Existence, Penalty Bounds and Optimal Realisation at a Maximiser of the Wasserstein-Doubled Difference on the Lift

In the setting of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation, assume that $(\Omega,\mathcal{F},P)$ is rich, which Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (v) does not assume, and 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 (i) and Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains (ii); 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 (ii).

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 (iv) 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

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

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

$$c_{0}=\delta^{-1}\bigl(b-b'-u(\mu_{0})+v(\mu_{0})+2\delta\,\mathcal{E}(\mu_{0})\bigr)-e_{0}.$$

Write $\mathcal{D}^{\Lambda}$ for the preimage of $\mathcal{D}$ under the law map, and 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)(i) 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)(ii) 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 (i) 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)(iii) 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

$$\alpha_{n}\,W_{2}(\hat{\mu}_{n},\hat{\nu}_{n})^{2}$$

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

4. (Realisation by an optimally coupled pair)(iv) Let $\alpha\in\mathbb{R}$ be positive and let $(\hat{\mu},\hat{\nu})\in\mathcal{D}\times\mathcal{D}$ satisfy $\Psi_{\alpha}(\hat{\mu},\hat{\nu})=M(\alpha)$. Then there are $\hat{X},\hat{Y}\in\mathcal{D}^{\Lambda}$ with $\mathcal{L}(\hat{X})=\hat{\mu}$, $\mathcal{L}(\hat{Y})=\hat{\nu}$ and

$$\lVert\hat{X}-\hat{Y}\rVert_{L^{2}}=W_{2}(\hat{\mu},\hat{\nu}),$$

and for any such pair the law of the pairing $(\hat{X},\hat{Y})$ is an optimal coupling of $\hat{\mu}$ and $\hat{\nu}$ and

$$u^{-}_{\delta}(\mathcal{L}(X))-v^{+}_{\delta}(\mathcal{L}(Y))-\tfrac{\alpha}{2}\lVert X-Y\rVert_{L^{2}}^{2}\ \le\ u^{-}_{\delta}(\hat{\mu})-v^{+}_{\delta}(\hat{\nu})-\tfrac{\alpha}{2}\lVert\hat{X}-\hat{Y}\rVert_{L^{2}}^{2}$$

for all $X,Y\in\mathcal{D}^{\Lambda}$.

5. (The two one-variable extrema)(v) Let $\alpha$, $(\hat{\mu},\hat{\nu})$ and $\hat{X},\hat{Y}$ be as in claim 4. Then the function $\mathcal{D}^{\Lambda}\to\mathbb{R}$ with value $u^{-}_{\delta}(\mathcal{L}(X))-\tfrac{\alpha}{2}\lVert X-\hat{Y}\rVert_{L^{2}}^{2}$ at $X$ attains at $\hat{X}$ a value at least its value at every point of $\mathcal{D}^{\Lambda}$, and the function $\mathcal{D}^{\Lambda}\to\mathbb{R}$ with value $v^{+}_{\delta}(\mathcal{L}(Y))+\tfrac{\alpha}{2}\lVert\hat{X}-Y\rVert_{L^{2}}^{2}$ at $Y$ attains at $\hat{Y}$ a value at most its value at every point of $\mathcal{D}^{\Lambda}$. In particular the first has a local maximum at $\hat{X}$ relative to $\mathcal{D}^{\Lambda}$ and the second a local minimum at $\hat{Y}$ relative to $\mathcal{D}^{\Lambda}$, in the metric space $(L^{2}(\Omega;\mathbb{R}^{d}),d_{L^{2}})$.