The Delta-Envelopes of a Function on the Wasserstein Space Relative to a Penalty Pair

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 penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$, with a nonnegative $C$ as in Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §bound, let $u:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ and let $\delta\in\mathbb{R}$ be positive. The set $\mathcal{D}$ is nonempty, containing the nonempty set $\mathcal{D}_{\Sigma}$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §nonempty, and is regarded as a subset of the metric space $(\mathcal{P}_{2}(\mathbb{R}^{d}),W_{2})$ of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §measures, with the second moment $M_{2}$; The Coordinate Fields of a Coupling, and the Second Moment as a Lipschitz Function of the Wasserstein Distance is read with $m=d$; that a real-valued function is bounded above, or below, near each point of a nonempty subset, and its upper and lower semicontinuous envelopes, are as defined there, always for this metric. The functions $u-\delta\mathcal{E}$ and $u+\delta\mathcal{E}$ on $\mathcal{D}$ are those with values $u(\nu)-\delta\,\mathcal{E}(\nu)$ and $u(\nu)+\delta\,\mathcal{E}(\nu)$ at $\nu\in\mathcal{D}$.

1. (The envelope $u^{-}_{\delta}$)¶ Suppose $u$ is bounded above near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$. Then $u-\delta\mathcal{E}$ is bounded above near each point of $\mathcal{D}$. Indeed, let $\mu\in\mathcal{D}$ and let $c\in\mathbb{R}$ and positive $R\in\mathbb{R}$ be such that $u(\nu)\le c$ for every $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ with $W_{2}(\nu,\mu)\le R$. Let $\nu\in\mathcal{D}$ satisfy $W_{2}(\nu,\mu)\le R$. By Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §bound one has $-C(1+M_{2}(\nu))\le\mathcal{E}(\nu)$, so $0\le\mathcal{E}(\nu)+C(1+M_{2}(\nu))$ and hence $-\mathcal{E}(\nu)\le C(1+M_{2}(\nu))$, by claim 3 of Elementary Arithmetic in an Ordered Field used in both directions. Also $M_{2}(\nu)\le(\sqrt{M_{2}(\mu)}+R)^{2}$ by The Coordinate Fields of a Coupling, and the Second Moment as a Lipschitz Function of the Wasserstein Distance §ball, whence $C(1+M_{2}(\nu))\le C\bigl(1+(\sqrt{M_{2}(\mu)}+R)^{2}\bigr)$ by the compatibility of the order with addition (an axiom of Ordered Field) and claim 5 of Elementary Arithmetic in an Ordered Field applied with the nonnegative multiplier $C$. Multiplying the resulting bound $-\mathcal{E}(\nu)\le C(1+(\sqrt{M_{2}(\mu)}+R)^{2})$ by the nonnegative $\delta$, again by claim 5 of Elementary Arithmetic in an Ordered Field, and using $\delta(-s)=-(\delta s)$ from claim 2 of Zero Products and Elementary Identities in a Field, gives $-\delta\,\mathcal{E}(\nu)\le\delta\,C(1+(\sqrt{M_{2}(\mu)}+R)^{2})$. Adding this to $u(\nu)\le c$, by claim 3 of Elementary Arithmetic in an Ordered Field and the compatibility of the order with addition, gives

a real number not depending on $\nu$. The upper semicontinuous envelope of $u-\delta\mathcal{E}$ on $\mathcal{D}$ is therefore defined, and we write

2. (The envelope $u^{+}_{\delta}$)¶ Suppose $u$ is bounded below near each point of $\mathcal{P}_{2}(\mathbb{R}^{d})$. Then $u+\delta\mathcal{E}$ is bounded below near each point of $\mathcal{D}$: let $\mu\in\mathcal{D}$, and let $c'\in\mathbb{R}$ and positive $R\in\mathbb{R}$ be such that $c'\le u(\nu)$ for every $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ with $W_{2}(\nu,\mu)\le R$. For such a $\nu$ lying in $\mathcal{D}$, Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §bound and The Coordinate Fields of a Coupling, and the Second Moment as a Lipschitz Function of the Wasserstein Distance §ball give $-C(1+(\sqrt{M_{2}(\mu)}+R)^{2})\le-C(1+M_{2}(\nu))\le\mathcal{E}(\nu)$, the first inequality by claim 3 of Elementary Arithmetic in an Ordered Field used in both directions on the bound $C(1+M_{2}(\nu))\le C(1+(\sqrt{M_{2}(\mu)}+R)^{2})$ established in clause 1. Multiplying by the nonnegative $\delta$ as in clause 1 and adding $c'\le u(\nu)$ gives

a real number not depending on $\nu$ on the left. The lower semicontinuous envelope of $u+\delta\mathcal{E}$ on $\mathcal{D}$ is therefore defined, and we write

¶ The two functions $u^{-}_{\delta}$ and $u^{+}_{\delta}$ are together called the $\delta$-envelopes of $u$ relative to the penalty pair.