Bounds, Monotonicity and Borwein-Preiss Perturbed Maximisers of the Doubled Difference in the Noise Wasserstein Distance for a Noise-Closed Penalty Pair

In the setting of First-Order Equations on the Noise Wasserstein Space Relative to a Noise Penalty Pair: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a noise-closed noise penalty pair on $\mathcal{P}^{a}_{\rho}$, and fix $e_{0}\in\mathbb{R}$ with $e_{0}\le\mathcal{E}(\mu)$ for every $\mu\in\mathcal{D}$, as provided by Basic Properties of a Noise-Closed Noise Penalty Pair: Lower Bound, Lower Semicontinuity, Complete Sublevel Sets and Bounded Distances §bounded-below. The reference measure $\rho$ belongs to $\mathcal{P}^{a}_{\rho}$ 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 §reference, and $W_{a}$ is a metric 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 §metric, so that $W_{a}(\mu,\nu)$ is a nonnegative real number for all $\mu,\nu\in\mathcal{P}^{a}_{\rho}$. Convergent series of real numbers are those of that definition. Let $u,v:\mathcal{D}\to\mathbb{R}$ and $b,b'\in\mathbb{R}$ satisfy $u(\mu)\le b$ and $b'\le v(\mu)$ for every $\mu\in\mathcal{D}$. By Basic Properties of the Delta-Envelopes on the Noise Wasserstein Space, and the Envelopes of Bounded Functions for a Noise-Closed Penalty Pair §growth, $u$ has penalty-subordinate growth from above and $v$ from below, so that for positive $\delta\in\mathbb{R}$ the $\delta$-envelopes $u^{-}_{\delta}$ and $v^{+}_{\delta}$, functions on $\mathcal{D}$, are defined. For positive $\delta,\alpha\in\mathbb{R}$ let $\Psi_{\delta,\alpha}:\mathcal{D}\times\mathcal{D}\to\mathbb{R}$ have the value

$$\Psi_{\delta,\alpha}(\mu,\nu)=u^{-}_{\delta}(\mu)-v^{+}_{\delta}(\nu)-\tfrac{\alpha}{2}\,W_{a}(\mu,\nu)^{2}$$

at $(\mu,\nu)$; the set $\mathcal{D}\times\mathcal{D}$ is nonempty because $\mathcal{D}$ contains $\mathcal{D}_{\Sigma}$ (Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair), which is nonempty by Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §nonempty. Then the following hold.

1. (Bounds)¶ For all positive $\delta,\alpha\in\mathbb{R}$ and all $(\mu,\nu)\in\mathcal{D}\times\mathcal{D}$,

$$\Psi_{\delta,\alpha}(\mu,\nu)\le b-b'-\delta\bigl(\mathcal{E}(\mu)+\mathcal{E}(\nu)\bigr)\le b-b'-2\delta e_{0};$$

so the nonempty set of values of $\Psi_{\delta,\alpha}$ is bounded above, and its supremum, written $M(\delta,\alpha)$ in this and the remaining clauses, is a real number.

2. (Diagonal)¶ For all positive $\delta,\alpha\in\mathbb{R}$ and every $\mu\in\mathcal{D}$, $u(\mu)-v(\mu)-2\delta\,\mathcal{E}(\mu)\le M(\delta,\alpha)$.

3. (Decreasing the penalty weight)¶ Let $\delta,\delta',\alpha,\tau\in\mathbb{R}$ satisfy $0<\delta'<\delta$, $0<\alpha$ and $0\le\tau$, and let $(\mu,\nu)\in\mathcal{D}\times\mathcal{D}$ satisfy $M(\delta,\alpha)-\tau\le\Psi_{\delta,\alpha}(\mu,\nu)$. Then

$$M(\delta,\alpha)-\tau+(\delta-\delta')\bigl(\mathcal{E}(\mu)+\mathcal{E}(\nu)\bigr)\le M(\delta',\alpha).$$

4. (Decreasing the doubling strength)¶ Let $\delta,\alpha,\alpha',\tau\in\mathbb{R}$ satisfy $0<\delta$, $0<\alpha'<\alpha$ and $0\le\tau$, and let $(\mu,\nu)\in\mathcal{D}\times\mathcal{D}$ satisfy $M(\delta,\alpha)-\tau\le\Psi_{\delta,\alpha}(\mu,\nu)$. Then

$$M(\delta,\alpha)-\tau+\tfrac{\alpha-\alpha'}{2}\,W_{a}(\mu,\nu)^{2}\le M(\delta,\alpha').$$

5. (Perturbed maximisers)¶ Let $\delta\in\mathbb{R}$ be positive. ¶There are $K,B\in\mathbb{R}$ with $0\le B$, depending on $\delta$, $u$, $v$, $b$, $b'$ and $e_{0}$ but not on $\alpha$ or $\tau$, with $W_{a}(\sigma,\rho)\le B$ for every $\sigma\in\mathcal{D}$ with $\mathcal{E}(\sigma)\le K$, such that the following holds for all $\alpha,\tau\in\mathbb{R}$ with $0<\alpha$ and $0<\tau<1$. There are $(\hat{\mu},\hat{\nu})\in\mathcal{D}\times\mathcal{D}$, sequences $(\mu_{k})_{k\in\mathbb{N}}$ and $(\nu_{k})_{k\in\mathbb{N}}$ in $\mathcal{D}$, and positive real numbers $c_{k}$ ($k\in\mathbb{N}$) whose series converges with $\sum_{k=1}^{\infty}c_{k}\le\tau$, such that: ¶$\mathcal{E}(\hat{\mu})\le K$, $\mathcal{E}(\hat{\nu})\le K$, and $\mathcal{E}(\mu_{k})\le K$ and $\mathcal{E}(\nu_{k})\le K$ for every $k\in\mathbb{N}$; ¶$M(\delta,\alpha)-\tau\le\Psi_{\delta,\alpha}(\hat{\mu},\hat{\nu})$; ¶for every $(\mu,\nu)\in\mathcal{P}^{a}_{\rho}\times\mathcal{P}^{a}_{\rho}$ the series $\sum_{k=1}^{\infty}c_{k}\bigl(W_{a}(\mu,\mu_{k})^{2}+W_{a}(\nu,\nu_{k})^{2}\bigr)$ converges; and, writing $\Phi(\mu,\nu)$ for $\Psi_{\delta,\alpha}(\mu,\nu)$ minus the sum of that series ($(\mu,\nu)\in\mathcal{D}\times\mathcal{D}$),

$$\Phi(\mu,\nu)<\Phi(\hat{\mu},\hat{\nu})\qquad\text{for every }(\mu,\nu)\in\mathcal{D}\times\mathcal{D}\text{ with }(\mu,\nu)\ne(\hat{\mu},\hat{\nu}).$$