Existence, duality, and cyclical monotonicity for weak transport costs
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Abstract
The optimal weak transport problem has recently been introduced by Gozlan et al. (J Funct Anal 273(11):3327–3405, 2017). We provide general existence and duality results for these problems on arbitrary Polish spaces, as well as a necessary and sufficient optimality criterion in the spirit of cyclical monotonicity. As an application we extend the Brenier–Strassen Theorem of Gozlan and Juillet (On a mixture of brenier and strassen theorems. arXiv:1808.02681, 2018) to general probability measures on \(\mathbb {R}^d\) under minimal assumptions. A driving idea behind our proofs is to consider the set of transport plans with a new (‘adapted’) topology which seems better suited for the weak transport problem and allows to carry out arguments which are close to the proofs in the classical setup.
Mathematics Subject Classification
60G42 90C46 58E301 Introduction
1.1 Notation
1.2 Literature
The initial works of Gozlan et al. [24, 25] are mainly motivated by applications to geometric inequalities. Indeed, particular costs of the form (1.1) were already considered by Marton [29, 30] and Talagrand [40, 41]. Further papers directly related to [25] include [21, 23, 36, 37, 38]. Notably the weak transport problem (1.1) also yields a natural framework to investigate a number of related problems: it appears in the recursive formulation of the causal transport problem [7], in [1, 2, 6, 15] it is used to investigate martingale optimal transport problems, in [3] it is applied to prove stability of pricing and hedging in mathematical finance, it appears in the characterization of optimal mechanism for the multiple good monopolist [19] and motivates the investigation of linear transfers in [17]. A more classical example is given by entropyregularized optimal transport (i.e. the Schrödinger problem); see [28] and the references therein.
1.3 Main results
We will establish analogues of three fundamental facts in optimal transport theory: existence of optimizers, duality, and characterization of optimizers through ccyclical monotonicity. We make the important comment, that these concepts (in particular existence and duality) have been previously studied for the weak transport problem. However, the results available so far may be too restrictive for certain applications.
Our goal is to establish these results at a level of generality that mimics the framework usually considered in the optimal transport literature (i.e. lower bounded, lower semicontinuous cost function). We emphasize that this extension is in fact required to treat specific examples of interest, cf. Sect. 1.3.4 below.
We briefly hint at the novel viewpoint which makes this extension possible: In a nutshell, the technicalities of the weak transport problem appear intricate and tedious since kernels \((\pi _x)_x\) are notoriously ill behaved with respect to weak convergence of measures on \({\mathcal {P}}(X\times Y)\). In the present paper we circumvent this difficulty by embedding \({\mathcal {P}}(X\times Y)\) into the bigger space \({\mathcal {P}}(X\times {\mathcal {P}}(Y))\). This idea is borrowed from the investigation of process distances (cf. [4, 5, 33]) and will allow us to carry out proofs that closely resemble familiar arguments from classical optimal transport.
1.3.1 Primal existence
As a first contribution we will establish in Sect. 2 the following basic existence results.
Theorem 1.1
Notably, Gozlan et.al. prove existence of minimizers under the assumption that \(\pi \mapsto \int C(x, \pi _x)\, d\mu (x)\) is continuous on the set of all transport plans with first marginal \(\mu \), whereas our aim is to establish existence based on properties of the function C. We also note that Theorem 1.1 was first established by Alibert et al. [2] in the case where X, Y are compact spaces.
In fact the assumptions of Theorem 1.1 may be more restrictive than they initially appear. Indeed, as the cost function defined in (1.5) below is not lower semicontinuous with respect to weak convergence, we will need to employ a refined version of Theorem 1.1 to carry out our application in Theorem 1.4 below.
Given a compatible metric \(d_Y\) on the Polish space Y and \(t\in [1,\infty )\), we write \({\mathcal {P}}_{d_Y}^t(Y)\) for the set of probability measures \(\nu \in {\mathcal {P}}(Y)\) such that \(\int d_Y(y,y_0)^t\, \nu (dy)<\infty \) for some (and then any) \(y_0\in Y\) and denote the tWasserstein metric on \({\mathcal {P}}_{d_Y}^t(Y)\) by \({\mathcal {W}}_t\) (see e.g. [42, Chapter 7]). In the sequel we make the convention that, whenever we refer to \(\mathcal {P}_{d_Y}^t(Y)\), it is assumed that this set is equipped with the topology generated by \({\mathcal {W}}_t\). On the other hand, regarding the Polish space X, we fix from now on a compatible bounded metric \(d_X\).
Theorem 1.2
We emphasize that Theorem 1.1 is a special case of Theorem 1.2. To see this, just take \(d_Y\) to be a compatible bounded metric. We also note that if C is strictly convex in the second argument and \(V(\mu ,\nu )<\infty \), then the minimizer \(\pi ^*\in \Pi (\mu ,\nu )\) is unique. We report our proofs in Sect. 2.
1.3.2 Duality
Theorem 1.3
The proof of Theorem 1.3 is provided in Sect. 3. We also refer to this section for a comparison of earlier duality results of Gozlan et al. [25, Theorem 9.6] and Alibert et al. [2, Theorem 4.2].
1.3.3 Cmonotonicity
Besides primal existence and duality, another fundamental result in classical optimal transport is the characterization of optimality through the notion of cyclical monotonicity; see [22, 35] as well as the monographs [34, 42, 43]. More recently, variants of this ‘monotonicity priniciple’ have been applied in transport problems for finitely or infinitely many marginals [11, 18, 26, 32, 44], the martingale version of the optimal transport problem [12, 13, 31], the Skorokhod embedding problem [9] and the distribution constrained optimal stopping problem [14].
We provide in Definition 5.1 below, a concept analogous to cyclical monotonicity (which we call Cmonotonicity) for weak transport costs C. We show that every optimal transport plan is Cmonotone in a very general setup. Conversely, we have that every Cmonotone transport plan is optimal under certain regularity assumptions. See Theorems 5.3 and 5.6 respectively.
We note that related concepts already appeared in [6, Proposition 4.1] (where necessity of a 2step optimality condition is established) and in [23] (necessity in the case of compactly supported measures and a quadratic cost criterion). To the best of our knowledge, our sufficient criterion is the first of its kind for weak transport costs.
We remark that the 2step monotonicity principle for weak transport costs has already proved vital in [6] for the construction of a martingale counterpart to the Brenier theorem and the Benamou–Brenier formula. On the other hand, we conjecture that this monotonicity principle could be used in order to generalize [23] to nonquadratic costs.
1.3.4 A general Brenier–Strassen theorem
Theorem 1.4
Existence of \(\mu ^*\) and the expression (1.6) were first proved by Gozlan et al. [24] for \(d=1\) and by Alfonsi et al. [1] for arbitrary \(d\in \mathbb {N}\). Indeed a general version of (1.6), appealing to \({\mathcal {W}}_p\) and probabilities \(\mu , \nu \in {\mathcal {P}}^p(\mathbb R^d)\) is provided in [1]. All other statements in the above theorem were originally established by Gozlan and Juillet [23] under the assumption of compactly supported measures \(\mu , \nu \). The proof of Theorem 1.4 is given in Sect. 6.
Note added in revision In an updated version of [23], Gozlan and Juillet have also removed the compactness assumption in Theorem 1.4. Their proof is based on duality arguments and in particular differs from the one given here.
2 Existence of minimizers
A principal idea behind the proofs of this paper is to endow the set of transport plans \({\mathcal {P}}(X\times Y)\) with a topology that is finer than the usual weak topology and which appropriately accounts for the asymmetric role of X and Y in the context of weak transport. This can be formalized by embedding \(\mathcal P(X\times Y)\) into the bigger space \({\mathcal {P}}(X\times \mathcal P(Y))\). I.e., given a transport plan \(\pi \), we will consider its disintegration \((\pi _x)_{x\in X}\) (w.r.t. its first marginal) and view it as a Mongetype coupling in the larger space \({\mathcal {P}}(X\times {\mathcal {P}}(Y))\). It turns out that on this ‘extended’ space the minimization problems Theorems 1.1 and 1.2 can be handled more efficiently.
Lemma 2.1
Proof
2.1 Existence of minimizers
The purpose of this subsection is to establish Theorem 1.2, or more precisely, a strengthened version of it; see Theorem 2.9 below. To this end we need a number of auxiliary results.
We start by stressing that, in general, the embedding J is not continuous. In fact:
Example 2.2
On the bright side, J possesses a crucial feature: it maps relatively compact sets to relatively compact sets. We prove this in Lemma 2.6 below. But first we need to digress into the characterization of tightness on \({\mathcal {P}}(\mathcal P(Y))\) and subspaces thereof. The following can be found in [39, p. 178, Ch. II].
Lemma 2.3
A set \({\mathcal {A}}\subseteq {\mathcal {P}}({\mathcal {P}}(Y))\) is tight if and only if the set of its intensities \(I({\mathcal {A}})\) is tight in \({\mathcal {P}}(Y)\).
We need to refine Lemma 2.3 for our purposes, since we equip \({\mathcal {P}}_{d_Y}^t(Y)\) with the \(\mathcal W_t\)topology instead of the weak topology.
Lemma 2.4
A set \({\mathcal {A}}\subseteq {\mathcal {P}}^t_{{\mathcal {W}}_t}(\mathcal P_{d_Y}^t(Y))\) is relatively compact if and only if the set of its intensities \(I({\mathcal {A}})\) is relatively compact in \(\mathcal P_{d_Y}^t(Y)\).
The proof of Lemma 2.4 heavily relies on the following lemma, for which we include a proof for sake of completeness.
Lemma 2.5
Note that if (2.7) holds for some \(y'\in Y\) it automatically holds for any \(y' \in Y\).
Proof of Lemma 2.4
Proof of Lemma 2.5
Lemma 2.6
If \(\Pi \subseteq {\mathcal {P}}_{d}^t(X\times Y)\) is relatively compact then \(J(\Pi )\subseteq {\mathcal {P}}^t_{{\hat{d}}}(X\times \mathcal P_{d_Y}^t(Y))\) is relatively compact. Conversely, if \(\Lambda \in {\mathcal {P}}^t_{{\hat{d}}}(X\times {\mathcal {P}}_{d_Y}^t(Y))\) is relatively compact then \({\hat{I}}(\Lambda )\subseteq {\mathcal {P}}^t_d(X\times Y)\) is relatively compact.
Proof
Since continuous maps preserve relative compactness in Hausdorff spaces, we immediately deduce relative compactness of \({\hat{I}}(\Lambda )\), and the sets \(\Pi ^X\subseteq {\mathcal {P}}(X)\) and \(\Pi ^Y\subseteq {\mathcal {P}}_{d_Y}^t(Y)\) consisting respectively of the X and Ymarginals of the elements in \(\Pi \).
Denote now respectively by \(\Pi _J^X\subseteq {\mathcal {P}}(X)\) and \(\Pi _J^Y\subseteq {\mathcal {P}}_{{\mathcal {W}}_t}^t({\mathcal {P}}_{d_Y}^t(Y))\) the set of X and \({\mathcal {P}}(Y)\)marginals of the elements in \(J(\Pi )\). Clearly \(\Pi _J^X=\Pi ^X\). By Lemma 2.4, the set \(\Pi _J^Y\) is relatively compact in \({\mathcal {P}}_{\mathcal W_t}^t({\mathcal {P}}_{d_Y}^t(Y))\) if and only if the set \(I(\Pi _J^Y)\) is relatively compact in \({\mathcal {P}}_{d_Y}^t(Y)\). However, if m is equal to the \({\mathcal {P}}(Y)\)marginal of \(J(\pi )\), then I(m) is equal to the Ymarginal of \(\pi \). It follows that \(I(\Pi _J^Y)\subseteq \Pi ^Y\) is relatively compact and so is \(\Pi _J^Y\). Since the marginals of \(J(\Pi )\) are relatively compact, we conclude that \(J(\Pi )\) itself is relatively compact. \(\square \)
It is convenient to introduce the following assumptions, which we will often require:
Definition 2.7
 C is lower semicontinuous with respect to the product topology of$$\begin{aligned} (X,d_X) \times ({\mathcal {P}}^t_{d_Y}(Y), {\mathcal {W}}_t), \end{aligned}$$

C is bounded from below.
We now show that under Condition (A+) the cost functional defining the weak transport problem is lower semicontinuous:
Proposition 2.8
Proof
We are finally ready to provide our main existence result:
Theorem 2.9
Proof
The existence of minimizers in \(\Lambda \) and \(\Pi \) are direct consequences of their compactness and the lower semicontinuity of the objective functionals (Proposition 2.8).
Of course Theorems 1.1 and 1.2 are particular cases of the second half of Theorem 2.9. More generally: if A is compact in \({\mathcal {P}}(X)\) and B is compact in \((\mathcal P^t_{d_Y}(Y),{\mathcal {W}}_t)\), then \(\Pi :=\bigcup _{\mu \in A,\nu \in B}\Pi (\mu ,\nu )\) is compact in \({\mathcal {P}}_{d}^{t}(X\times Y)\) and Theorem 2.9 applies.
3 Duality
Theorem 3.1
Remark 3.2
A proof of Theorem 1.3 can be obtained by means of [25, Theorem 9.6], since we may verify the hypotheses therein thanks to our Proposition 2.8. We prefer to obtain the slightly stronger Theorem 3.1 via selfcontained arguments. The primal–dual equality (3.3) was obtained in [2, Theorem 4.2] in the case when X, Y are compact spaces.
Proof of Theorem 3.1
If for all \(x\in X\) the map \(C(x,\cdot )\) is convex,
4 On the restriction property
The restriction property of optimal transport roughly states that if a coupling is optimal, then the conditioning of the coupling to a subset is also optimal given its marginals. This property fails for weak optimal transport, as we illustrate with a simple example:
Example 4.1
However, we can state the following positive result.^{1}
Proposition 4.2
Suppose that \(\pi \) is optimal between the marginals \(\mu \) and \(\nu \), \(V(\mu ,\nu )<\infty \), and that \(C(x,\cdot )\) is convex. Let \(0 \le {\tilde{\mu }}\le \mu \) be a nonnegative measure such that \(0\not \equiv {\tilde{\mu }} \) and define \({\hat{\mu }} = {\tilde{\mu }}/{\tilde{\mu }}(X)\). Then \({\hat{\pi }}(dx,dy):={\hat{\mu }}(dx) \pi _x(dy)\) is optimal between its marginals.
Proof
5 CMonotonicity for weak transport costs
Cyclical monotonicity plays a crucial role in classical optimal transport [22, 35]. This has inspired similar development for weak transport costs in [6, 23]:
Definition 5.1
We first show that Cmonotonicity is necessary for optimality under minimal assumptions. We then provide strengthened assumptions under which Cmonotonicity is sufficient.
5.1 Cmonotonicity: necessity
Lemma 5.2
 (a)For every \(i = 1,\ldots , n\) there is a \(\mu _i\)null set \(A_i \subseteq X_i\) s.t.$$\begin{aligned} B \subseteq \bigcup _{i=1}^n {{\,\mathrm{proj}\,}}_{X_i}^{1}(A_i). \end{aligned}$$
 (b)
There exists a coupling \(\pi \in \Pi (\mu _1,\ldots ,\mu _n)\) with \(\pi (B) > 0\).
The previous lemma is originally stated only for Borel sets, but the same proof technique also works for analytic sets.
Our main result, concerning the necessity of Cmonotonicity is the following:
Theorem 5.3
Let C be jointly measurable and \(C(x,\cdot )\) be convex and lower semicontinuous for all x. Assume that \(\pi ^*\) is optimal for \(V(\mu ,\nu )\) and \(V(\mu ,\nu )<\infty \). Then \(\pi ^*\) is Cmonotone.
Proof
 (1)
\({\tilde{\pi }}\in \Pi (\mu ,\nu )\),
 (2)
\(\int \mu ( dx)C(x,\pi ^*_x) > \int \mu ( dx)C(x, {\tilde{\pi }}_x )\).
We conclude that no measure Q with the stated properties exists. By Lemma 5.2, we obtain that \(D_N\) is contained in a set of the form \(\bigcup _{k=1}^N {{\,\mathrm{proj}\,}}_k^{1}(M_N)\) where \(\mu (M_N) = 0\) and \({{\,\mathrm{proj}\,}}_k\) denotes the projection from \(X^N\) to its kth component. Since \(N\in {\mathbb {N}}\) was arbitrary, we can define the set \(\Gamma := (\bigcup _{N\in \mathbb {N}} M_N)^C\) with \(\mu (\Gamma ) = 1\), which has the desired property. \(\square \)
Lemma 5.4
Proof of Lemma 5.4

\(\sigma (\vec a) \in {\hat{A}}\)
 for each i, j such that \( 0\le i<j \le N\) it holds$$\begin{aligned} a_i = a_j \implies \sigma (i) < \sigma (j). \end{aligned}$$
Lemma 5.5
Proof
5.2 Cmonotonicity: sufficiency
The conditions under which Theorem 5.3 holds are rather mild. If we assume further continuity properties of C, the next theorem establishes that Cmonotonicity is also a sufficient criterion for optimality, resembling the classical case. For weak transport costs, we don’t know of any comparable result in the literature.
We recall that, for the given compatible complete metric \(d_Y\) on Y, we denote by \({\mathcal {W}}_1\) the 1Wasserstein distance [42, Chapter 7].
Theorem 5.6
In the proof we will use the following auxiliary result, which we will establish subsequently:
Lemma 5.7
Proof of Theorem 5.6
Proof of Lemma 5.7
6 On the Brenier–Strassen theorem of Gozlan and Juillet
Lemma 6.1
Proof
We now provide the proof of Theorem 1.4, in which case \(\theta (\cdot )=\cdot ^2\):
Proof of Theorem 1.4
Footnotes
 1.
In a preliminary version of this article the restriction property Proposition 4.2 was used to derive Theorem 1.4 from the compact version given by Gozlan and Juillet [23]. Following the insightful suggestion of the anonymous referee, we now give a more self contained argument that does not require Proposition 4.2 / [23]. We have decided to keep Proposition 4.2 since it might be of some independent interest.
 2.
In fact one obtains \( \max _{i\in \{1,\ldots ,N\}} d_{{\mathcal {N}}}(g(\vec a)(\vec a)_i,g(\vec b)(\vec b)_i) \le \max _{i \in \{1,\ldots ,N\}} d_{{\mathcal {N}}}(a_i,b_i)\), for \(d_{{\mathcal {N}}}\) the metric on \({{\mathcal {N}}}\) that we recall in Lemma 5.5.
 3.
A vector \(v = (v_i)_{i=1}^N\in {\mathcal {N}}^N\) is increasing if for any \(1 \le i < j \le N\) we have \(v_i \le v_j\), where inequality here is meant in the lexicographic order on \(\mathcal {N}\).
Notes
Acknowledgements
Open access funding provided by University of Vienna.
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