Existence, Penalty Bounds and the Least Penalty at a Maximiser of the Wasserstein-Doubled Difference

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a Wasserstein-coercive penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$. The set $\mathcal{D}$ contains the nonempty $\mathcal{D}_{\Sigma}$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §pair and Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §nonempty; fix $\mu_{0}\in\mathcal{D}$, 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 Wasserstein-Coercive Penalty Pair §bounded-below.

Let $\delta\in\mathbb{R}$ be positive and let $u,v:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ be such that $u$ is upper semicontinuous and $v$ lower semicontinuous on $\mathcal{P}_{2}(\mathbb{R}^{d})$ relative to $\mathcal{P}_{2}(\mathbb{R}^{d})$ in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$, and such that $u(\sigma)\le b$ and $b'\le v(\sigma)$ for every $\sigma\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and some $b,b'\in\mathbb{R}$. By Basic Properties of a Wasserstein-Coercive Penalty Pair §envelopes the $\delta$-envelope $u^{-}_{\delta}$ of $u$ and the $\delta$-envelope $v^{+}_{\delta}$ of $v$ relative to the penalty pair are defined and satisfy $u^{-}_{\delta}=u-\delta\mathcal{E}$ and $v^{+}_{\delta}=v+\delta\mathcal{E}$ on $\mathcal{D}$.

For positive $\alpha\in\mathbb{R}$ let $\Psi_{\alpha}:\mathcal{D}\times\mathcal{D}\to\mathbb{R}$ be the function with value

let $M(\alpha)$ be the supremum of its values, and put

Let $2^{n}$ be the natural power; the letter $k$, a dimension in Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, is not used here. Then the following hold.

1. (A maximiser exists)¶ For every positive $\alpha\in\mathbb{R}$ the supremum $M(\alpha)$ is a real number and there is $(\hat{\mu},\hat{\nu})\in\mathcal{D}\times\mathcal{D}$ with $\Psi_{\alpha}(\hat{\mu},\hat{\nu})=M(\alpha)$.

2. (The penalty is bounded at a near-maximiser)¶ Let $\alpha\in\mathbb{R}$ be positive, let $\eta\in\mathbb{R}$ be nonnegative and let $(\hat{\mu},\hat{\nu})\in\mathcal{D}\times\mathcal{D}$ satisfy $\Psi_{\alpha}(\mu_{0},\mu_{0})-\eta\le\Psi_{\alpha}(\hat{\mu},\hat{\nu})$. Then $\mathcal{E}(\hat{\mu})\le c_{0}+\delta^{-1}\eta$ and $\mathcal{E}(\hat{\nu})\le c_{0}+\delta^{-1}\eta$, and if $R\in\mathbb{R}$ is as in Basic Properties of a Wasserstein-Coercive Penalty Pair §moment for the level $c=c_{0}+\delta^{-1}\eta$ then $M_{2}(\hat{\mu})\le R$ and $M_{2}(\hat{\nu})\le R$. Neither bound depends on $\alpha$.

3. (The penalised distance vanishes along a doubling sequence)¶ Let $\alpha_{0}\in\mathbb{R}$ be positive and let $\alpha_{n}=2^{n}\alpha_{0}$ for $n\in\mathbb{N}$. For each $n$ let $(\hat{\mu}_{n},\hat{\nu}_{n})\in\mathcal{D}\times\mathcal{D}$ satisfy $\Psi_{\alpha_{n}}(\hat{\mu}_{n},\hat{\nu}_{n})=M(\alpha_{n})$. Then the sequence whose $n$-th term is

converges to $0$; in particular so does the sequence whose $n$-th term is $W_{2}(\hat{\mu}_{n},\hat{\nu}_{n})$.

4. (The penalty attains a least value)¶ There is $\mu_{\min}\in\mathcal{D}$ with $\mathcal{E}(\mu_{\min})\le\mathcal{E}(\sigma)$ for every $\sigma\in\mathcal{D}$.