Proof of Basic Properties of a Noise-Closed Noise Penalty Pair: Lower Bound, Lower Semicontinuity, Complete Sublevel Sets and Bounded Distances

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

We work in the setting of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation. Elementary order and arithmetic of real numbers (including squares of nonnegative reals and the Archimedean property) is carried by The Real Numbers: Standing Notation and Background §background, in force through Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §background and Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §background. Throughout, $(\mathcal{P}^{a}_{\rho},W_{a})$ is a metric space 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 §metric, and $W_{a}$ is symmetric on $\mathcal{P}^{a}_{\rho}$ 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 §symmetry. By Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair, $\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$ and $\mathcal{E}:\mathcal{D}\to\mathbb{R}$, so $\mathcal{D}_{c}\subseteq\mathcal{P}^{a}_{\rho}$ for every $c\in\mathbb{R}$.

Claim 1 (bounded below). Step 1. Apply condition 2 of the noise-closed property with level $0$: there is $B_{0}\in\mathbb{R}$ with $W_{a}(\mu,\rho)\le B_{0}$ for every $\mu\in\mathcal{D}$ with $\mathcal{E}(\mu)\le0$. By Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §bound there is a nonnegative $C\in\mathbb{R}$ with $-C\bigl(1+W_{a}(\mu,\rho)^{2}\bigr)\le\mathcal{E}(\mu)$ for every $\mu\in\mathcal{D}$. Put $e_{0}=-C(1+B_{0}^{2})$; since $C\ge0$ and $1+B_{0}^{2}>0$, we have $e_{0}\le0$.

Step 2. Let $\mu\in\mathcal{D}$. If $\mathcal{E}(\mu)\le0$, then $0\le W_{a}(\mu,\rho)\le B_{0}\le|B_{0}|$ by Step 1, so claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, applied to the nonnegative numbers $W_{a}(\mu,\rho)$ and $|B_{0}|$, gives $W_{a}(\mu,\rho)^{2}\le|B_{0}|^{2}=B_{0}^{2}$; hence $1+W_{a}(\mu,\rho)^{2}\le1+B_{0}^{2}$ and, $C$ being nonnegative, $e_{0}=-C(1+B_{0}^{2})\le-C\bigl(1+W_{a}(\mu,\rho)^{2}\bigr)\le\mathcal{E}(\mu)$. If $\mathcal{E}(\mu)>0$, then $e_{0}\le0<\mathcal{E}(\mu)$. In both cases $e_{0}\le\mathcal{E}(\mu)$.

Claim 2 (lower semicontinuity). We verify Lower Semicontinuous Function on a Subset of a Metric Space for the metric space $(\mathcal{P}^{a}_{\rho},W_{a})$, the subset $A=\mathcal{D}$ and $u=\mathcal{E}$. Let $x\in\mathcal{D}$ and let $\varepsilon\in\mathbb{R}$ be positive; we argue by contradiction and suppose that no positive $\delta$ has the required property.

Step 1. Then for every $n\in\mathbb{N}$, applying the supposition to $\delta=\tfrac1n$ (positive, as $n$ is positive in $\mathbb{R}$ by The Real Numbers: Standing Notation and Background §numbers), there is $y_{n}\in\mathcal{D}$ with $W_{a}(x,y_{n})<\tfrac1n$ and $\mathcal{E}(y_{n})\le\mathcal{E}(x)-\varepsilon$; we choose one such $y_{n}$ for each $n$, obtaining a sequence $(y_{n})_{n\in\mathbb{N}}$ in $\mathcal{D}$.

Step 2. The sequence $(y_{n})$ converges to $x$ in $(\mathcal{P}^{a}_{\rho},W_{a})$ in the sense of Convergent Sequence in a Metric Space: given a positive $\varepsilon'$, claim 3 of The Archimedean Property of the Real Numbers gives $N\in\mathbb{N}$ with $0<\tfrac1N<\varepsilon'$, and for every $n\ge N$ we have $\tfrac1n\le\tfrac1N$, hence $W_{a}(y_{n},x)=W_{a}(x,y_{n})<\tfrac1n<\varepsilon'$ by symmetry of $W_{a}$.

Step 3. Put $c=\mathcal{E}(x)-\varepsilon$. Every $y_{n}$ lies in $\mathcal{D}$ with $\mathcal{E}(y_{n})\le c$, and $(y_{n})$ converges to $x\in\mathcal{P}^{a}_{\rho}$, so condition 1 of the noise-closed property at level $c$ gives $\mathcal{E}(x)\le c=\mathcal{E}(x)-\varepsilon$, that is $\varepsilon\le0$, contradicting $0<\varepsilon$. Hence some positive $\delta$ has the property that every $y\in\mathcal{D}$ with $W_{a}(x,y)<\delta$ satisfies $\mathcal{E}(x)-\varepsilon<\mathcal{E}(y)$. As $x\in\mathcal{D}$ and $\varepsilon$ were arbitrary, $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$.

Claim 3 (complete sublevel sets). Fix $c\in\mathbb{R}$.

Step 1. By claim 1 of The Restriction of a Metric to a Subset Induces the Subspace Topology, applied to the metric space $(\mathcal{P}^{a}_{\rho},W_{a})$ and the subset $\mathcal{D}_{c}$, the restriction of $W_{a}$ to $\mathcal{D}_{c}\times\mathcal{D}_{c}$ is a metric on $\mathcal{D}_{c}$, so $(\mathcal{D}_{c},W_{a})$ is a metric space.

Step 2. Let $(\mu_{n})_{n\in\mathbb{N}}$ be a Cauchy sequence in $(\mathcal{D}_{c},W_{a})$. Since the restricted metric takes the same values $W_{a}(\mu_{m},\mu_{\ell})$, the sequence is also a Cauchy sequence in $(\mathcal{P}^{a}_{\rho},W_{a})$, so by Completeness of the Quadratic Wasserstein Space and of the Noise Wasserstein Space over a Hilbert Space §noise-complete it converges in $(\mathcal{P}^{a}_{\rho},W_{a})$ to some $\mu\in\mathcal{P}^{a}_{\rho}$.

Step 3. Each $\mu_{n}$ lies in $\mathcal{D}$ with $\mathcal{E}(\mu_{n})\le c$, so condition 1 of the noise-closed property gives $\mu\in\mathcal{D}$ and $\mathcal{E}(\mu)\le c$, that is $\mu\in\mathcal{D}_{c}$. The distances $W_{a}(\mu_{n},\mu)$ are the same in $(\mathcal{D}_{c},W_{a})$, so by Convergent Sequence in a Metric Space the sequence converges to the point $\mu$ of $\mathcal{D}_{c}$ in $(\mathcal{D}_{c},W_{a})$. Hence $(\mathcal{D}_{c},W_{a})$ is complete.

Claim 4 (bounded distances). Let $c,B$ be as in the claim and let $\mu,\nu\in\mathcal{D}_{c}$. The three measures $\mu$, $\rho$ and $\nu$ belong to $\mathcal{P}^{a}_{\rho}$, the first and last because $\mathcal{D}_{c}\subseteq\mathcal{P}^{a}_{\rho}$ and $\rho$ 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 §reference. The triangle inequality 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 §triangle, applied with $\mu,\rho,\nu$ in place of its $\mu,\nu,\lambda$, and the symmetry of $W_{a}$ give

$$W_{a}(\mu,\nu)\le W_{a}(\mu,\rho)+W_{a}(\rho,\nu)=W_{a}(\mu,\rho)+W_{a}(\nu,\rho)\le B+B=2B,$$

the last step by the hypothesis on $B$, applied to $\mu$ and to $\nu$. $\blacksquare$