)=1\cdot\log1=0$, because $\log1=\log(1\cdot1)=\log1+\log1$ by The Natural Logarithm. So $\phi\circ1$ is the zero function $\mathbf{1}_{\emptyset}$, integrable with integral $\gamma(\emptyset)=0$ by The Integral of an Indicator Function is the Measure of the Set, and the claim follows from Relative Entropy of Probability Measures §relative-entropy.

The measure $\gamma_{c'}$ lies in $\mathcal{P}_{2}(X)$ by Moments of a Diagonal Gaussian Measure on a Hilbert Space: Coordinate Covariances, Finite Second Moment, and Exponential Moments of the Squared Norm §moment, and $m_{k}(\gamma_{c'})=c'_{k}$ by Moments of a Diagonal Gaussian Measure on a Hilbert Space: Coordinate Covariances, Finite Second Moment, and Exponential Moments of the Squared Norm §coordinates (with $j=k$), $c'$ being a variance sequence by Step 1. So $\sum_{k}|d_{k}m_{k}(\gamma_{c'})|=\sum_{k}|d_{k}|c'_{k}$ converges by (2), and Step 3 applies to $\mu=\gamma_{c'}$: as $\gamma_{c'}$ has finite relative entropy with respect to $\gamma_{c'}$, it has finite relative entropy with respect to $\gamma_{c}$. By The Entropy Domain of a Diagonal Gaussian Reference Measure Has the Noise Map Property §inclusion, applied with the variance sequence $c$, the reference measure $\rho=\gamma_{c}$ and the given $\kappa$, we get $\gamma_{c'}\in\mathcal{P}^{a}_{\rho}$.

Likewise $\gamma_{c}\in\mathcal{P}_{2}(X)$ with $m_{k}(\gamma_{c})=c_{k}$ by the same two claims, $\sum_{k}|d_{k}|c_{k}$ converges, and Step 3 applied to $\mu=\gamma_{c}$ shows that $\gamma_{c}$ has finite relative entropy with respect to $\gamma_{c'}$. By The Entropy Domain of a Diagonal Gaussian Reference Measure Has the Noise Map Property §inclusion, applied with the variance sequence $c'$, the reference measure $\rho'=\gamma_{c'}$ and $\kappa'$ (Step 1), $\gamma_{c}\in\mathcal{P}^{a}_{\rho'}$.

Now let $\mu\in\mathcal{P}^{a}_{\rho}$. Since also $\gamma_{c'}\in\mathcal{P}^{a}_{\rho}$, The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §connected (with reference measure $\rho$) shows that $(\mu,\gamma_{c'})=(\mu,\rho')$ is noise-connected, i.e. $\mu\in\mathcal{P}^{a}_{\rho'}$ by The Measures Noise-Connected to the Reference Measure §space read with reference measure $\rho'$. Conversely let $\mu\in\mathcal{P}^{a}_{\rho'}$. Since $\gamma_{c}\in\mathcal{P}^{a}_{\rho'}$, The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §connected with reference measure $\rho'$ (which lies in $\mathcal{P}_{2}(X)$, as shown above) shows that $(\mu,\gamma_{c})=(\mu,\rho)$ is noise-connected, i.e. $\mu\in\mathcal{P}^{a}_{\rho}$ by The Measures Noise-Connected to the Reference Measure §space. Hence $\mathcal{P}^{a}_{\rho'}=\mathcal{P}^{a}_{\rho}$, proving claim 2.

Step 5 (Claim 4, and $\mathcal{D}'=\mathcal{D}$). Let $\mu\in\mathcal{P}^{a}_{\rho}$. Then $\mu\in\mathcal{P}_{2}(X)$ by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §moments, and by Diagonal Quadratic Profiles on the Noise Wasserstein Space: Moment Bounds, a Lipschitz Estimate, the Gradient Field, and Noise Intrinsic Test Functions §profile (with $b=d$) the series $\sum_{k}d_{k}m_{k}(\mu)$ converges absolutely, with $\frac12\sum_{k}d_{k}m_{k}(\mu)=\Phi_{d}(\mu)$. So Step 3 applies to $\mu$: a member $\mu$ of $\mathcal{P}^{a}_{\rho}$ has finite relative entropy with respect to $\gamma_{c}$ if and only if it has finite relative entropy with respect to $\gamma_{c'}$, and then

$$H(\mu\,|\,\gamma_{c'})=H(\mu\,|\,\gamma_{c})+\Phi_{d}(\mu)+\frac12\sum_{k=1}^{\infty}\log\frac{c'_{k}}{c_{k}}.\tag{7}$$

By The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §penalty-domain, $\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$ (with $\kappa$); read relative to $\gamma_{c'}$ (with $\kappa'$), the same clause gives $\mathcal{D}'\subseteq\mathcal{P}^{a}_{\rho'}$, which equals $\mathcal{P}^{a}_{\rho}$ by Step 4. Hence every member of $\mathcal{D}$ lies in $\mathcal{P}^{a}_{\rho}$ and has finite relative entropy with respect to $\gamma_{c'}$, so lies in $\mathcal{D}'$; and every member of $\mathcal{D}'$ lies in $\mathcal{P}^{a}_{\rho}$ and has finite relative entropy with respect to $\gamma_{c}$, so lies in $\mathcal{D}$. Thus $\mathcal{D}'=\mathcal{D}$, the first half of claim 3. For $\mu\in\mathcal{D}$, (7) is the first identity of claim 4; multiplying it by $\beta$ and using $\mathcal{E}(\mu)=\beta H(\mu\,|\,\gamma_{c})$ and $\mathcal{E}'(\mu)=\beta H(\mu\,|\,\gamma_{c'})$ from The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §pair (the latter defined since $\mu\in\mathcal{D}'$) gives the second. This proves claim 4.

Step 6 (The score: $\mathcal{D}'_{\Sigma}=\mathcal{D}_{\Sigma}$ and claim 5). Let $\nu\in\mathcal{D}=\mathcal{D}'$; then $\nu\in\mathcal{P}^{a}_{\rho}\subseteq\mathcal{P}_{2}(X)$ by Step 5 and The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §moments, and the classes $x_{k}\in L^{2}(\nu)$ of Step 0 are available. Two facts are used.

(a) Shifting a score. Let $k\in\mathbb{N}$, $g\in L^{2}(\nu)$ and $\varphi\in\mathcal{F}C^{1}_{b}(X)$. By bilinearity of the inner product, Step 0, (0) and the linearity of the integral, Linearity and Monotonicity of the Lebesgue Integral §integrable (the functions involved being integrable with respect to $\nu$ as recorded in The Relative Score with Respect to a Diagonal Gaussian Measure on a Hilbert Space),

$$\langle g+d_{k}x_{k},\varphi\rangle_{L^{2}(\nu)}-\int_{X}\Bigl(\frac{x_{k}}{c'_{k}}\varphi(x)-\partial_{k}\varphi(x)\Bigr)\nu(dx)=\langle g,\varphi\rangle_{L^{2}(\nu)}-\int_{X}\Bigl(\frac{x_{k}}{c_{k}}\varphi(x)-\partial_{k}\varphi(x)\Bigr)\nu(dx),$$

since $d_{k}\int_{X}x_{k}\varphi(x)\,\nu(dx)-\int_{X}x_{k}\varphi(x)\bigl(c'^{-1}_{k}-c^{-1}_{k}\bigr)\nu(dx)=0$. Hence, by The Relative Score with Respect to a Diagonal Gaussian Measure on a Hilbert Space §score: if $(\zeta_{k})_{k}$ is the relative score of $\nu$ with respect to $\gamma_{c}$, then $(\zeta_{k}+d_{k}x_{k})_{k}$ is the relative score of $\nu$ with respect to $\gamma_{c'}$ (take $g=\zeta_{k}$); and if $(\zeta'_{k})_{k}$ is the relative score of $\nu$ with respect to $\gamma_{c'}$, then $(\zeta'_{k}-d_{k}x_{k})_{k}$ is the relative score of $\nu$ with respect to $\gamma_{c}$ (take $g=\zeta'_{k}-d_{k}x_{k}$). The relative score is unique when it exists, as recorded in that definition.

(b) The Fisher bound. For $u,v$ in a real inner product space, $\lVert u+v\rVert^{2}=\lVert u\rVert^{2}+2\langle u,v\rangle+\lVert v\rVert^{2}$ and $0\le\lVert u-v\rVert^{2}=\lVert u\rVert^{2}-2\langle u,v\rangle+\lVert v\rVert^{2}$, whence $\lVert u+v\rVert^{2}\le2\lVert u\rVert^{2}+2\lVert v\rVert^{2}$. With Step 0, for every $g\in L^{2}(\nu)$, every $s\in\{1,-1\}$ and every $k$,

$$a_{k}\lVert g+s\,d_{k}x_{k}\rVert_{L^{2}(\nu)}^{2}\le2a_{k}\lVert g\rVert_{L^{2}(\nu)}^{2}+2a_{k}d_{k}^{2}\,m_{k}(\nu).$$

The series $\sum_{k}a_{k}d_{k}^{2}m_{k}(\nu)$ 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 $t_{k}=a_{k}d_{k}^{2}\ge0$ and $M=B^{2}$: indeed $t_{k}a_{k}=(|d_{k}|a_{k})^{2}\le B^{2}$, and $0\le t_{k}c_{k}=(|d_{k}|a_{k})\,|d_{k}|c_{k}\le B\,|d_{k}|c_{k}$, so $\sum_{k}t_{k}c_{k}$ converges by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §comparison and Elementary Properties of Series of Real Numbers §linearity.

Now let $\nu\in\mathcal{D}_{\Sigma}$, with relative score $(\zeta_{k})_{k}$ with respect to $\gamma_{c}$ and finite Fisher information relative to $\gamma_{c}$ with weights $a$, i.e. $\sum_{k}a_{k}\lVert\zeta_{k}\rVert_{L^{2}(\nu)}^{2}$ converges (Weight Sequences and the Weighted Fisher Information Relative to a Diagonal Gaussian Measure on a Hilbert Space §information). By (a), $\nu$ has the relative score $(\zeta_{k}+d_{k}x_{k})_{k}$ with respect to $\gamma_{c'}$; by (b) with $g=\zeta_{k}$, $s=1$, Elementary Properties of Series of Real Numbers §linearity and Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §comparison, the series $\sum_{k}a_{k}\lVert\zeta_{k}+d_{k}x_{k}\rVert_{L^{2}(\nu)}^{2}$ converges, so $\nu$ has finite Fisher information relative to $\gamma_{c'}$ with weights $a$. As $\nu\in\mathcal{D}=\mathcal{D}'$, The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §score-domain (read relative to $\gamma_{c'}$) gives $\nu\in\mathcal{D}'_{\Sigma}$. Conversely, let $\nu\in\mathcal{D}'_{\Sigma}$, with relative score $(\zeta'_{k})_{k}$ with respect to $\gamma_{c'}$ and $\sum_{k}a_{k}\lVert\zeta'_{k}\rVert_{L^{2}(\nu)}^{2}$ convergent. By (a), $(\zeta'_{k}-d_{k}x_{k})_{k}$ is the relative score of $\nu$ with respect to $\gamma_{c}$, and by (b) with $g=\zeta'_{k}$, $s=-1$, the series $\sum_{k}a_{k}\lVert\zeta'_{k}-d_{k}x_{k}\rVert_{L^{2}(\nu)}^{2}$ converges; as $\nu\in\mathcal{D}'=\mathcal{D}$, $\nu\in\mathcal{D}_{\Sigma}$ by The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §score-domain. Hence $\mathcal{D}'_{\Sigma}=\mathcal{D}_{\Sigma}$, completing claim 3.

For claim 5, let $\nu\in\mathcal{D}_{\Sigma}$ with relative score $(\zeta_{k})_{k}$ with respect to $\gamma_{c}$. By the preceding paragraph its relative score with respect to $\gamma_{c'}$ is $(\zeta_{k}+d_{k}x_{k})_{k}$, the first assertion. By The Noise Score Field of a Measure of Finite Weighted Fisher Information: Existence, Norm, Pairing with Noise Gradients, Head Approximation and Tangency §field, read with $c'$ in place of $c$, $Z'^{a}_{\nu}$ is the unique element of $L^{2}(\nu;X^{a})$ whose coordinate along $f_{k}$ is $a_{k}^{1/2}(\zeta_{k}+d_{k}x_{k})$ for every $k$. On the other hand, by The Noise Score Field of a Measure of Finite Weighted Fisher Information: Existence, Norm, Pairing with Noise Gradients, Head Approximation and Tangency §field the coordinate of $Z^{a}_{\nu}$ along $f_{k}$ is $a_{k}^{1/2}\zeta_{k}$, and by Diagonal Quadratic Profiles on the Noise Wasserstein Space: Moment Bounds, a Lipschitz Estimate, the Gradient Field, and Noise Intrinsic Test Functions §field (with $b=d$, $\nu\in\mathcal{P}^{a}_{\rho}$) that of $V_{d}(\nu)$ is the class of $x\mapsto a_{k}^{1/2}d_{k}x_{k}$, that is $a_{k}^{1/2}d_{k}x_{k}$; by the additivity of coordinates in The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §coordinates, the coordinate of $Z^{a}_{\nu}+V_{d}(\nu)$ along $f_{k}$ is $a_{k}^{1/2}\zeta_{k}+a_{k}^{1/2}d_{k}x_{k}=a_{k}^{1/2}(\zeta_{k}+d_{k}x_{k})$. By the uniqueness just quoted, $Z'^{a}_{\nu}=Z^{a}_{\nu}+V_{d}(\nu)$. Finally, by The Gaussian Entropy Pair on the Noise Wasserstein Space: Relative Entropy and the Noise Score Field §pair and the vector space axioms of $L^{2}(\nu;X^{a})$ (The Space of Square-Integrable Hilbert-Valued Maps is a Real Hilbert Space: Coordinates and Synthesis §hilbert),

$$\Sigma'(\nu)=\beta Z'^{a}_{\nu}=\beta Z^{a}_{\nu}+\beta V_{d}(\nu)=\Sigma(\nu)+\beta V_{d}(\nu).$$

This proves claim 5 and completes the proof.