Weak monotone rearrangement on the line

TL;DR

This paper provides a complete geometric characterization of the weak monotone rearrangement between measures on the real line, expanding understanding in optimal transport and its applications.

Contribution

It offers a comprehensive geometric description of the weak monotone rearrangement, complementing previous results and enhancing theoretical understanding.

Findings

Complete geometric characterization of weak monotone rearrangement

Extension of classical monotone rearrangement theory

Implications for optimal transport and related fields

Abstract

Weak optimal transport has been recently introduced by Gozlan et al. The original motivation stems from the theory of geometric inequalities; further applications concern numerics of martingale optimal transport and stability in mathematical finance. In this note we provide a complete geometric characterization of the 'weak' version of the classical monotone rearrangement between measures on the real line, complementing earlier results of Alfonsi, Corbetta, and Jourdain.

Figures

Equations

$$W_{\theta }(\mu ,\nu ):={\inf }_{\pi \in \Pi (\mu ,\nu )}{\int }_{\mathbb{R}\times \mathbb{R}}\theta (x-y)\pi (dx,dy),$$

$$V_{\theta }(\mu ,\nu ):={\inf }_{{\mu }^{\ast }{\le }_c\nu }{W}_{\theta }(\mu ,{\mu }^{\ast }).$$

$$u_{\eta }(y):={\int }_{\mathbb{R}}|x-y|\eta (dx)\le {\int }_{\mathbb{R}}|x-y|\nu (dx)=:u_{\nu }(y),$$

$$\displaystyle\textstyle V_{\theta}(\mu,\nu)=\inf_{\pi\in\Pi(\mu,\nu)}\int_{\mathbb{R}^{d}}\theta\Big{(}x-\int_{\mathbb{R}^{d}}y\pi_{x}(dy)\Big{)}\mu(dx),$$

$$\pi (A\times B)={\int }_A\mu (dx){\pi }_{T(x)}^M(B).$$

$${\pi }^M={\sum }_n{\pi }_{|I_n\times {\bar{I}}_n}^M+(\text{mbox}Id,\text{mbox}Id)({\mu }_{|F}).$$

$$V_{\theta }(\mu ,\nu )={\inf }_{\mu {\le }_c{\nu }^{\ast }}{W}_{\theta }({\nu }^{\ast },\nu );$$

$$\theta (x)\le c(1+|x{|}^{\rho })\forall x\in {\mathbb{R}}^d.$$

$$T(x):={\int }_{\mathbb{R}}y{\pi }_x(dy),$$

$$\begin{array}{r}\text{int}(\text{co}(\mathrm{supp}({\pi }_x)))\cap \text{co}(\mathrm{supp}({\pi }_y))\cup \text{int}(\text{co}(\mathrm{supp}({\pi }_y)))\cap \text{co}(\mathrm{supp}({\pi }_x))\ne \varnothing .\end{array}$$

$$\mu ((a,c))>0,\mu ((c,b))>0,$$

$$\text{int}(\text{co}(\mathrm{supp}({\pi }_x)))\cap \text{co}(\mathrm{supp}({\pi }_y))\cup \text{int}(\text{co}(\mathrm{supp}({\pi }_y)))\cap \text{co}(\mathrm{supp}({\pi }_x))=\varnothing .$$

$$\begin{array}{r}\mathrm{supp}({\pi }_x)\subseteq (-\infty ,d]\text{for μ-a.e. }x<c,\\ \mathrm{supp}({\pi }_y)\subseteq [d,\infty )\text{ for μ-a.e. }y>c.\end{array}$$

$$1$$

$$=\pi ((-\infty ,d_{-})\times (-\infty ,d_{-}]\cup \{c\}\times [d_{-},d_{+}]\cup (d_{+},\infty )\times [d_{+},\infty )),$$

$${\int }_{\mathbb{R}}|x-d_{-}|\mu (dx)$$

$$={\int }_{(a,d_{-})}|y-d_{-}|{\pi }_x(dy)\mu (dx)+{\int }_{[d_{-},b)}|y-d_{-}|{\pi }_x(dy)\mu (dx)$$

$$={\int }_{\mathbb{R}}|x-d_{-}|\nu (dx).\text{qed}$$

$$p_{\alpha }+q_{\alpha }=p+q\forall \alpha \in [0,1],$$

$$[\beta ,1]\ni \alpha \mapsto {\int }_{\mathbb{R}}z{p}_{\alpha }(dz),[\beta ,1]\ni \alpha \mapsto {\int }_{\mathbb{R}}z{q}_{\alpha }(dz)$$

$$s_{\alpha }:=\inf \{x\in \mathbb{R}:F_p(x)\ge \alpha \},t_{\alpha }:=\inf \{x\in \mathbb{R}:F_q(x)\ge \alpha \},$$

$${\tilde{p}}_{\alpha }:=p{|}_{(-\infty ,s_{\alpha })}+(\alpha -F_p(s_{\alpha }-)){\delta }_{\{s_{\alpha }\}},{\tilde{q}}_{\alpha }:=q{|}_{(-\infty ,t_{\alpha })}+(\alpha -F_q(t_{\alpha }-)){\delta }_{\{t_{\alpha }\}}.$$

$${\alpha }_1:=p([c_1,+\infty ))>0\text{ and }{\alpha }_2:=q((-\infty ,c_2])>0.$$

$${\int }_{\mathbb{R}}z{p}_{{\alpha }^{\prime }}(dz)-{\int }_{\mathbb{R}}z{p}_{\alpha }(dz)$$

$$\ge ({\alpha }^{\prime }-\alpha )(c_1-c_2)>0,$$

$${\sum }_{i=1}^N\theta \left(x_i-\int y{\pi }_{x_i}(dy)\right)\le {\sum }_{i=1}^{N}C\left(x_i-\int y{m}_i(dy)\right).$$

$$\displaystyle\textstyle\theta\Big{(}x-\int_{\mathbb{R}}z\pi^{*}_{x}(dy)\Big{)}+\theta\Big{(}y-\int_{\mathbb{R}}z\pi^{*}_{y}(dz)\Big{)}\leq\theta\Big{(}x-\int_{\mathbb{R}}zp_{1}(dz)\Big{)}+\theta\Big{(}y-\int_{\mathbb{R}}zp_{2}(dz)\Big{)}.$$

$$p_1^{\alpha }=(1-\alpha ){\pi }_x^{\ast }+\alpha {\pi }_y^{\ast },p_2^{\alpha }=\alpha {\pi }_x^{\ast }+(1-\alpha ){\pi }_y^{\ast }.$$

$$\displaystyle\textstyle\Big{(}\partial_{+}\theta(x-T(x))-\partial_{+}\theta(y-T(y))\Big{)}(T(x)-T(y))\geq 0,$$

$$\displaystyle\textstyle\Big{(}x-T(x)-y+T(y)\Big{)}(T(x)-T(y))\geq 0.$$

$$x<y,T(x)=\tilde{x}<\tilde{y}=T(y),x-T(x)>y-T(y),$$

$${\pi }_x^{\ast }+{\pi }_y^{\ast }={\bar{\pi }}_{T(x)}+{\bar{\pi }}_{T(y)}=p_{\alpha }+q_{\alpha }.$$

Full text

In this note we provide a complete geometric characterization of the weak version of the classical monotone rearrangement between measures on the real line, complementing earlier results of Alfonsi, Corbetta, and Jourdain.

1. Introduction

Recently, there has been a growing interest in weak transport problems as introduced by Gozlan et al [15]. While the original motivation mainly stems from applications to geometric inequalities (cf. the works of Marton [17, 16] and Talagrand [21, 22]), weak transport problems appear also in a number of further topics, including martingale optimal transport [2, 4, 10, 6], the causal transport problem [7, 1], and stability in math. finance [5].

1.1. Framework and main results

Write $\Pi(\mu,\nu)$ for the set of couplings between $\mu,\nu\in\mathcal{P}(\mathbb{R}^{d})$. Starting with the seminal article of Gangbo-McCann [12] problems of the form

[TABLE]

where $\theta\colon\mathbb{R}^{d}\rightarrow\mathbb{R}$ denotes a convex function have received particular attention in optimal transport. The pendant in weak optimal transport consists in

[TABLE]

Here $\leq_{c}$ denotes the convex order, i.e. $\mu\leq_{c}\nu$ iff $\int\varphi\,d\mu\leq\int\varphi\,d\nu$ for all convex $\varphi:\mathbb{R}^{d}\to\mathbb{R}$.

The problem (1.2), and in particular its one dimensional version, is investigated in [2, 15, 14, 19, 18, 20, 11, 13, 8, 5]. The main purpose of this note is to give a complete geometric characterization of the optimizer $\mu^{*}$ in one dimension.

Definition 1.1**.**

Fix $\mu,\nu\in\mathcal{P}_{1}(\mathbb{R})$. We call a function $S\colon\mathbb{R}\rightarrow\mathbb{R}$ admissible if it satisfies

1. (i)

$S$* is increasing,* 2. (ii)

$S$* is 1-Lipschitz,* 3. (iii)

$S(\mu)\leq_{c}\nu$.

Theorem 1.2**.**

Let $\mu,\nu\in\mathcal{P}_{1}(\mathbb{R})$. There exists an admissible $T$ ($\mu$-a.s. unique) which is maximal in the sense that $S(\mu)\leq_{c}T(\mu)$ for every other admissible $S$.

If $\theta\colon\mathbb{R}\rightarrow\mathbb{R}$ is convex then $\mu^{*}:=T(\mu)$ is an optimizer of (1.2). If $\theta\colon\mathbb{R}\rightarrow\mathbb{R}$ is strictly convex and $V_{\theta}(\mu,\nu)$ is finite, $T(\mu)$ is the unique optimizer of (1.2).

We call the ($\mu$-a.s. unique) map $T$ in Theorem 1.2 the weak monotone rearrangement.

A particular consequence of Theorem 1.2 is that the optimizer of (1.2) does not depend on the choice of the convex function $\theta$. We find this fact non-trivial as well as remarkable and highlight that it is not new: different independent proofs were given by Gozlan et al [15], Alfonsi, Corbetta, Jourdain [2] and Shu [20]. Alfonsi, Corbetta and Jourdain [3, Example 2.4] notice that this does not pertain in higher dimensions.

The map $T$ can be explicitly characterized in geometric terms using the notion of irreducibility introduced in [9]: Measures $\eta,\nu\in P_{1}(\mathbb{R})$ are in convex order iff

[TABLE]

and, by continuity, the set $U$ where this inequality is strict is open. Hence $U=\bigcup_{n}I_{n}$, where $(I_{n})$ is an at most countable family of disjoint open intervals; these intervals $I_{n}$ are called irreducible with respect to $(\eta,\nu)$.

Theorem 1.3**.**

The weak monotone rearrangement $T$ of $\mu,\nu\in\mathcal{P}_{1}(\mathbb{R})$ is the unique admissible map which has slope 1 on each interval $T^{-1}(I)$, where $I$ is irreducible wrt $(T(\mu),\nu)$.

Theorem 1.3 represents a necessary and sufficient condition for the optimality of the measure $T(\mu^{*})$ in (1.2). We note that the ‘necessary’ part was first obtained (using somewhat different phrasing) by Alfonsi, Corbetta, and Jourdain [2, Proposition 3.12]. We also refer the reader to the semi-explicit representation of $T$ and $T(\mu^{*})$ given in [2].

1.2. Connection with martingale transport plans

Intuitively, the irreducible intervals of $(\eta,\nu)$ are the components where we need to ‘expand’ $\eta$ in order to transform it into $\nu$. In this sense Theorem 1.3 asserts that the mass of $\mu$ can either concentrate between $\mu$ and $T(\mu)$, or it can expanded between $T(\mu)$ and $\nu$ (see Figure 1).

To make this precise, we recall from [15] that (1.2) can be reformulated as

[TABLE]

where $(\pi_{x})_{x\in\mathbb{R}^{d}}$ denotes a regular disintegration of the coupling $\pi$ wrt its first marginal $\mu$.

The set of optimizers of (1.4) is also straightforward to express in terms of $T$: Write $\Pi_{M}(\eta,\nu)$ for the set of martingale couplings (or martingale transport plans), i.e. $\pi\in\Pi(\eta,\nu)$ which satisfy barycenter$(\pi_{x})=x$, $\eta$-a.s. By Strassen’s theorem $\Pi_{M}(\eta,\nu)$ is nonempty iff $\eta\leq_{c}\nu$. Using this notation, $\pi\in\Pi(\mu,\nu)$ is optimal iff there exists a martingale coupling $\pi^{M}\in\Pi_{M}(T(\mu),\nu)$ such that $\pi$ is the concatenation of the transports described by $T$ and $\pi^{M}$:

[TABLE]

Any $\pi^{M}\in\Pi_{M}(\eta,\nu)$ can be decomposed based on the family of irreducible intervals $(I_{n})_{n}$: denoting $F:=(\cup_{n}I_{n})^{c}$ by [9, Appendix A] we have

[TABLE]

Plainly, (1.6) asserts that any martingale transport plan can move mass only within the individual irreducible intervals, whereas particles $x\in F$ have to stay put.

1.3. A reverse problem

Alfonsi, Corbetta, and Jourdain [2] proved that the same value is obtained when reversing the order of transport and convex order relaxation in (1.2), i.e.

[TABLE]

moreover they find ([2, Proposition 3.12]) a monotone mapping which is optimal between the optimizer of (1.7) and $\nu$ as well as between $\mu$ and the optimizer of (1.2).

As a counterpart to Theorems 1.2 and 1.3 we establish the following result, strengthening the connection between (1.2) and (1.7).

Theorem 1.4**.**

Let $\mu,\nu\in\mathcal{P}_{1}(\mathbb{R})$. Then there exists a unique $\leq_{c}$-smallest measure $\nu^{*}$, $\mu\leq_{c}\nu^{*}$, which can be pushed onto $\nu$ by an increasing 1-Lipschitz mapping. Moreover we have

1. (1)

If $\theta\colon\mathbb{R}\rightarrow\mathbb{R}$ is convex then $\nu^{*}$ is an optimizer of (1.7). If $\theta\colon\mathbb{R}\rightarrow\mathbb{R}$ is strictly convex and $V_{\theta}(\mu,\nu)$ is finite, $\nu^{*}$ is the unique optimizer of (1.7). 2. (2)

There exists a ($\nu^{*}$-unique) increasing 1-Lipschitz mapping $\tilde{T}$ which pushes $\nu^{*}$ onto $\nu$; $\tilde{T}$ has slope 1 on each interval $I$, where $I$ is irreducible wrt $(\mu,\nu^{*})$. 3. (3)

$\tilde{T}$* is the weak monotone rearrangement between $\mu$ and $\nu$.*

As in the previous section, (1.7) could be interpreted as a concatenation of a martingale transport with the weak monotone rearrangement.

1.4. An auxiliary result

We close this introductory section which an auxiliary result that will be important in the proofs of our results. Since it might be of independent interest we provide the $d$-dimensional version. We denote the topology induced by the $\rho$-Wasserstein distance on the space of probability measures on $\mathbb{R}^{d}$ by $\mathcal{W}_{\rho}$.

Theorem 1.5** (Stability).**

Let $1\leq\rho<\infty$, $(\mu^{k})_{k\in\mathbb{N}}\in\mathcal{P}_{\rho}(\mathbb{R}^{d})^{\mathbb{N}}$, $(\nu^{k})_{k\in\mathbb{N}}\in\mathcal{P}_{1}(\mathbb{R}^{d})^{\mathbb{N}}$ and $\theta\colon\mathbb{R}^{d}\rightarrow\mathbb{R}$ convex and such that for some constant $c>0$ it holds

[TABLE]

If $\mu^{k}\rightarrow\mu$ in $\mathcal{W}_{\rho}$ and $\nu^{k}\rightarrow\nu$ in $\mathcal{W}_{1}$, then $\lim_{k}V_{\theta}(\mu^{k},\nu^{k})=V_{\theta}(\mu,\nu)$. If additionally $\theta$ is strictly convex, we have that

1. (1)

$\arg\min_{\eta\leq_{c}\nu^{k}}W_{\theta}(\mu^{k},\nu^{k})\rightarrow\arg\min_{\eta\leq_{c}\nu}W_{\theta}(\mu,\nu)\quad\text{in }\mathcal{W}_{1}$, 2. (2)

the sequence of maps $T^{k}$, where $T^{k}(\mu)\leq_{c}\nu$ and $V_{\theta}(\mu,\nu^{k})=W_{\theta}(\mu,T^{k}(\mu))$, converges in $\mu$-probability to $T$, where $T(\mu)\leq_{c}\nu$ and $V_{\theta}(\mu,\nu)=W_{\theta}(\mu,\nu)$.

2. C-Monotonicity implies geometric characterization

In this part we prove the following

Theorem 2.1**.**

Let $\mu,\nu\in\mathcal{P}_{1}(\mathbb{R})$ and $\theta\colon\mathbb{R}\rightarrow\mathbb{R}$ strictly convex. If $V_{\theta}(\mu,\nu)$ yields a finite value, for any optimizer $\pi\in\Pi(\mu,\nu)$ of (1.4), the map

[TABLE]

is $\mu$-almost surely uniquely defined and is independent of the specific coupling $\pi$, it is admissible in the sense of Definition 1.1, and it has slope $1$ on $T^{-1}(I)$ if $I$ is an irreducible interval for $(T(\mu),\nu)$.

To prove Theorem 2.1, we need some further properties connected to irreducibility:

Lemma 2.2**.**

Suppose $\{u_{\mu}<u_{\nu}\}=:(a,b)$. Then for any $\pi\in\Pi_{M}(\mu,\nu)$, any regular disintegration $(\pi_{x})_{x\in\mathbb{R}}$ wrt $\mu$ and any $c\in(a,b)$ such that $\mu((a,c))>0,\,\mu((c,b))>0$, there are $x\in(a,c]$, $y\in[c,b)$, $x\neq y$, such that the supports of $\pi_{x}$ and $\pi_{y}$ overlap, i.e.

[TABLE]

Proof.

To show this assertion, we assume the opposite. So there exist $c\in\{u_{\mu}<u_{\nu}\}$, $\pi\in\Pi_{M}(\mu,\nu)$ with fixed disintegration $(\pi_{x})_{x\in\mathbb{R}}$ wrt $\mu$ and

[TABLE]

so that for all $x,y$ with $a<x\leq c\leq y<b$, $x\neq y$, we have

[TABLE]

Since $\pi$ is a martingale coupling, and by (2.2), there exists $d\in(a,b)$ with

[TABLE]

Write $d_{+}$ for the largest and $d_{-}$ for the smallest $d$ such that (2.3) holds. Note then that $d_{-},d_{+}\in(a,b)$. We have either $\operatorname{supp}(\pi_{c})\subseteq[d_{-},d_{+}]$ or $\mu(\{c\})=0$, which in any case implies $\mu([c\wedge d_{-},c\vee d_{+}]\setminus\{c\})=0$. Thus, we infer

[TABLE]

and we conclude by contradicting $u_{\mu}(d_{-})<u_{\nu}(d_{-})$ since

[TABLE]

Lemma 2.3**.**

Let $p,q\in\mathcal{P}_{1}(\mathbb{R})$ have overlapping supports (cf. (2.1)). Then there exists a continuous map $[0,1]\ni\alpha\mapsto(p_{\alpha},q_{\alpha})\in\mathcal{P}(\mathbb{R})\times\mathcal{P}(\mathbb{R})$ such that

[TABLE]

and such that for some $\beta\in(0,1)$ the functions

[TABLE]

are strictly decreasing and increasing, respectively.

Proof.

Let $\alpha\in[0,1]$ and define the inverse distribution functions by

[TABLE]

where $F_{p}$ and $F_{q}$ denote the cumulative distribution functions of $p$ and $q$, respectively. Define two auxiliary measures

[TABLE]

Defining probability measures $p_{\alpha}$ and $q_{\alpha}$ by $p_{\alpha}:=\tilde{p}_{\alpha}+\tilde{q}_{1-\alpha}$ and $q_{\alpha}:=p+q-p_{\alpha}$, yields (2.4) and continuity of $\alpha\mapsto(p_{\alpha},q_{\alpha})$. Since $p$ and $q$ satisfy (2.1), we find constants $c_{1},c_{2}\in\operatorname{supp}(p)\cap\operatorname{supp}(q)$ with $c_{1}>c_{2}$ such that

[TABLE]

Let $\alpha_{3}<\alpha_{1}\wedge\alpha_{2}$, then for any $1-\alpha_{3}\leq\alpha<\alpha^{\prime}\leq 1$ we have

[TABLE]

and conclude that for $\beta:=1-\alpha_{3}$ the maps defined in (2.5) are strictly monotone. ∎

An important tool in the proof of Theorem 2.1 is $C$-monotonicity, a concept which was introduced for the weak optimal transport problem in [6, 13, 8].

Definition 2.4** ($C$-monotonicity).**

A coupling $\pi\in\Pi(\mu,\nu)$ is $C$-monotone if there exists a measurable set $\Gamma\subseteq X$ with $\mu(\Gamma)=1$, such that for any finite number of points $x_{1},\dots,x_{N}$ in $\Gamma$ and measures $m_{1},\dots,m_{N}$ with $\sum_{i=1}^{N}m_{i}=\sum_{i=1}^{N}\pi_{x_{i}}$

[TABLE]

Proof of Theorem 2.1.

Let $\pi^{*}$ be optimal for the weak optimal transport problem (1.4). By the monotonicity principle [8, Theorem 5.2] $\pi^{*}$ is $C$-monotone, therefore, there exists a set $\Gamma\subseteq\mathbb{R}$ with $\mu(\Gamma)=1$ and such that for all $x,y\in\Gamma,\,p_{1},p_{2}\in\mathcal{P}_{1}(\mathbb{R})$ satisfying $\pi^{*}_{x}+\pi^{*}_{y}=p_{1}+p_{2},$ we have

[TABLE]

As an immediate consequence, we find that the map $T(x)=\int_{\mathbb{R}}y\pi^{*}_{x}(dy)$ is $\mu$-almost surely increasing. Letting $x,y\in\Gamma$, for any $\alpha\in[0,1]$ we define

[TABLE]

Plugging $p^{\alpha}_{1}$ and $p^{\alpha}_{2}$ into (2.6) and computing the righthand-side derivative yields

[TABLE]

which is by strict convexity of $\theta$ equivalent to

[TABLE]

Hence, $|x-y|\geq|T(x)-T(y)|$ and $T$ is $\mu$-a.e. $1$-Lipschitz.

Further, we have that $T(\mu)\leq_{c}\nu$ and $\bar{\pi}:=(T,id)(\pi^{*})\in\Pi_{M}(T(\mu),\nu)$. Without loss of generality, we can assume $\bar{\pi}_{T(x)}=\pi^{*}_{x},\,\mu\text{-a.e.}$ Let $(I_{k})_{k\in\mathbb{N}}$ be the intervals given by the decomposition of $(T(\mu),\nu)$ into irreducible intervals. Assume that there is an interval $I_{k}$ so that on $T^{-1}(I_{k})$ the map $T$ does not have $\mu$-a.s. slope 1. Then Lemma 2.2 provides two points $\tilde{x},\tilde{y}$ in $T(\Gamma\cap I_{k})$ and two corresponding points $x,y\in\Gamma\cap I_{k}$ such that

[TABLE]

and the overlapping condition (2.1) holds for $\bar{\pi}_{T(x)}=\pi^{*}_{x}$, $\bar{\pi}_{T(y)}=\pi^{*}_{y}$. Lemma 2.3 allows us to define measures $p_{\alpha}$ and $q_{\alpha}$ on $\mathbb{R}$ such that

[TABLE]

For a graphical depiction compare with Figure 2. Hence,

[TABLE]

are strictly monotone, continuous maps on $[\beta,1]$. Therefore, we find $\alpha\in(\beta,1)$ with

[TABLE]

By strict convexity of $\theta$ we find

[TABLE]

which then yields a contradiction to $C$-monotonicity:

[TABLE]

3. Sufficiency of the geometric characterization

Naturally the question arises whether any map $T$ satisfying the properties in Theorem 2.1 must be optimal. The aim of this section is to establish this:

Theorem 3.1**.**

Let $\mu,\nu\in\mathcal{P}_{1}(\mathbb{R})$. Then any coupling $\pi\in\Pi(\mu,\nu)$ for which $T(x):=\int_{\mathbb{R}}y\pi_{x}(dy)$ is admissible (in the sense of Definition 1.1) with slope $1$ on each interval $T^{-1}(I)$, where $I$ is irreducible wrt $(T(\mu),\nu)$, is optimal for (1.4), i.e.,

[TABLE]

The proof is based on dual optimizers and their explicit representation. As long as $T$ is strictly increasing, [19, Theorem 2.1] provides dual optimizers to $V_{\theta}(\mu,T(\mu))$. Investigating dual optimizers further, we are able to show here $V_{\theta}(\mu,T(\mu))=V_{\theta}(\mu,\nu)$. First, Lemma 3.2 helps us to carefully approximate the increasing map $T$ with strictly increasing maps $T_{\varepsilon}$.

Lemma 3.2**.**

Let $T\colon\mathbb{R}\rightarrow\mathbb{R}$ be an increasing map, with $T-id$ decreasing. Then, for any $\varepsilon>0$ there is a strictly increasing map $T_{\varepsilon}\colon\mathbb{R}\rightarrow\mathbb{R}$, with $T_{\varepsilon}-id$ decreasing, such that

[TABLE]

and $T$ is affine with slope $1$ on an interval $I$ if and only if $T_{\varepsilon}$ is affine with slope $1$ on $I$.

Proof.

Since $T$ is increasing we know that the pre-image of any point under $T$ corresponds to an interval. Therefore, we can find at most countable many, disjoint intervals $(I_{k})_{k\in\mathbb{N}}$ of finite length, where $T(I_{k})$ is a singleton and $T$ is strictly increasing on the complement, i.e., on $\bigcap_{k}I_{k}^{c}$. For any $\varepsilon>0$, we define $g_{\varepsilon}\colon\mathbb{R}\rightarrow\mathbb{R}$

[TABLE]

Then the map $T_{\varepsilon}(x):=T(x)+\int_{-\infty}^{x}[g_{\varepsilon}(y)\wedge 1]\,dy$ satisfies the desired properties. ∎

Let $T\colon\mathbb{R}\rightarrow\mathbb{R}$ be an increasing 1-Lipschitz function. Then $T$ induces a unique decomposition of $\mathbb{R}$ into at most countably many maximal, closed, disjoint intervals $(I_{k})_{k}$ and a ($G_{\delta}$-set) $G$ such that for all $k\in\mathbb{N}$ the map $T|_{I_{k}}$ is affine with slope 1 and $T|_{G}$ is properly contractive, i.e., for any two points $x,y\in G$ we have $|T(x)-T(y)|<|x-y|$. Below we call the intervals $(I_{k})_{k}$ irreducible wrt $T$.

Proof of Theorem 3.1.

The convex function $\theta$ can be approximated by a pointwise-increasing sequence of Lipschitz convex functions $(\theta_{n})_{n\in\mathbb{N}}$, e.g.

[TABLE]

By monotone convergence we find111Recall that by [14] the optimizer of the weak transport problem does not depend on the convex cost. $\sup_{n}V_{\theta_{n}}(\mu,\nu)=V_{\theta}(\mu,\nu)$. Indeed, if $\pi^{n}$ optimizes $V_{\theta_{n}}(\mu,\nu)$ and assuming wlog that $\pi_{n}\to\pi$, then

[TABLE]

by [8, Proposition 2.8]. Thus the claim follows by taking the supremum in $m$.

From this, it suffices to consider the case when $\theta$ is Lipschitz continuous. By Lemma 3.2 we find for any $\varepsilon>0$ a strictly increasing map $T_{\varepsilon}$, such that $T_{\varepsilon}-id$ is decreasing, the decompositions of $T$ and $T_{\varepsilon}$ match, and $|T_{\varepsilon}-T|_{\infty}\leq\varepsilon.$ Then [19, Theorem 2.1] provides a convex, Lipschitz continuous function $f_{\varepsilon}\colon\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x\in\mathbb{R}$

[TABLE]

which is even affine on the parts where $T_{\varepsilon}$ is affine. Write $S_{\varepsilon}(x)=\int_{-\infty}^{x}[g_{\varepsilon}(z)\wedge 1]dz$, where $g_{\varepsilon}$ is defined as in (3.1) so $T_{\varepsilon}=T+S_{\varepsilon}$. In the following we will show that $f_{\varepsilon}$ is a dual optimizer of the coupling $\pi^{\varepsilon}$ defined as the push-forward measure of $\pi$ by the function

[TABLE]

First, we compute the barycenters of $\pi^{\varepsilon}_{x}$:

[TABLE]

and $T_{\varepsilon}(\mu)\leq_{c}\operatorname{proj}_{2}(\pi^{\varepsilon})=:\nu_{\varepsilon}$. Given the sets $(I_{k})_{k\in\mathbb{N}}$ and $F$ from the decomposition of $(T(\mu),\nu)$ into irreducible intervals, we find the sets $(I_{k}^{\varepsilon})_{k\in\mathbb{N}}$ and $F_{\varepsilon}$ from the decomposition of $(T_{\varepsilon}(\mu),\nu_{\varepsilon})$ into irreducible intervals by setting

[TABLE]

In view of the structure of martingale couplings, see [9, Theorem A.4], we find that for $\mu$-a.e. $x\in T_{\varepsilon}^{-1}(I^{\varepsilon}_{k})$ we have $\operatorname{supp}(\pi_{x}^{\varepsilon})\subseteq I^{\varepsilon}_{k}$ and $\pi_{x}^{\varepsilon}=\delta_{T(x)+S_{\varepsilon}(x)}$ $\mu$-a.e. on $F_{\varepsilon}$. Since the decompositions of $T$ and $(T(\mu),\nu)$ are complementary, we infer the same for the decompositions of the map $T_{\varepsilon}$ and the pair $(T_{\varepsilon}(\mu),\nu_{\varepsilon})$. The next computation establishes duality of the pair $(\pi^{\varepsilon},f_{\varepsilon})$, where we use affinity of $f_{\varepsilon}$ on the irreducible components of $(T_{\varepsilon}(\mu),\nu_{\varepsilon})$:

[TABLE]

This easily proves that $\pi^{\varepsilon}$ is optimal for the optimal weak transport problem $V_{\theta}(\mu,\nu_{\varepsilon})$. Drawing the limit for $\varepsilon\searrow 0$, we observe

[TABLE]

As $\theta$ is Lipschitz, we can apply stability Theorem 1.5 and obtain optimality of $\pi$. ∎

4. Geometry of the weak monotone rearrangement

We can summarize Theorems 2.1 and 3.1 as follows: There exists an admissible map $T$ with slope $1$ on $T^{-1}(I)$ whenever $I$ is an irreducible interval wrt $(T(\mu),\nu)$, such that $\pi$ is optimal for (1.4) iff $T(x)=\int y\pi_{x}(dy)$ ($\mu$-a.s.). We now show that this map is the weak monotone rearrangement and is therefore the maximum in convex order of the set222We abuse terminology here, meaning maximum of $\{S(\mu):S\in M(\mu,\nu)\}$.

[TABLE]

Heuristically speaking, if the maximum in convex order of the set $M(\mu,\nu)$ is again given by an increasing, 1-Lipschitz map, then this map is as close as possible to a shifted identity. In turn, this is favourable when trying to find the minimum in convex order of

[TABLE]

which gives reason to why there should exist a single optimizer to (1.4) for all convex $\theta$.

As preparation to establishing Theorem 1.3, we prove Lemma 4.1 and Lemma 4.2.

Lemma 4.1**.**

Let $\mu\in\mathcal{P}_{1}(\mathbb{R})$, $T,S\colon\mathbb{R}\rightarrow\mathbb{R}$ be increasing maps with

[TABLE]

then the maximum (wrt the convex order) of $T(\mu)$ and $S(\mu)$, which is uniquely determined by its potential functions, is again given by an increasing map. If in addition, the maps are $L$-Lipschitz with $L>0$, then the maximum is also given by an $L$-Lipschitz map.

Proof.

The maximum of $T(\mu)$ and $S(\mu)$ wrt. the convex order is uniquely determined by the maximum of its potential functions, i.e. $u_{S(\mu)}\vee u_{T(\mu)}=u_{S(\mu)\vee T(\mu)}$. The right-hand side derivative of the potential function can be expressed by the cumulative distribution function, namely $\partial_{+}u_{\mu}(x)=2F_{\mu}(x)-1$. By continuity of the potential functions, we find a partition of $\mathbb{R}$ into at most countably many disjoint intervals $(I_{k})_{k\in\mathbb{N}}$, where $u_{T(\mu)}=u_{S(\mu)}$ on $\partial I_{k}$, and restricted onto $I_{k}$ one of the following holds true:

1. (a)

$u_{S(\mu)}|_{I_{k}}\geq u_{T(\mu)}|_{I_{k}}$, 2. (b)

$u_{S(\mu)}|_{I_{k}}\leq u_{T(\mu)}|_{I_{k}}$.

Suppose wlog (a) holds, then

[TABLE]

By monotonicity, we can define $T^{*}$ on $\tilde{I}_{k}:=(T^{-1}(l_{k})\vee S^{-1}(l_{k}),T^{-1}(r_{k})\vee S^{-1}(r_{k})]$333We use the convention that the maximum of the empty set equals $-\infty$. by

[TABLE]

Hence, $T^{*}$ is an increasing map, $F_{T^{*}(\mu)}=F_{S(\mu)\vee T(\mu)}$ and $S(\mu)\vee T(\mu)$ is given by the map $T^{*}$. If $T$ and $S$ are in addition $L$-Lipschitz, it follows by construction that $T^{*}$ is $L$-Lipschitz. ∎

Lemma 4.2**.**

Let $\eta_{1}\leq_{c}\eta_{2}$ and $T_{2}(\eta_{2})\leq_{c}T_{1}(\eta_{1})$, where $T_{1},T_{2}$ are increasing and $1$-Lipschitz. Then $(id-T_{1})(\eta_{1})\leq_{c}(id-T_{2})(\eta_{2})$.

In particular, if $T$ and $S$ are increasing $1$-Lipschitz maps s.t. $\int_{\mathbb{R}}T(x)\mu(dx)=\int_{\mathbb{R}}S(x)\mu(dx)$, and we denote by $R$ the increasing $1$-Lipschitz map with $R(\mu)=S(\mu)\vee T(\mu)$, which exists by Lemma 4.1, then $(id-R)(\mu)\leq_{c}(id-S)(\mu)\wedge(id-T)(\mu)$.

Proof.

By approximation, it suffices to settle the case when $\eta_{1},\eta_{2}$ are uniform measures on $n\in\mathbb{N}$ atoms. Let $x^{i}_{1}\leq x^{i}_{2}\leq\dots\leq x^{i}_{n}$ denote the atoms of $\eta_{i}$. Then the vector $z^{i}:=(T_{i}(x^{i}_{1}),\dots,T_{i}(x^{i}_{n}))$ is ordered in an increasing way. What is more, the vector $y^{i}:=(x^{i}_{1}-z^{i}_{1},\dots,x^{i}_{n}-z^{i}_{n})$ is likewise ordered increasingly, since $id-T_{i}$ is an increasing map. By e.g. [14, Proposition 2.6] we know that

[TABLE]

But then also $\sum_{\ell\leq k}x^{2}_{\ell}-z^{2}_{\ell}\leq\sum_{\ell\leq k}x^{1}_{\ell}-z^{1}_{\ell}$, so again by [14, Proposition 2.6] we conclude $(id-T_{1})(\eta_{1})\leq_{c}(id-T_{2})(\eta_{2})$. The second statement easily follows from the first one. ∎

Proof of Theorem 1.3.

Existence of an admissible map $T$ which has slope 1 on each interval $T^{-1}(I)$, where $I$ is irreducible wrt $(T(\mu),\nu)$, was already shown in Theorem 2.1. Therefore, it remains to show that the map is maximal. Denote by $T$ the map given by Theorem 2.1 associated with an optimizer to (1.4) and some strictly convex $\theta\colon\mathbb{R}\rightarrow\mathbb{R}$. Let $S$ be an arbitrary map in $M(\mu,\nu)$. Then Lemma 4.2 states that

[TABLE]

where $R$ is defined as the increasing, 1-Lipschitz map such that $R(\mu)=S(\mu)\vee T(\mu)$. Additionally to existence, strict convexity of $\theta$ ensures $\mu$-almost sure uniqueness of $T$ in the sense that for any optimal coupling $\pi$ we have $\textstyle\int_{\mathbb{R}}y\pi_{x}(dy)=T(x)\quad\mu\text{-a.s.}$ Thus, $R(\mu)=T(\mu)$ and $T=R$ $\mu$-almost surely. ∎

Proof of Theorem 1.2.

This is a direct consequence of Theorem 2.1, which provides existence of a map with the desired geometric properties, and Theorem 1.3, which provides the equivalence between the geometric properties and maximality. ∎

5. On the reverse problem of Alfonsi, Corbetta, and Jourdain

We aim to prove Theorem 1.4 pertaining the reverse problem (1.7).

Lemma 5.1**.**

Let $\eta_{1},\eta_{2}\in\mathcal{P}_{1}(\mathbb{R})$, $T_{1},T_{2}\colon\mathbb{R}\rightarrow\mathbb{R}$ be increasing maps with

[TABLE]

Denote the minimum in convex order of $\eta_{1}$ and $\eta_{2}$ by $\eta$. Then there exists an increasing map $T^{*}$ such that $T^{*}(\eta)=\nu$. If in addition, the maps are $L$-Lipschitz with $L>0$, then the same holds true for $T^{*}$.

Proof.

Suppose there exist increasing maps $T_{i}$ and measures $\eta_{i}$, $i=1,2$, such that $T_{1}(\eta_{1})=\nu=T(\eta_{2})$. Then the potential function of the minimum $\eta$ of $\eta_{1}$ and $\eta_{2}$ wrt the convex order is given by the convex hull of $u_{\eta_{1}}$ and $u_{\eta_{2}}$. The potential function $u_{\eta}$ completely specifies the cumulative distribution function through $\partial_{+}u_{\eta}=2F_{\eta}-1$. Thus, we can find a partition of $\mathbb{R}$ into countably many, disjoint intervals $I_{k}=[a_{k},b_{k})\cap\mathbb{R}$. For each $k\in\mathbb{N}$, we have $i\neq j\in\{1,2\}$ with $u_{\eta}(a_{k})=u_{\eta_{i}}(a_{k})$, $u_{\eta}(b_{k})=u_{\eta_{j}}(b_{k})$ such that one of the following holds

1. (a)

$u_{\eta}(x)=u_{\eta_{i}}(x)$ on $I_{k}$, 2. (b)

$F_{\eta}(a_{k})=F_{\eta}(b_{k}-)=F_{\eta_{j}}(b_{k}-)$ and $u_{\eta}(x)<u_{\eta_{1}}(x)\wedge u_{\eta_{2}}(x)$ on $(a_{k},b_{k})$.

According to this decomposition, we can define an increasing map $T^{*}$ via

[TABLE]

Note that $T^{*}$ is $L$-Lipschitz if $T_{i}$, $i=1,2$, are $L$-Lipschitz. Let $y\in\mathbb{R}$, due to the continuity of the maps $T_{1}$ and $T_{2}$, we can find points $x_{1},x_{2}\in\mathbb{R}$ with $p=F_{\nu}(y)$, $x_{1}=F^{-1}_{\eta_{1}}(p)$ $x_{2}=F^{-1}_{\eta_{2}}(p)$. Assume that $i=1,j=2$ with $x_{1}\in I_{k}$. If (a) holds, then we have $F_{\eta}(x_{1})=F_{\eta_{1}}(x_{1})$. Now presume that (b) holds, then $F_{\eta_{2}}(b_{k}-)=F_{\eta}(a_{k})\leq F_{\eta_{1}}(x_{1})$. Then

[TABLE]

Hence, by monotonicity of the map $T^{*}$ we conclude $F_{T^{*}(\eta)}=F_{\nu}$. ∎

Proof of Theorem 1.4.

Define $\nu^{*}$ via the weak monotone rearrangement $T$ between $\mu$ and $\nu$, as $\mu$ on the contraction parts of $T$ and accordingly shifted $\nu$ on the affine (irreducible) intervals such that $\tilde{T}(\nu^{*})=\nu$. Let $\eta\geq_{c}\mu$. Then for any strictly convex $\theta\colon\mathbb{R}\rightarrow\mathbb{R}$ and coupling $\pi^{2}\in\Pi(\eta,\nu)$ we have $\int_{\mathbb{R}\times\mathbb{R}}\theta(y-z)\pi^{2}(dy,dz)\geq\int_{\mathbb{R}}\theta(y-\int_{\mathbb{R}}z\pi^{2}_{y}(dz))\eta(dy),$ with equality iff $\pi^{2}$ is actually given by a map. Hence, if the optimizer $\pi^{2}$ of $W_{\theta}(\eta,\nu)$ is not given by a map and $\pi^{1}\in\Pi_{M}(\mu,\eta)$, we have

[TABLE]

Thus, by the structure of the weak monotone rearrangement, we deduce optimality of $\nu^{*}$ for Problem (1.7). To show uniqueness of (1.7), assume that $\eta$ attains the minimum of (1.7) and the optimizer of $W_{\theta}(\eta,\nu)$ is given by the map $R$. For any martingale coupling $\pi^{1}\in\Pi_{M}(\mu,\eta)$ we define a map by

[TABLE]

Then by optimality $L(\mu)=T(\mu)$ and, in particular,

[TABLE]

which shows $L=T$ $\mu$-almost surely. By strict convexity, we have

[TABLE]

Since $\pi^{1}$ was arbitrary in $\Pi_{M}(\mu,\eta)$ we get that $R$ is affine with slope $1$ on $I$, whenever $I$ is an irreducible interval wrt $(\mu,\eta)$. Therefore $\eta$ and $\nu^{*}$ restricted to $I$ coincide. Hence, $\eta=\nu^{*}$.

We finally show that $\nu^{*}$ is minimal in the convex order as stated. By Lemma 5.1, we can assume $\mu\leq_{c}\eta\leq_{c}\nu^{*}$ and that $\eta$ can be pushed forward onto $\nu$ via an increasing $1$-Lipschitz map $S$. It follows by Lemma 4.2 that $(id-S)(\eta)\leq_{c}(id-\tilde{T})(\nu^{*})$, so

[TABLE]

and by the uniqueness obtained above we deduce $\eta=\nu^{*}$. ∎

6. Stability of barycentric weak transport problems in multiple dimensions

The final part of the article is concerned with stability of the weak optimal transport problem under barycentric costs, see (1.4). Unlike in the rest of the article we work here on $\mathbb{R}^{d}$. The final aim is to prove Theorem 1.5.

One surprising aspect of this result is that we only require $\nu^{k}\rightarrow\nu$ in $\mathcal{W}_{1}$ and not necessarily in $\mathcal{W}_{\rho}$. This relates to the conditional expectation in (1.4) being ‘inside of $\theta$.’ We first prove an illuminating intermediate result:

Proposition 6.1**.**

Let $1\leq\rho<\infty$, $(\mu^{k})_{k\in\mathbb{N}}\in\mathcal{P}_{\rho}(\mathbb{R}^{d})^{\mathbb{N}}$, $(\nu^{k})_{k\in\mathbb{N}}\in\mathcal{P}_{\rho}(\mathbb{R}^{d})^{\mathbb{N}}$ and $\theta\colon\mathbb{R}^{d}\rightarrow\mathbb{R}$ convex and satisfying the growth condition (1.8). Suppose that $\mu^{k}\rightarrow\mu$ and $\nu^{k}\rightarrow\nu$ in $\mathcal{W}_{\rho}$, and that $\eta\leq_{c}\nu$. Then there exist $\eta^{k}\leq_{c}\nu^{k}$ such that

• (i)

$\eta^{k}\rightarrow\eta$* in $\mathcal{W}_{\rho}$,*

• (ii)

$\lim_{k}W_{\theta}(\mu^{k},\eta^{k})=W_{\theta}(\mu,\eta).$**

Proof.

It is well-known that $(i)$ together with the stated convergence of the $\mu^{k}$’s implies $(ii)$, so we proceed to prove the former. Let $\pi^{k}$ be an optimal coupling attaining $W_{\rho}(\nu,\nu^{k})$. Let $M$ be any martingale coupling with first marginal $\eta$ and second marginal $\nu$, the existence of which is guaranteed by the assumption $\eta\leq_{c}\nu$ together with Strassen’s theorem. We convene on the notation $M(dx,dy)$ and $\pi(dy,dz)$, and define the measure

[TABLE]

This measure has $\eta$, $\nu$ and $\nu^{k}$ as first, second and third marginals. We next define $R^{k}(x)$ as the conditional expectation under $P$ of the third variable given the first one, namely

[TABLE]

Next we introduce $\eta^{k}:=R^{k}(\eta)$ so by definition $\eta^{k}\leq_{c}\nu^{k}$. Finally

[TABLE]

by the martingale property and two applications of Jensen’s inequality. The desired conclusion follows. ∎

Remark 6.2*.*

In the context of the previous proposition, if $\eta$ is supported in finitely many atoms, then the condition that $\nu^{k}\rightarrow\nu$ in $\mathcal{W}_{\rho}$ can be relaxed to convergence in $\mathcal{W}_{1}$. To wit, if $\eta=\sum_{i=1}^{\ell}\alpha_{i}\delta_{x^{i}}$, one can take $\rho=1$ in (6.1) and prove

[TABLE]

so taking $\rho$-power and integrating w.r.t. $\eta$ we get $W_{\rho}(\eta,\eta^{k})^{\rho}\leq K\,W_{1}(\nu,\nu^{k})^{\rho}\to 0$.

The previous remark shows that we need to reduce to the finite-support setting. We carry to this in the next two lemmas:

Lemma 6.3**.**

Let $\eta\in\mathcal{P}_{1}(\mathbb{R}^{d})$. Then for any $\varepsilon>0$ there is a compactly supported, positive measures $\tilde{\eta}$ with

[TABLE]

Proof.

We first partition $\mathbb{R}^{d}$ into countable, disjoint $d$-dimensional cubes $(Q^{\delta}_{k})_{k\in\mathbb{N}}$ of length $\delta>0$. Define an approximation $\eta^{\delta}$ of $\eta$ by

[TABLE]

Note that $\eta^{\delta}\leq_{c}\eta$ and $\eta^{\delta}\rightarrow\eta$ in $\mathcal{W}_{1}$ when $\delta\searrow 0$. If there exists an approximation $\eta^{\delta}$ such that the assertion holds, then it is straightforward to construct the corresponding measure for $\eta$, which in turn satisfies the assertion with respect to $\eta$. Wlog, we may assume that

[TABLE]

where $\bar{z}$ denotes the barycenter of $\eta$. Then we can find $\delta>0$ such that

[TABLE]

Let $z^{\delta}_{n_{1}},\ldots,z^{\delta}_{n_{2d}}$ span $\mathbb{R}^{d}$ in the sense above and

[TABLE]

For any $\varepsilon>0$ there is a $\tilde{\varepsilon}\in(0,\varepsilon)$ such that

[TABLE]

Besides, there exists a compact set $K\subseteq\mathbb{R}^{d}$ such that

[TABLE]

and $z^{\delta}_{n_{1}},\ldots,z^{\delta}_{n_{2d}}\in K$. Therefore, we find $(\tilde{\alpha}_{i})_{i=1}^{2d}\in\mathbb{R}_{+}^{2d}$ with

[TABLE]

If $\varepsilon$ is chosen smaller than $\eta^{\delta}(z_{n_{i}}^{\delta})$ for all $i=1,\ldots,2d$, we can define the $\tilde{\eta}^{\delta}$ via

[TABLE]

Lemma 6.4**.**

Let $\mu,\eta\in\mathcal{P}_{\rho}(\mathbb{R}^{d})$ and $\theta\colon\mathbb{R}^{d}\rightarrow\mathbb{R}$ convex satisfying the growth condition (1.8). Then there exists a sequence $(\eta^{k})_{k\in\mathbb{N}}$ of finitely supported measures with $\eta^{k}\leq_{c}\eta$, $\eta^{k}\rightarrow\eta$ in $\mathcal{W}_{\rho}$ and $W_{\theta}(\mu,\eta^{k})\rightarrow W_{\theta}(\mu,\eta)$.

Proof.

For any $\varepsilon>0$ we find a compact set $K_{\varepsilon}\subseteq\operatorname{supp}(\eta)$ such that $\int_{K_{\varepsilon}^{c}}|y|^{\rho}\eta(dy)<\varepsilon$. For any $\delta>0$, the set $K_{\varepsilon}$ can be covered by finitely many, disjoint sets $(A^{\varepsilon,\delta}_{i})_{i=1}^{N_{\varepsilon,\delta}}$ with diameter smaller than $\delta$ and $K_{\varepsilon}=\bigcup_{i}A^{\varepsilon,\delta}_{i}$. Define the measure $\eta^{\varepsilon,\delta}$ by

[TABLE]

where the points $z_{\varepsilon}$ and $(z_{i}^{\varepsilon,\delta})_{i=1}^{N_{\varepsilon,\delta}}$ are given by

[TABLE]

By construction, there exists a martingale coupling between $\eta^{\varepsilon,\delta}$ and $\eta$, thus, $\eta^{\varepsilon,\delta}\leq_{c}\eta$. Note that $\theta$ restricted to $K_{\varepsilon}$ is Lipschitz continuous. Then drawing the limit $\delta\rightarrow 0$ yields

[TABLE]

and by convexity, $\textstyle|z_{\varepsilon}|^{\rho}\eta(K_{\varepsilon}^{c})\leq\int_{K_{\varepsilon}^{c}}|y|^{\rho}\eta(dy).$ Choosing $\delta(\varepsilon)$ sufficiently small, we have

[TABLE]

By the growth condition (1.8) and stability, we obtain $W_{\theta}(\mu,\eta^{\varepsilon,\delta(\varepsilon)})\to W_{\theta}(\mu,\eta)$. ∎

We can now prove a version of Proposition 6.1 under weaker assumptions:

Lemma 6.5**.**

Let $(\nu^{k})_{k\in\mathbb{N}}$ be a sequence in $\mathcal{P}_{1}(\mathbb{R}^{d})$ and let $(\mu^{k})_{k\in\mathbb{N}}$ be a sequence $\mathcal{P}_{\rho}(\mathbb{R}^{d})$ with $\nu^{k}\rightarrow\nu$ in $\mathcal{W}_{1}$, $\mu^{k}\rightarrow\mu$ in $\mathcal{W}_{\rho}$, where $\rho\geq 1$, and let $\theta\colon\mathbb{R}^{d}\rightarrow\mathbb{R}$ be a convex functions satisfying the growth constraint (1.8). Then for any $\eta\leq_{c}\nu$ we find a sequence of $\eta^{k}\leq_{c}\nu^{k}$ such that $W_{\theta}(\mu^{k},\eta^{k})\rightarrow W_{\theta}(\mu,\eta)$ and $\eta^{k}\rightarrow\eta$ in $\mathcal{W}_{1}$.

Proof.

Wlog assume that $\theta$ is positive. By Lemma 6.3, we can find for any $\varepsilon>0$ a compactly supported $\hat{\eta}=\tilde{\eta}+(1-\tilde{\eta}(\mathbb{R}^{d}))\delta_{\bar{z}}\in\mathcal{P}_{\rho}(\mathbb{R}^{d})$ with $\hat{\eta}\leq_{c}\eta$, $\mathcal{W}_{1}(\hat{\eta},\eta)<\varepsilon$ and

[TABLE]

where $\bar{z}:=\int_{\mathbb{R}^{d}}y\eta(dy)$. Using stability of classical optimal transport, see [23, Theorem 5.20], we may assume that $|W_{\theta}(\mu,\hat{\eta})-W_{\theta}(\mu,\eta)|<\varepsilon$. By Lemma 6.4 we may reduce to the case of finitely supported $\hat{\eta}$. We conclude the proof with Remark 6.2. ∎

Finally we can give the pending proof of Theorem 1.5:

Proof of Theorem 1.5.

Lower-semicontinuity of the map $(\mu,\nu)\mapsto V_{\theta}(\mu,\nu)$ follows from [8, Theorem 1.3]. By [8, Lemma 6.1] we have

[TABLE]

where the infimum is even attained for a measure $\eta\leq_{c}\nu$. By Lemma 6.5 we find a sequence $\eta^{k}\leq_{c}\nu^{k}$, so that again using [8, Lemma 6.1] we find

[TABLE]

If $\theta$ is strictly convex, the infimum in (6.4) is attained by a unique probability measure $\eta^{k}\leq_{c}\nu^{k}$,444The uniqueness of $\eta^{k}$, $T^{k}$ and $\pi^{k}$ was already shown in [2, Theorem 2.1] for $|\cdot|^{\rho}$, $\rho>1$ which in turn is the push-forward of $\mu^{k}$ under a $\mu^{k}$-uniquely defined map $T^{k}$. Moreover, the $W_{\theta}$-optimal transport plan $\pi^{k}\in\Pi(\mu^{k},\nu^{k})$ is uniquely determined by $\mu^{k}(dx)\delta_{T^{k}(x)}(dy)$: Suppose the contrary and let $T^{\prime}(x):=\int_{\mathbb{R}^{d}}y\pi^{k}_{x}(dy)$, then $T^{\prime}(\mu^{k})\leq_{c}\nu$ and we find the contradiction

[TABLE]

Hence, by convergence of the values of $V_{\theta}$ and tightness of $(\eta_{k})_{k\in\mathbb{N}}$, we deduce the convergence of the $\eta^{k}$ to the optimal $\eta\leq_{c}\nu$ in $\mathcal{W}_{1}$. Suppose that $\mu^{k}=\mu$ for all $k\in\mathbb{N}$, then due to the uniqueness of the optimal transport maps $T$ between $T$ and $T(\mu$), we can apply Theorem [23, Corollary 5.23] and obtain convergence of the transport maps $T^{k}$ to $T$. ∎