Existence and Basic Estimates at a Maximiser of the Gauge-Doubled Difference on the Wasserstein Space

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a coercive penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$, let $\rho$ be the centred heat gauge, a metric on $\mathcal{P}_{2}(\mathbb{R}^{d})$ by The Centred Heat Gauge: Metric Properties, Comparison with the Wasserstein Distance, Behaviour Under Translations, the Squared Gauge as a Test Function, and the Polarisation Inequality (i), and let $M_{2}$ be the second moment. The set $\mathcal{D}$ contains $\mathcal{D}_{\Sigma}$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains (i), which is nonempty by 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 Coercive Penalty Pair: a Lower Bound for the Penalty, Semicontinuity, and Exact Delta-Envelopes (i).

Let $\delta\in\mathbb{R}$ be positive and let $u,v:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ be continuous on $\mathcal{P}_{2}(\mathbb{R}^{d})$ in $(\mathcal{P}_{2}(\mathbb{R}^{d}),\rho)$; fix $b,b'\in\mathbb{R}$ with $u(\sigma)\le b$ and $b'\le v(\sigma)$ for every $\sigma\in\mathcal{P}_{2}(\mathbb{R}^{d})$. By Continuity, Semicontinuity and Lipschitz Bounds Pass from the Centred Heat Gauge to the Wasserstein Distance (i), read with $A=\mathcal{P}_{2}(\mathbb{R}^{d})$, the functions $u$ and $v$ are continuous, so that by Basic Properties of a Coercive Penalty Pair: a Lower Bound for the Penalty, Semicontinuity, and Exact Delta-Envelopes (iv) the $\delta$-envelope $u^{-}_{\delta}$ of $u$ and the $\delta$-envelope $v^{+}_{\delta}$ of $v$ relative to the penalty pair are defined, with values $u(\nu)-\delta\mathcal{E}(\nu)$ and $v(\nu)+\delta\mathcal{E}(\nu)$ at $\nu\in\mathcal{D}$. For positive $\alpha\in\mathbb{R}$ let $\Phi_{\alpha}:\mathcal{D}\times\mathcal{D}\to\mathbb{R}$ be the function with value

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

at $(\mu,\nu)\in\mathcal{D}\times\mathcal{D}$, and put

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

Then the following hold.

1. (A maximiser exists)(i) For every positive $\alpha\in\mathbb{R}$ there is $(\hat{\mu},\hat{\nu})\in\mathcal{D}\times\mathcal{D}$ such that $\Phi_{\alpha}(\mu,\nu)\le\Phi_{\alpha}(\hat{\mu},\hat{\nu})$ for every $(\mu,\nu)\in\mathcal{D}\times\mathcal{D}$.

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 $\Phi_{\alpha}(\mu_{0},\mu_{0})-\eta\le\Phi_{\alpha}(\hat{\mu},\hat{\nu})$. Then

$$\delta\,\mathcal{E}(\hat{\mu})+\delta\,\mathcal{E}(\hat{\nu})\le b-b'-u(\mu_{0})+v(\mu_{0})+2\delta\,\mathcal{E}(\mu_{0})+\eta,$$

and consequently $\mathcal{E}(\hat{\mu})\le c_{0}+\delta^{-1}\eta$ and $\mathcal{E}(\hat{\nu})\le c_{0}+\delta^{-1}\eta$.

3. (The second moment is bounded at a near-maximiser)(iii) In the situation of claim 2, let $R\in\mathbb{R}$ be as in Coercive Penalty Pairs on the Wasserstein Space (iii) for the level $c=c_{0}+\delta^{-1}\eta$. Then $M_{2}(\hat{\mu})\le R$ and $M_{2}(\hat{\nu})\le R$. In particular $R$ does not depend on $\alpha$.

4. (The gauge at a maximiser under a Lipschitz bound)(iv) Let $L\in\mathbb{R}$ be nonnegative and suppose that

$$|u(\mu)-u(\nu)|\le L\,\rho(\mu,\nu),\qquad|v(\mu)-v(\nu)|\le L\,\rho(\mu,\nu)\qquad\bigl(\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})\bigr).$$

Let $\alpha\in\mathbb{R}$ be positive and let $(\hat{\mu},\hat{\nu})\in\mathcal{D}\times\mathcal{D}$ satisfy $\Phi_{\alpha}(\mu,\nu)\le\Phi_{\alpha}(\hat{\mu},\hat{\nu})$ for every $(\mu,\nu)\in\mathcal{D}\times\mathcal{D}$. Then

$$\alpha\,\rho(\hat{\mu},\hat{\nu})\le2L,\qquad\alpha\,\rho(\hat{\mu},\hat{\nu})^{2}\le4L^{2}\alpha^{-1}.$$