Duality and quotient spaces of generalized Wasserstein spaces
aa r X i v : . [ m a t h . M G ] J un DUALITY AND QUOTIENT SPACES OF GENERALIZEDWASSERSTEIN SPACES
NHAN-PHU CHUNG AND THANH-SON TRINH
Abstract.
In this article, using ideas of Liero, Mielke and Savar´e in [21] we establisha Kantorovich duality for generalized Wasserstein distances W a,b on a generalized Polishmetric space, introduced by Picolli and Rossi in [24]. As a consequence, we give anotherproof that W a,b coincide with flat metrics which is a main result of [25], and thereforewe get a result of independent interest that Ä M ( X ) , W a,b ä is a geodesic space for everyPolish metric space X . We also prove that ( M G ( X ) , W a,bp ) is isometric isomorphism to( M ( X/G ) , W a,bp ) for isometric actions of a compact group G on a Polish metric space X ;and several results of Gromov-Hausdorrf convergence and equivariant Gromov-Hausdorffconvergence of generalized Wasserstein spaces. The latter results were proved for standardWasserstein spaces in [22],[14] and [8] respectively. Introduction
The Monge-Kantorovich’s balanced optimal transport problem has been studied exten-sively after pioneer works of Kantorovich on 1940s [17, 18]. In connection with this problem,Wassertein distances in the space of probability measures are powerful tools to study gradi-ent flows and partial differential equations [1] and theory of Ricci curvature bounded belowfor general metric-measure spaces [22, 28].Recently, unbalanced optimal transport problems and various generalized Wassersteindistances on the space of finite measures have been introduced and investigated by numerousauthors [7, 20, 21, 24]. In [24], Piccoli and Rossi defined a generalized Wassertein distance W a,bp ( µ, ν ), combining the usual Wasserstein distance and L -distance. After that, they alsoproved the generalized Benamou-Breiner formula for W a,bp and showed that the generalizedWasserstein distance W , coincides with the flat metric [25]. As natural we would ask whichother properties of standard Wasserstein distances still hold for generalized Wassersteindistances W a,bp .In this article, our first result is the Kantorovich duality for the distance W a,b . In [21],Liero, Mielke and Savar´e established Kantorovich duality for various Entropy-Transportproblems where entropy functions satisfy coercive conditions. As our nonsmooth entropyfunction F ( s ) = a | − s | is not superlinear and the cost function b.d ( · , · ) does not havecompact sublevels when X is a general Polish metric space we can not get the Kantorovichduality in our setting directly from [21]. However, inspiring from their methods we canprove that Date : June 11, 2019.
Theorem 1.1.
Let X be a Polish metric space. For any µ , µ ∈ M ( X ) , we have W a,b ( µ , µ ) = sup ( ϕ ,ϕ ) ∈ Φ W X i Z X I ( ϕ i ( x )) dµ i ( x ) , where I ( ϕ ) = inf s ≥ ( sϕ + a | − s | ) for ϕ ∈ R , and Φ W := { ( ϕ , ϕ ) ∈ C b ( X ) × C b ( X ) | ϕ ( x )+ ϕ ( y ) ≤ b.d ( x, y ) and ϕ ( x ) , ϕ ( y ) ≥ − a, ∀ x, y ∈ X } . As a consequence, we get a version of Kantorovich-Rubinstein theorem for generalizedWasserstein distance W a,b , which is a main result of [25] and is proved by a different methodthere. Theorem 1.2.
Let ( X, d ) be a Polish metric space, Then for every a, b > , µ, ν ∈ M ( X ) we have W a,b ( µ, ν ) = sup ¶ Z X f d ( µ − ν ) : f ∈ F © , where F := ¶ f ∈ C b ( X ) , k f k ∞ ≤ a, k f k Lip ≤ b © . And from that we get the following result which is independent of interest.
Corollary 1.3.
Let ( X, d ) be a Polish metric space and let a, b > . Then Ä M ( X ) , W a,b ä isa geodesic space. On the other hand, in [22] Lott and Villani established an isometric isomorphism forthe Wasserstein spaces P G ( X ) of G -invariant elements in P ( X ) and P ( X/G ), where
X/G is the quotient space of X induced from an isometric action of a compact group G on acompact metric space X . Later, this result is extended for general metric spaces X in [14].Our second result is its version for generalized Wasserstein distances W a,bp . Theorem 1.4.
Let a compact group G act on the right of a locally compact Polish metricspace ( X, d ) by isometries. Let p : X → X/G be the natural quotient map and numbers a, b > , p ≥ . Then(1) the map p ♯ : M p ( X ) → M p ( X/G ) is onto and furthermore for every ν ∗ ∈ M p ( X/G ) we can find µ ∈ M Gp ( X ) such that p ♯ µ = ν ∗ ;(2) W a,bp ( p ♯ µ, p ♯ ν ) ≤ W a,bp ( µ, ν ) for every µ, ν ∈ M ( X ) ;(3) the map p ♯ : ( M G ( X ) , W a,bp ) → ( M ( X/G ) , W a,bp ) is an isometry;(4) the map p ♯ : ( M Gp ( X ) , W a,bp ) → ( M p ( X/G ) , W a,bp ) is an isometry. Lastly, we prove Gromov-Hausdorff convergence of the generalized Wasserstein spaces andequivariant Gromov-Hausdorff convergence for induced actions on generalized Wassersteinspaces. These results have been established for standard Wasserstein spaces in [22] and [8]respectively.
Theorem 1.5.
Let { ( X n , d n ) } be a sequence of bounded, Polish metric spaces and C > . If { ( X n , d n ) } converges in the Gromov-Hausdorff topology to a bounded, Polish met-ric space ( X, d ) then ¶Ä M Cp ( X n ) , W a,bp ä© converges in the Gromov-Hausdorff topology to Ä M Cp ( X ) , W a,bp ä for every a, b > , p ≥ . UALITY AND QUOTIENTS SPACES OF GENERALIZED WASSERSTEIN SPACES 3
Theorem 1.6.
Let { α n } be a sequence of continuous actions of a topological group G onbounded, Polish metric spaces { X n } and let C > , p ≥ , a, b > . Let ( α n ) ♯ be the inducedaction of α n on the space ( M Cp ( X n ) , W a,bp ) , for every n ∈ N . If lim n →∞ d mGH ( α n , α ) = 0 for some continuous action α of G on a bounded, Polish metric space X . We denote by α ♯ the induced action of α on the space ( M Cp ( X n ) , W a,bp ) . Then lim n →∞ d mGH Ä ( α n ) ♯ , α ♯ ä = 0 . The paper is organized as following. In section 2, we review generalized Wassersteindistances and equivariant Gromov-Hausdorff distance. In section 3, we will prove theorem1.5 and theorem 1.6. Theorem 1.4 will be proved in section 4. Finally, in section 5 we willprove theorem 1.1, theorem 1.2 and corollary 1.3. Furthermore, in this last section we alsopresent some interesting consequences of theorem 1.2, and several results of optimal plansand dual optimal for W a,b . Acknowledgements:
Part of this paper was carried out when N. P. Chung visitedUniversity of Science, Vietnam National University at Hochiminh city on January 2019.He is grateful to Dang Duc Trong for his warm hospitality. The authors were partiallysupported by the National Research Foundation of Korea (NRF) grants funded by theKorea government (No. NRF- 2016R1A5A1008055 , No. NRF-2016R1D1A1B03931922 andNo. NRF-2019R1C1C1007107). We also thank Benedetto Piccoli and Francesco Rossi forpointing [15] out to us. 2.
Preliminaries
Notations, Wasserstein spaces and generalized Wasserstein spaces.
First, we review notations we use in the paper and recall the definitions of Wassersteindistances and some of their properties. For more details, readers can see [29, 30].Let (
X, d ) be a metric space. We denote by M ( X ) and P ( X ) the sets of all nonnegativeBorel measures with finite mass and all probability Borel measures, respectively.Given a Borel measure µ , we denote its mass by | µ | := µ ( X ). A set M ⊂ M ( X ) isbounded if sup µ ∈ M | µ | < ∞ , and it is tight if for every ε >
0, there exists a compact subset K ε of X such that for all µ ∈ M , we have µ ( X \ K ε ) ≤ ε .For every µ, ν ∈ M ( X ), we say that µ is absolutely continuous with respect to ν andwrite µ ≪ ν if ν ( A ) = 0 yields µ ( A ) = 0 for every Borel subset A of X . We call that µ and ν are mutually singular and write µ ⊥ ν if there exists a Borel subset B of X such that µ ( B ) = ν ( X \ B ) = 0. We write µ ≤ ν if for all Borel subset A of X we have µ ( A ) ≤ ν ( A ). Theorem 2.1. (Prokhorov’s theorem) Let ( X, d ) be a metric space. If a subset M ⊂ M ( X ) is bounded and tight then M is relatively compact under the weak*- topology. For every p ≥
1, we denote by M p ( X ) (reps. P p ( X )) the space of all measures µ ∈ M ( X )(reps. P ( X )) with finite p -moment, i.e. there is some (and therefore any) x ∈ X such that Z X d p ( x, x ) dµ ( x ) < ∞ . NHAN-PHU CHUNG AND THANH-SON TRINH
For every measures µ, ν ∈ M ( X ), a Borel probability measure π on X × X is called atransference plan between µ and ν if | µ | π ( A × X ) = µ ( A ) and | ν | π ( X × B ) = ν ( B ) , for every Borel subsets A, B of X . We denote the set of all transference plan between µ and ν by Π( µ, ν ).Given measures µ, ν ∈ M p ( X ) with the same mass, i.e. | µ | = | ν | . The Wassersteindistance between µ and ν is defined by W p ( µ, ν ) := Ç | µ | inf π ∈ Π( µ,ν ) J p ( π ) å /p , where J p ( π ) = R X × X d p ( x, y ) dπ ( x, y ). For each µ, ν ∈ M ( X ) with | µ | = | ν | , we denote byOpt p ( µ, ν ) the set of all π ∈ Π( µ, ν ) such that W pp ( µ, ν ) = | µ | J p ( π ). If ( X, d ) is a Polishmetric space, i.e. (
X, d ) is complete and separable then Opt p ( µ, ν ) is nonempty. This resultfollows from [29, Theorem 1.3] by setting µ ∗ = µ/ | µ | , ν ∗ = ν/ | ν | ∈ P p ( X ).Let ( X, d X ) and ( Y, d Y ) be metric spaces and f : X → Y be a Borel map. We denote by f ♯ µ ∈ M ( Y ) the push-forward measure defined by f ♯ µ ( B ) := µ Ä f − ( B ) ä , for every Borel subset B of Y .We now review the definitions of the generalized Wasserstein distances introduced byPiccoli and Rossi in [24]. Note that although in [24] the authors only presented for the case X = R d their methods work for a general Polish metric space X . Definition 2.2.
Let X be a Polish metric space and let a, b > , p ≥
1. For every µ, ν ∈ M ( X ), the generalized Wasserstein distance W a,bp between µ and ν is defined by W a,bp ( µ, ν ) := inf e µ, e ν ∈ M p ( X ) | e µ | = | e ν | C ( e µ, e ν ) , where C ( e µ, e ν ) = a | µ − e µ | + a | ν − e ν | + b W p ( e µ, e ν ) . Proposition 2.3. ( [24, Proposition 1] ) If X is a Polish metric space then Ä M ( X ) , W a,bp ä isa metric space. Moreover, there exists e µ, e ν ∈ M p ( X ) such that | e µ | = | e ν | , e µ ≤ µ, e ν ≤ ν and W a,bp ( µ, ν ) = C ( e µ, e ν ) . If measures e µ, e ν ∈ M p ( X ) with the same mass such that W a,bp ( µ, ν ) = C ( e µ, e ν ) then wesay that ( e µ, e ν ) is an optimal for W a,bp ( µ, ν ). Proposition 2.4. ( [24, Proposition 4] ) If ( X, d ) is a Polish metric space then Ä M ( X ) , W a,bp ä is a complete metric space. Equivariant Gromov-Hausdorff distances for group actions.
First, we recall the definition of Gromov-Hausdorff distance between two metric spaces.For more details, see standard references [6, 27].
UALITY AND QUOTIENTS SPACES OF GENERALIZED WASSERSTEIN SPACES 5
Let (
X, d ) be a metric space. For every ε >
0, the ε -neighborhood of a subset S of X ,denoted by B ε ( S ), is defined as B ε ( S ) = S x ∈ S B ε ( x ). Definition 2.5.
Let X and Y be subsets of a metric space ( Z, d ). The Hausdorff distancebetween X and Y , denoted by d H ( X, Y ) is defined as follow d H ( X, Y ) := inf { ε > | X ⊂ B ε ( Y ) and Y ⊂ B ε ( X ) } . Definition 2.6.
Let X and Y be metric spaces. The Gromov-Hausdorff distance d GH ( X, Y )is the infimum of r >
Z, d ) and subspaces X ′ and Y ′ of Z which are isometric to X and Y respectively such that d H ( X ′ , Y ′ ) < r . Definition 2.7.
Given two bounded metric spaces ( X , d ) , ( X , d ). An ε -Gromov-Hausdorffapproximation from X to X is a map f : X → X such that(i) For every x , x ′ ∈ X then | d ( f ( x ) , f ( x ′ )) − d ( x , x ′ ) | ≤ ε (ii) For every x ∈ X , there exists x ∈ X such that d ( f ( x ) , x ) ≤ ε .If f is an ε -Gromov-Hausdorff approximation from X to X then it has an approximateinverse f ′ : X → X which is a 3 ε -Gromov-Hausdorff approximation from X to X .To see this, we will construct f ′ as follows. Let x ∈ X , choose x ∈ X such that d ( x , f ( x )) ≤ ε , since the second condition of definition of f . Setting f ′ ( x ) = x then f ′ is a 3 ε -Gromov-Hausdorff approximation from X to X . Moreover, it is clear that d ( x , f ′ ( f ( x ))) ≤ ε and d ( x , f ( f ′ ( x ))) ≤ ε for every x ∈ X , x ∈ X . Now we review the equivariant Gromov-Hausdorff distances. They were introduced firstby Fukaya in [10–13] for isometric actions. After that they have been studied further forgeneral actions [2, 8, 9, 19].Let (
X, d ) be a metric space. The C distance between the maps f, g : ( X, d ) → ( X, d ) isdefined by d sup ( f, g ) := sup x ∈ X d ( f ( x ) , g ( x )) . Definition 2.8.
Let α and β be continuous actions of G on metric spaces ( X, d X ) and( Y, d Y ) respectively. A map f : G y X → G y Y is an ε -GH approximation from α to β if f : X → Y is an ε -isometry satisfying that d sup ( f ◦ α g , β g ◦ f ) ≤ ε for every g ∈ G . If f is measurable we say that f is an ε -measurable GH approximation. Definition 2.9.
Let α and β be continuous actions of G on metric spaces ( X, d X ) and( Y, d Y ) respectively. The equivariant GH-distance d GH and d mGH between α and β aredefined by d GH ( α, β ) := inf { ε > ∃ ε -GH approximations f : G y X → G y Y and g : G y Y → G y X } ,d mGH ( α, β ) := inf { ε > ∃ ε -measurable GH approximations f : G y X → G y Y and g : G y Y → G y X } , and is ∞ if the infimum does not exist. NHAN-PHU CHUNG AND THANH-SON TRINH Gromov-Hausdorff convergences for generalized Wasserstein spaces
In this section, we will prove theorem 1.5 and theorem 1.6.Let X be a Polish metric space and let C > p ≥
1. We denote by M Cp ( X ) thespace of all measures µ ∈ M p ( X ) such that | µ | ≤ C . Note that when X is bounded then M p ( X ) = M ( X ) for every p ≥ Lemma 3.1.
Let ( X , d ) and ( X , d ) be two bounded, Polish metric spaces and C > .If f : ( X , d ) → ( X , d ) is an ε -Gromov-Hausdorff approximation and measurable then f ♯ : Ä M Cp ( X ) , W a,bp ä → Ä M Cp ( X ) , W a,bp ä is an e ε -Gromov-Hausdorff approximation, where e ε = 8 bC /p ε + b Ä pC Ä diam( X ) p − + diam( X ) p − ä ε ä /p . Proof.
Given µ, ν ∈ M Cp ( X ) and let ( e µ, e ν ) ∈ M p ( X ) × M p ( X ) be an optimal for W a,bp ( µ, ν )such that | e µ | = | e ν | and e µ ≤ µ, e ν ≤ ν .Setting µ := f ♯ e µ and ν := f ♯ e ν then µ ≤ f ♯ µ, ν ≤ f ♯ ν and | µ | = | ν | . Therefore W a,bp ( f ♯ µ, f ♯ ν ) ≤ a | f ♯ µ − µ | + a | f ♯ ν − ν | + bW p ( µ, ν ) . Let π be an optimal transference plan between e µ and e ν . Define π := ( f × f ) ♯ π . Then π ∈ Π ( f ♯ e µ, f ♯ e ν ). Therefore, W pp ( f ♯ e µ, f ♯ e ν ) ≤ | f ♯ e µ | Z X × X d p ( x , y ) dπ ( x , y ) = | e µ | Z X × X d p ( f ( x ) , f ( y )) dπ ( x , y ) . Applying the mean value theorem for the function t p , t ≥ | x p − y p | ≤ p | x − y | max ¶ x p − , y p − © ≤ p | x − y | Ä x p − + y p − ä . So for all x , y ∈ X , | d p ( f ( x ) , f ( y )) − d p ( x , y ) | ≤≤ p | d ( f ( x ) , f ( y )) − d ( x , y ) | Ä d p − ( f ( x ) , f ( y )) + d p − ( x , y ) ä ≤ pM ε, where M = diam ( X ) p − + diam ( X ) p − . Therefore, W pp ( f ♯ e µ, f ♯ e ν ) ≤ | e µ | Z X × X d p ( x , y ) dπ ( x , y ) + | e µ | pM ε = W pp ( e µ, e ν ) + | e µ | pM ε. Since | e µ | ≤ | µ | ≤ C , one has W p ( f ♯ e µ, f ♯ e ν ) ≤ W p ( e µ, e ν ) + ( pCM ε ) /p . Moreover, as | f ♯ µ − f ♯ e µ | + | f ♯ ν − f ♯ e ν | = | µ − e µ | + | ν − e ν | we obtain W a,bp ( f ♯ µ, f ♯ ν ) ≤ a | µ − e µ | + a | ν − e ν | + bW p ( e µ, e ν ) + b ( pCM ε ) /p = W a,bp ( µ, ν ) + b ( pCM ε ) /p . Now, using [8, Lemma 4.1] there exists a measurable function f ′ which is a 9 ε -Gromov-Hausdorff approximation from X to X , and d ( x , ( f ′ ◦ f ) ( x )) ≤ ε for all x ∈ X and d ( x , ( f ◦ f ′ ) ( x )) ≤ ε for all x ∈ X . UALITY AND QUOTIENTS SPACES OF GENERALIZED WASSERSTEIN SPACES 7
Using the same argument as above we get that W a,bp Ä f ′ ♯ ( f ♯ µ ) , f ′ ♯ ( f ♯ ν ) ä ≤ W a,bp ( f ♯ µ, f ♯ ν ) + b (9 pCM ε ) /p . Since | µ | = | f ′ ♯ ( f ♯ µ ) | we have, W a,bp Ä µ, f ′ ♯ ( f ♯ µ ) ä ≤ bW p Ä µ, f ′ ♯ ( f ♯ µ ) ä . By [8, Lemma 2.8] weget that W pp Ä µ, f ′ ♯ ( f ♯ µ ) ä ≤ | µ | Z X d p ( x , f ′ ( f ( x ))) dµ ( x ) ≤ | µ | (4 ε ) p ≤ C (4 ε ) p . And thus, W a,bp Ä µ, f ′ ♯ ( f ♯ µ ) ä ≤ bC /p ε. Similarly, W a,bp Ä ν, f ′ ♯ ( f ♯ ν ) ä ≤ bC /p ε .Therefore, W a,bp ( µ, ν ) ≤ W a,bp Ä f ′ ♯ ( f ♯ µ ) , f ′ ♯ ( f ♯ ν ) ä + W a,bp Ä µ, f ′ ♯ ( f ♯ µ ) ä + W a,bp Ä ν, f ′ ♯ ( f ♯ ν ) ä ≤ W a,bp ( f ♯ µ, f ♯ ν ) + b (9 pCM ε ) /p + 8 bC /p ε. And hence (cid:12)(cid:12)(cid:12) W a,bp ( f ♯ µ, f ♯ ν ) − W a,bp ( µ, ν ) (cid:12)(cid:12)(cid:12) ≤ b (9 pCM ε ) /p + 8 bC /p ε = e ε. (3.1)Moreover, for all µ ∈ M Cp ( X ), let µ = f ′ ♯ µ we will prove that W a,bp ( µ , f ♯ µ ) ≤ e ε . Since | µ | = | f ♯ Ä f ′ ♯ µ ä | we have W a,bp ( µ , f ♯ µ ) ≤ bW p Ä µ , f ♯ Ä f ′ ♯ µ ää . Applying [8, Lemma 2.8]again we obtain W pp Ä µ , f ♯ Ä f ′ ♯ µ ää ≤ | µ | Z X d p ( x , f ( f ′ ( x ))) dµ ( x ) ≤ | µ | (4 ε ) p ≤ C (4 ε ) p . Therefore W a,bp ( µ , f ♯ µ ) ≤ bC /p ε ≤ e ε. (3.2)Combining (3.1) and (3.2) we have f ♯ is an e ε -Gromov-Hausdorff approximation. (cid:3) Proof of Theorem 1.5.
Since { ( X n , d n ) } converges in the Gromov-Hausdorff topology to( X, d ), there exists a sequence of ε n -approximations f n : X n → X with lim n →∞ ε n = 0.By [8, Lemma 4.1], there is a sequence of functions f ∗ n that is measurable and 5 ε n -Gromov-Hausdorff approximations from X n to X . Using lemma 3.1 we get the result. (cid:3) Lemma 3.2.
Let α , α be actions of a topological group G on bounded, Polish metricspaces ( X , d ) , ( X , d ) , respectively and let C > . If f : X → X is an ε -measurableGH approximation from α to α then for every p ≥ , the map f ♯ : Ä M Cp ( X ) , W a,bp ä → Ä M Cp ( X ) , W a,bp ä is an e ε -measurable GH approximation from ( α ) ♯ to ( α ) ♯ where e ε = 8 bC /p ε + b Ä pC Ä diam( X ) p − + diam( X ) p − ä ε ä /p . NHAN-PHU CHUNG AND THANH-SON TRINH
Proof.
By lemma 3.1 we get that f ♯ is an e ε -Gromov-Hausdorff approximation. Therefore,to finish the proof, we only need to check that d sup Ä f ♯ ◦ ( α ) ♯,g , ( α ) ♯,g ◦ f ♯ ä ≤ e ε. Let µ ∈ M Cp ( X ) , g ∈ G . Since d sup ( f ◦ α ,g , α ,g ◦ f ) ≤ ε and (cid:12)(cid:12)(cid:12) ( f ◦ α ,g ) ♯ µ (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ( α ,g ◦ f ) ♯ µ (cid:12)(cid:12)(cid:12) = | µ | , applying [8, Lemma 2.8] we obtain W pp Ä ( f ◦ α ,g ) ♯ µ , ( α ,g ◦ f ) ♯ µ ä ≤ | µ | Z X d p (( f ◦ α ,g ) ( x ) , ( α ,g ◦ f ) ( x )) dµ ( x ) ≤ C ε p . As (cid:12)(cid:12)(cid:12) ( f ◦ α ,g ) ♯ µ (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ( α ,g ◦ f ) ♯ µ (cid:12)(cid:12)(cid:12) we have W a,bp ÄÄ f ♯ ◦ ( α ) ♯,g ä ( µ ) , Ä ( α ) ♯,g ◦ f ♯ ä ( µ ) ä = W a,bp Ä ( f ◦ α ,g ) ♯ µ , ( α ,g ◦ f ) ♯ µ ä ≤ bW p Ä ( f ◦ α ,g ) ♯ µ , ( α ,g ◦ f ) ♯ µ ä ≤ bC /p ε ≤ e ε. (cid:3) Proof of Theorem 1.6.
This theorem follows from lemma 3.2. (cid:3)
Remark 3.3.
From [8, Lemma 4.2] , we see that if α n , α are isometric actions then theconclusion of theorem 1.6 is also true for d GH instead of d mGH . The quotient maps of generalized Wasserstein spaces
Let X be a locally compact space. A continuous map f : X → X is proper if f − ( K )is compact for every compact K ⊂ X . A continuous action of a locally compact group G to the right on X is proper if the map α : G × X → X × X , defined by ( g, x ) ( xg, x )is proper. Let a locally compact group G act continuously and properly to the right on X .For x ∈ X the orbit G ( x ) is defined by G ( x ) = { y ∈ X | ∃ g ∈ G : y = xg } . We denote X/G the orbit space with the relation ∼ is defined by x ∼ y iff ∃ g ∈ G : y = xg . Asthe action is continuous and proper, the orbit space X/G is Hausdorff and locally compact[4, Chapter III, § § X/G .Let λ be a left Haar measure on G and p : X → X/G be the natural quotient map. Let f ∈ C c ( X ) and x ∈ X . As the action is proper, the function G → C , g f ( xg ) is in C c ( G ).Then we can define the map f : X → C by f ( x ) := R G f ( xg ) dλ ( g ) for every x ∈ X . Since λ is left invariant we get that f ( xh ) = f ( x ) for every x ∈ X and h ∈ G . Therefore wecan define the map f ∗ : X/G → C by f ∗ ( p ( x )) = R G f ( xg ) dλ ( g ), for every x ∈ X . It isnot difficult to see that the function f is continuous and hence f ∗ is continuous on X/G as the map p is an open map. As supp( f ) ⊂ Y for some compact subset Y of X , one hassupp( f ∗ ) ⊂ p ( Y ), a compact subset of X/G and hence f ∗ ∈ C c ( X/G ). As a consequence,
UALITY AND QUOTIENTS SPACES OF GENERALIZED WASSERSTEIN SPACES 9 we can define a linear function Φ : C c ( X ) → C c ( X/G ) by Φ( f ) = f ∗ and Φ( f ) ≥ f ≥
0. Applying Riesz representation theorem we get that for a Borel measure ν on X/G there is a Borel measure b ν on X such that b ν ( f ) = ν ( f ∗ ) for every f ∈ C c ( X ).On the other hand, for every x ∈ X, h ∈ G one has( R h − f ) ( x ) = Z G R h − f ( xg ) dλ ( g ) = ∆( h ) Z G f ( xg ) dλ ( g ) = ∆( h ) f ( x ) , where R h f ( x ) = f ( xh ) for every x ∈ X, h ∈ G , and ∆ : G → (0 , ∞ ) is the right-handmodular function of G , i.e. R G s ( gh − ) dλ ( g ) = ∆( h ) R G s ( g ) dλ ( g ) for every h ∈ G , s ∈ L ( G, λ ).Therefore for every f ∈ C c ( X ) , h ∈ G one has ( R h f ) ∗ = ∆( h − ) f ∗ and hence b ν ( R h − f ) = ∆( h ) b ν ( f ) . Furthermore, we have the following result.
Lemma 4.1. ( [5, Chapter 7, §
2, Proposition 4] or [31, Theorem 7.3.3] ) Let a locally compactgroup G act properly on the right of a locally compact space X and let λ be a left Haarmeasure of G . Then(1) Given a Borel measure ν on X/G there exists a unique Borel measure b ν on X such that b ν ( f ) = ν ( f ∗ ) for every f ∈ C c ( X ) . In addition to, one has b ν ( R h − f ) =∆( h ) b ν ( f ) , for every h ∈ G .(2) Conversely, let µ be a Borel measure on X such that µ ( R h − f ) = ∆( h ) µ ( f ) , forevery f ∈ C c ( X ) , h ∈ G . Then there exists a unique Borel measure µ b on X/G suchthat µ = c µ b . The measure µ b in the previous lemma is called the quotient of µ and λ and is denotedby µ/λ .If the acting group G is unimodular, i.e. ∆( h ) = 1 for every h ∈ G , then we get that forevery Borel measure ν of X/G there exists a unique G -invariant measure b ν on X such that b ν ( f ) = ν ( f ∗ ) for every f ∈ C c ( X ). Furthermore, if G is compact and λ is the normalizedHaar measure of G then for every G -invariant measure µ of G , the quotient measure µ/λ coincides with the push forward measure p ♯ µ where p : X → X/G is the natural quotientmap [31, Proposition 7.3.5]. Therefore in this case the map p ♯ : M G ( X ) → M ( X/G ) isbijective, where M G ( X ) is the space of all µ ∈ M ( X ) which is G -invariant. A measure µ ∈ M ( X ) is G -invariant if µ ( Ag ) = µ ( A ) for every g ∈ G and Borel subset A of X .From now on, the acting group G is compact and let G act on the right of a locallycompact complete separable metric space ( X, d ) by isometries. An element of the quotient
X/G will be denoted by x ∗ = p ( x ). On X/G we define the followings distance d X/G ( x ∗ , y ∗ ) := inf g ∈ G d X ( gx, y ) = inf g,h ∈ G d X ( gx, hy ) . Since G is compact, for every x, y ∈ X , there exists g ∈ G such that d X/G ( x ∗ , y ∗ ) = d X ( gx, y ) . As (
X, d ) is a locally compact complete separable metric space, so is (
X/G, d
X/G ). For every p ≥
1, we denote M Gp ( X ) = M p ( X ) ∩ M G ( X ). Let λ be the normalized Haarmeasure of G . For every x ∈ X , the measure λ x := R G δ xg dλ ( g ) is the unique G -invariantprobability measure satisfying p ♯ λ x = δ x ∗ . As λ x = λ y whenever there is some g ∈ G such that x = yg , the map from X/G to P ( X ) defined by x ∗ λ x is well defined andmeasurable. Therefore, for every µ ∗ ∈ P ( X/G ) we can define a G -invariant measure c µ ∗ on X by c µ ∗ := R X/G λ x dµ ∗ ( x ∗ ).Now we are ready to prove theorem 1.4. Proof of Theorem 1.4. (1) We need to check that if µ ∈ M p ( X ) then p ♯ µ ∈ M p ( X/G ). Since µ ∈ M p ( X ) thereexists x ∈ X such that R X d pX ( x , x ) dµ ( x ) < ∞ . So Z X/G d pX/G ( x ∗ , x ∗ ) d ( p ♯ µ ) ( x ∗ ) = Z X d pX/G ( x ∗ , p ( x )) dµ ( x ) ≤ Z X d pX ( x , x ) dµ ( x ) < ∞ . Therefore p ♯ µ ∈ M p ( X/G ).Now we prove that p ♯ : M p ( X ) → M p ( X/G ) is onto. Let ν ∗ ∈ M p ( X/G ). Then thereexists x ∗ ∈ X/G such that R X/G d ∗ ( x ∗ , x ∗ ) p dν ∗ ( x ∗ ) < ∞ . Applying [14, Theorem 3.2] we getthat W p ( “ ν ∗ | “ ν ∗ | , λ x ) = W p ( ν ∗ | ν ∗ | , δ x ∗ ) and therefore Ä Z X d p ( x, x ) d c ν ∗ ( x ) ä /p = | c ν ∗ | /p W p ( c ν ∗ | c ν ∗ | , δ x ) ≤ | c ν ∗ | /p ( W p ( c ν ∗ | c ν ∗ | , λ x ) + W p ( λ x , δ x ))= | c ν ∗ | /p ( W p ( ν ∗ | ν ∗ | , δ x ∗ ) + W p ( Z G δ x g dλ ( g ) , δ x )) ≤ Ä Z X/G d ∗ ( x ∗ , x ∗ ) p dν ∗ ( x ∗ ) ä /p + | c ν ∗ | /p sup g ∈ G d ( x g, x ) < ∞ . Hence c ν ∗ ∈ M p ( X ). As we also have c ν ∗ ∈ M G ( X ) and p ♯ c ν ∗ = ν ∗ we get that the map p ♯ : M p ( X ) → M p ( X/G ) is onto.(2) Let µ, ν ∈ M ( X ) and ( γ , γ ) ∈ M p ( X ) × M p ( X ) be an optimal for W a,bp ( µ, ν ) suchthat | γ | = | γ | and γ ≤ µ, γ ≤ ν . Let π be an optimal transference between γ and γ .Define π = ( p × p ) ♯ π then π ∈ Π ( p ♯ γ , p ♯ γ ). So that W pp ( p ♯ γ , p ♯ γ ) ≤ | p ♯ γ | Z ( X/G ) × ( X/G ) d pX/G ( x ∗ , y ∗ ) dπ ( x ∗ , y ∗ )= | γ | Z X × X d pX/G ( p ( x ) , p ( y )) dπ ( x, y ) ≤ | γ | Z X × X d pX ( x, y ) dπ ( x, y )= W pp ( γ , γ ) . UALITY AND QUOTIENTS SPACES OF GENERALIZED WASSERSTEIN SPACES 11
Since γ ≤ µ, γ ≤ ν and | γ | = | γ | we get p ♯ γ ≤ p ♯ µ, p ♯ γ ≤ p ♯ ν and | p ♯ γ | = | p ♯ γ | . Onthe other hand, we also have p ♯ γ i ∈ M p ( X/G ) , i = 1 ,
2. Therefore, W a,bp ( p ♯ µ, p ♯ ν ) ≤ a | p ♯ µ − p ♯ γ | + a | p ♯ ν − p ♯ γ | + bW p ( p ♯ γ , p ♯ γ ) ≤ a | µ − γ | + a | ν − γ | + bW p ( γ , γ )= W a,bp ( µ, ν ) . (3) As p ♯ : M G ( X ) → M ( X/G ) is onto and from (2), it is sufficient to prove that W a,bp ( p ♯ µ, p ♯ ν ) ≥ W a,bp ( µ, ν ) for every µ, ν ∈ M G ( X ) . Let µ, ν ∈ M G ( X ) and put µ ∗ = p ♯ µ, ν ∗ = p ♯ ν ∈ M ( X/G ). For every x ∈ X , we define a G -invariant measure λ x ∈ P ( X ) by λ x := R G δ xg dλ ( g ) . We set µ ∗ := µ ∗ / | µ ∗ | , ν ∗ := ν ∗ / | ν ∗ | and define measures c µ ∗ , c ν ∗ ∈ P ( X ) as follows c µ ∗ = Z X/G λ x dµ ∗ ( x ∗ ) and c ν ∗ = Z X/G λ x dν ∗ ( x ∗ ) . (4.1)Then c µ ∗ , c ν ∗ are the G -invariant probability measures on X such that p ♯ c µ ∗ = µ ∗ and p ♯ c ν ∗ = ν ∗ . Moreover, if we put µ = µ/ | µ | = µ/ | µ ∗ | and ν = ν/ | ν | = ν/ | ν ∗ | then µ , ν arealso the G -invariant probability measures on X satisfying p ♯ µ = µ ∗ , p ♯ ν = ν ∗ . Therefore, c µ ∗ = µ , c ν ∗ = ν , and hence µ = | µ ∗ | c µ ∗ and ν = | ν ∗ | c ν ∗ . Thus, we only need to prove that W a,bp ( µ ∗ , ν ∗ ) ≥ W a,bp Ä | µ ∗ | c µ ∗ , | ν ∗ | c ν ∗ ä . We choose anoptimal ( e µ ∗ , e ν ∗ ) ∈ M p ( X/G ) × M p ( X/G ) for W a,bp ( µ ∗ , ν ∗ ) such that | e µ ∗ | = | e ν ∗ | , e µ ∗ ≤ µ ∗ , e ν ∗ ≤ ν ∗ then W a,bp ( µ ∗ , ν ∗ ) = a | µ ∗ − e µ ∗ | + a | ν ∗ − e ν ∗ | + bW p ( e µ ∗ , e ν ∗ ) . Putting e µ ∗ := e µ ∗ / | e µ ∗ | , e ν ∗ := e ν ∗ / | e ν ∗ | and we define e µ ∗ = Z X/G λ x d e µ ∗ ( x ∗ ) and e ν ∗ = Z X/G λ x d e ν ∗ ( x ∗ ) . (4.2)Then e µ ∗ , e ν ∗ are the G -invariant probability measures on X such that p ♯ e µ ∗ = e µ ∗ , p ♯ e ν ∗ = e ν ∗ .Therefore, using [14, Theorem 3.2] we get that W p ( e µ ∗ , e ν ∗ ) = W p ( e µ ∗ , e ν ∗ ) . So W p ( e µ ∗ , e ν ∗ ) = | e µ ∗ | /p W p ( e µ ∗ , e ν ∗ ) = | e µ ∗ | /p W p ( e µ ∗ , e ν ∗ ) = W p ( | e µ ∗ | e µ ∗ , | e ν ∗ | e ν ∗ ) . Moreover, since (4.1) and (4.2), for every Borel subset A of X one has | e µ ∗ | e µ ∗ ( A ) = | e µ ∗ | Z X/G λ x ( A ) d e µ ∗ ( x ∗ ) = Z X/G λ x ( A ) d | e µ ∗ | e µ ∗ ( x ∗ ) ≤ Z X/G λ x ( A ) d | µ ∗ | µ ∗ ( x ∗ ) = | µ ∗ | c µ ∗ ( A ) . So | e µ ∗ | e µ ∗ ≤ | µ ∗ | c µ ∗ . As e µ ∗ ∈ M p ( X/G ) we get that so is e µ ∗ and therefore similar to theproof of (1) we have e µ ∗ ∈ M p ( X ). Hence | e µ ∗ | e µ ∗ ∈ M p ( X ). Similarly, | e ν ∗ | e ν ∗ ≤ | ν ∗ | c ν ∗ and | e ν ∗ | e ν ∗ ∈ M p ( X ). On the other hand, | µ ∗ − e µ ∗ | = (cid:12)(cid:12)(cid:12) | µ ∗ | µ ∗ − | e µ ∗ | e µ ∗ (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) | µ ∗ | c µ ∗ − | e µ ∗ | e µ ∗ (cid:12)(cid:12)(cid:12) and | ν ∗ − e ν ∗ | = (cid:12)(cid:12)(cid:12) | ν ∗ | ν ∗ − | e ν ∗ | e ν ∗ (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) | ν ∗ | c ν ∗ − | e ν ∗ | e ν ∗ (cid:12)(cid:12)(cid:12) . It implies that W a,bp ( µ ∗ , ν ∗ ) = a (cid:12)(cid:12)(cid:12) | µ ∗ | c µ ∗ − | e µ ∗ | e µ ∗ (cid:12)(cid:12)(cid:12) + a (cid:12)(cid:12)(cid:12) | ν ∗ | c ν ∗ − | f ν ∗ | e ν ∗ (cid:12)(cid:12)(cid:12) + bW p Ä | f µ ∗ | e µ ∗ , | f ν ∗ | e ν ∗ ä . Therefore we obtain W a,bp ( µ ∗ , ν ∗ ) ≥ W a,bp Ä | µ ∗ | c µ ∗ , | ν ∗ | c ν ∗ ä = W a,bp Ä c µ ∗ , c ν ∗ ä . (4) follows from (1) and (3). (cid:3) The dual formulation for the W a,b distance In this section, we will study a dual formulation for the generalized Wasserstein W a,b distance and its consequences. Before proving theorem 1.1, let us recall some preparationresults.Let F : [0 , ∞ ) → (0 , ∞ ) be a convex and lower semicontinuous function. We definefunction F ◦ : R → [ −∞ , ∞ ] by F ◦ ( ϕ ) := inf s ≥ Ä ϕs + F ( s ) ä for every ϕ ∈ R . As F is convexthe map x F ( x ) − F (0) x − is increasing in (0 , ∞ ) and hence we define F ′∞ := lim s →∞ F ( s ) s =sup s> F ( s ) − F (0) s . Now we define the functional F : M ( X ) × M ( X ) → [0 , ∞ ] by F ( γ | µ ) := Z X F ( f ) dµ + F ′∞ γ ⊥ ( X ) , where γ = f µ + γ ⊥ is the Lebesgue decomposition of γ with respect to µ . Theorem 5.1. ( [21, Theorem 2.7 and Remark 2.8] ) Let F : [0 , ∞ ) → (0 , ∞ ) be a convex andlower semicontinuous function and X be a Polish metric space. Then for every γ, µ ∈ M ( X ) ,we have F ( γ | µ ) = sup ¶ Z X F ◦ ( ϕ ) dµ − Z X ϕdγ : ϕ, F ◦ ( ϕ ) ∈ C b ( X ) © . We now give the proof of the easy part of theorem 1.1.
Lemma 5.2.
Suppose X is a Polish metric space. For every µ , µ ∈ M ( X ) , we have W a,b ( µ , µ ) ≥ sup ( ϕ ,ϕ ) ∈ Φ W X i Z X I ( ϕ i ( x )) dµ i ( x ) . Proof.
Let µ , µ ∈ M ( X ). Let ( γ , γ ) be an optimal for W a,b ( µ , µ ) such that | γ | = | γ | and γ i ≤ µ i , i = 1 ,
2. Then W a,b ( µ , µ ) = a | µ − γ | + a | µ − γ | + b W ( γ , γ ) . Let π be an optimal transference between γ and γ . We define γ := | γ | π then γ , γ arethe marginals of γ and W ( γ , γ ) = R X × X d ( x, y ) dγ ( x, y ) . For each i ∈ { , } , since γ i ≤ µ i , by Radon-Nikodym theorem we get that there exists ameasurable function f i : X → [0 ,
1] such that γ i = f i µ i . From this, we have a | µ i − γ i | = Z X a (1 − f i ( x )) dµ i ( x ) . Now, for every ( ϕ , ϕ ) ∈ Φ W we get that W a,b ( µ , µ ) = X i Z X a (1 − f i ( x )) dµ i ( x ) + b Z X × X d ( x, y ) dγ ( x, y ) ≥ X i Z X a (1 − f i ( x )) dµ i ( x ) + Z X × X ( ϕ ( x ) + ϕ ( y )) dγ ( x, y ) UALITY AND QUOTIENTS SPACES OF GENERALIZED WASSERSTEIN SPACES 13 = X i Z X ( a (1 − f i ( x )) + f i ( x ) ϕ i ( x )) dµ i ( x ) . Furthermore, as 0 ≤ f i ( x ) ≤ x ∈ X , we have I ( ϕ i ( x )) = inf s ≥ ( sϕ i ( x ) + a | − s | ) ≤ f i ( x ) ϕ i ( x ) + a (1 − f i ( x )) , i = 1 , . Therefore, W a,b ( µ , µ ) ≥ P i R X I ( ϕ i ( x )) dµ i ( x ) , for all ( ϕ , ϕ ) ∈ Φ W . (cid:3) Lemma 5.3. If X is a Polish metric space then for every µ , µ ∈ M ( X ) we have W a,b ( µ , µ ) = inf γ ∈ M ß a | µ − γ | + a | µ − γ | + b Z X × X d ( x, y ) dγ ( x, y ) ™ , where γ , γ are the marginals of γ and M = ß γ ∈ M ( X × X ) | Z X × X d ( x, y ) dγ ( x, y ) < ∞ ™ .Proof. For any γ ∈ M , let γ and γ be the marginals of γ . Then | γ | = | γ | and γ i ∈ M ( X ) , i = 1 ,
2. Therefore a | µ − γ | + a | µ − γ | + b Z X × X d ( x, y ) dγ ( x, y ) ≥ a | µ − γ | + a | µ − γ | + b W ( γ , γ ) ≥ W a,b ( µ , µ ) . So inf γ ∈ M ¶ a | µ − γ | + a | µ − γ | + b R X × X d ( x, y ) dγ ( x, y ) © ≥ W a,b ( µ , µ ) . Conversely, let ( e µ , e µ ) ∈ M ( X ) × M ( X ) be an optimal for W a,b ( µ , µ ) and let e π bean optimal transference plan between e µ and e µ . Then we get that W a,b ( µ , µ ) = a | µ − e µ | + a | µ − e µ | + b | e µ | Z X × X d ( x, y ) d e π ( x, y ) . We now define e γ := | e µ | e π then e µ , e µ are the marginals of e γ and | e µ | Z X × X d ( x, y ) d e π ( x, y ) = Z X × X d ( x, y ) d e γ ( x, y ) . Thus, W a,b ( µ , µ ) = a | µ − e µ | + a | µ − e µ | + b Z X × X d ( x, y ) d e γ ( x, y ) ≥ inf γ ∈ M ß a | µ − γ | + a | µ − γ | + b Z X × X d ( x, y ) dγ ( x, y ) ™ . (cid:3) Remark 5.4.
From Proposition 2.3 we also have W a,b ( µ , µ ) = inf γ ∈ M ≤ ( µ ,µ ) ß a | µ − γ | + a | µ − γ | + b Z X × X d ( x, y ) dγ ( x, y ) ™ , where M ≤ ( µ , µ ) = { γ ∈ M : γ i ≤ µ i , i = 1 , } .Proof of Theorem 1.1. We will prove this theorem in two steps. In the first step, we consider X is compact. We will prove for a general Polish metric space X in step 2. Step 1. X is a compact metric space. For any µ , µ ∈ M ( X ), using lemma 5.3 we obtain W a,b ( µ , µ ) = inf γ ∈ M ß a | µ − γ | + a | µ − γ | + b Z X × X d ( x, y ) dγ ( x, y ) ™ . (5.1)where γ , γ are the marginals of γ .For each i ∈ { , } , let γ i = f i µ i + γ ⊥ i be the Lebesgue decomposition of γ i with respect to µ i . Then we have | µ i − γ i | = (cid:12)(cid:12)(cid:12) ( f i − µ i + γ ⊥ i (cid:12)(cid:12)(cid:12) = Z X | − f i | dµ i + γ ⊥ i ( X ) . Applying theorem 5.1 with F ( s ) = a | − s | , F o ( ϕ ) = I ( ϕ ) and F ( γ i | µ i ) = a | µ i − γ i | we getthat a | µ i − γ i | = sup ß Z X I ( ϕ i ( x )) dµ i ( x ) − Z X ϕ i ( x ) dγ i ( x ) | ϕ i , I ( ϕ i ) ∈ C b ( X ) ™ . Observe that, for every ϕ ∈ R we have I ( ϕ ) = inf s ≥ ( sϕ + a | − s | ) = a if ϕ > aϕ if − a ≤ ϕ ≤ a −∞ otherwise . It implies that a | µ i − γ i | = sup ß Z X I ( ϕ i ( x )) dµ i ( x ) − Z X ϕ i ( x ) dγ i ( x ) | ϕ i ∈ C b ( X ) and ϕ i ( x ) ≥ − a, ∀ x ∈ X ™ . (5.2)Since (5.1) and (5.2) we obtain W a,b ( µ , µ ) = inf γ ∈ M sup ( ϕ ,ϕ ) ∈ Φ (X i Z X I ( ϕ i ( x )) dµ i ( x ) + Z X × X ( b.d ( x, y ) − ϕ ( x ) − ϕ ( y )) dγ ( x, y ) ) , where Φ := {{ ϕ , ϕ ) ∈ C b ( X ) × C b ( X ) | ϕ i ( x ) ≥ − a, ∀ x ∈ X, i = 1 , } . We define thefunction L : M × Φ → R by L ( γ, ( ϕ , ϕ )) = X i Z X I ( ϕ i ( x )) dµ i ( x ) + Z X × X ( b.d ( x, y ) − ϕ ( x ) − ϕ ( y )) dγ ( x, y ) , for every γ ∈ M and ( ϕ , ϕ ) ∈ Φ. Recall that a function g : Φ → R is concave if g ( tx +(1 − t ) y ) ≥ tg ( x )+(1 − t ) g ( y ) for every x, y ∈ Φ , t ∈ [0 , L ( · , ( ϕ , ϕ ))is convex, and L ( γ, · ) is concave as I ( ϕ i ) is concave. Observe that ϕ i ∈ C b ( X ) and using[30, Lemma 4.3] we obtain L ( · , ( ϕ , ϕ )) is lower semicontinuous in M endowed with theweak*- topology.Next, we will estimate inf γ ∈ M L ( γ, ( ϕ , ϕ )) for every ( ϕ , ϕ ) ∈ Φ.(1) If ϕ ( x ) + ϕ ( y ) ≤ b.d ( x, y ) for all x, y ∈ X theninf γ ∈ M Z X × X ( b.d ( x, y ) − ϕ ( x ) − ϕ ( y )) dγ ( x, y ) ≥ . UALITY AND QUOTIENTS SPACES OF GENERALIZED WASSERSTEIN SPACES 15
Furthermore, if let γ be the null measure, i.e. γ ( A ) = 0 for every Borel subset A of X , then R X × X ( b.d ( x, y ) − ϕ ( x ) − ϕ ( y )) dγ ( x, y ) = 0 . Therefore,inf γ ∈ M L ( γ, ( ϕ , ϕ )) = X i Z X I ( ϕ i ( x )) dµ i ( x ) . (2) If there exist x , y ∈ X such that ϕ ( x ) + ϕ ( y ) > b.d ( x , y ) then we choose γ = λδ ( x ,y ) for λ >
0. Then L ( γ, ( ϕ , ϕ )) = X i Z X I ( ϕ i ( x )) dµ i ( x ) + λ ( b.d ( x , y ) − ϕ ( x ) − ϕ ( y )) . Let λ → ∞ we get inf γ ∈ M L ( γ, ( ϕ , ϕ )) = −∞ . Therefore,inf γ ∈ M L ( γ, ( ϕ , ϕ )) = P i R X I ( ϕ i ) dµ i if ϕ ( x ) + ϕ ( y ) ≤ b.d ( x, y ) , ∀ x, y ∈ X −∞ otherwise . Hence sup ( ϕ ,ϕ ) ∈ Φ inf γ ∈ M L ( γ, ϕ ) = sup ( ϕ ,ϕ ) ∈ Φ W P i R X I ( ϕ i ( x )) dµ i ( x ) . So we only need to check thatinf γ ∈ M sup ( ϕ ,ϕ ) ∈ Φ L ( γ, ϕ ) = sup ( ϕ ,ϕ ) ∈ Φ inf γ ∈ M L ( γ, ϕ ) . As we always have inf γ ∈ M sup ( ϕ ,ϕ ) ∈ Φ L ( γ, ϕ ) ≥ sup ( ϕ ,ϕ ) ∈ Φ inf γ ∈ M L ( γ, ϕ ), we only need to check forthe case sup ( ϕ ,ϕ ) ∈ Φ inf γ ∈ M L ( γ, ϕ ) is finite. We choose C > sup ( ϕ ,ϕ ) ∈ Φ inf γ ∈ M L ( γ, ϕ ) and the constantfunction e ϕ = ( − a/ , − a/ ∈ Φ. Then we get that L ( γ, e ϕ ) = I ( − a/
2) ( µ ( X ) + µ ( X )) + b Z X × X d ( x, y ) dγ ( x, y ) + a.γ ( X × X )= − a µ ( X ) + µ ( X )) + b Z X × X d ( x, y ) dγ ( x, y ) + a.γ ( X × X ) . Hence the set P := { γ ∈ M : L ( γ, e ϕ ) ≤ C } is bounded in the sense that there exists K > γ ( X × X ) ≤ K for every γ ∈ P . As X is compact, the set P is compact in theweak*-topology. Therefore, using [21, Theorem 2.4] we obtaininf γ ∈ M sup ( ϕ ,ϕ ) ∈ Φ L ( γ, ( ϕ , ϕ )) = sup ( ϕ ,ϕ ) ∈ Φ inf γ ∈ M L ( γ, ( ϕ , ϕ )) . It implies thatinf γ ∈ M ß a | µ − γ | + a | µ − γ | + b Z X × X d ( x, y ) dγ ( x, y ) ™ = sup ( ϕ ,ϕ ) ∈ Φ W X i Z X I ( ϕ i ( x )) dµ i ( x ) . (5.3)Hence, the proof of step 1 is completed. Step 2. X is a Polish metric space. Since µ , µ ∈ M ( X ), for every ε >
0, there exist a compact set X ⊂ X such that µ i ( X \ X ) ≤ ε, i = 1 , . We define µ ∗ i := µ i | X , i = 1 ,
2, i.e. for all Borel subset A of X , µ ∗ i ( A ) = µ i ( A ∩ X ). Wechoose an optimal ( u , u ) for W a,b ( µ ∗ , µ ∗ ) such that | u | = | u | and u i ≤ µ ∗ i , i = 1 ,
2. Then W a,b ( µ ∗ , µ ∗ ) = a | µ ∗ − u | + a | µ ∗ − u | + b W ( u , u ) . Moreover, for each i ∈ { , } we have | µ ∗ i − u i | = µ ∗ i ( X ) − u i ( X )= µ i ( X ) − u i ( X )= µ i ( X ) − u i ( X ) − µ i ( X \ X ) ≥ | µ i − u i | − ε. Therefore, W a,b ( µ ∗ , µ ∗ ) ≥ a | µ − u | + a | µ − u | + b W ( u , u ) − aε ≥ W a,b ( µ , µ ) − aε. On the other hand, using lemma 5.3 we get that W a,b ( µ ∗ , µ ∗ ) = inf γ ∈ M ß a | µ ∗ − γ | + a | µ ∗ − γ | + b Z X × X d ( x, y ) dγ ( x, y ) ™ , where γ , γ are the marginals of γ and M = ß γ ∈ M ( X × X ) | Z X × X d ( x, y ) dγ ( x, y ) < ∞ ™ .Let M ∗ = { γ ∈ M | γ (( X × X ) \ ( X × X )) = 0 } then M ∗ ⊂ M and thus, W a,b ( µ ∗ , µ ∗ ) ≤ inf γ ∈ M ∗ ß a | µ ∗ − γ | + a | µ ∗ − γ | + b Z X × X d ( x, y ) dγ ( x, y ) ™ = inf γ ∈ M ∗ ® a | µ ∗ − γ | ( X ) + a | µ ∗ − γ | ( X ) + b Z X × X d ( x, y ) dγ ( x, y ) ´ . Since X is compact, using the identity (5.3) in step 1, we obtaininf γ ∈ M ∗ ® a | µ ∗ − γ | ( X ) + a | µ ∗ − γ | ( X ) + b Z X × X d ( x, y ) dγ ( x, y ) ´ = sup ( ϕ ,ϕ ) ∈ Φ ∗ W X i Z X I ( ϕ i ) dµ ∗ i , whereΦ ∗ W = { ( ϕ , ϕ ) ∈ C b ( X ) × C b ( X ) | ϕ ( x ) + ϕ ( y ) ≤ b.d ( x, y ) and ϕ ( x ) , ϕ ( y ) ≥ − a, ∀ x, y ∈ X } . In addition, there exists ( ϕ ∗ , ϕ ∗ ) ∈ Φ ∗ W such that X i Z X I ( ϕ ∗ i ( x )) dµ ∗ i ( x ) ≥ sup ( ϕ ,ϕ ) ∈ Φ ∗ W X i Z X I ( ϕ i ( x )) dµ ∗ i ( x ) − ε. Then we get that X i Z X I ( ϕ ∗ i ( x )) dµ ∗ i ( x ) ≥ W a,b ( µ , µ ) − (2 a + 1) ε. (5.4)Next, for each x ∈ X we define ϕ ( x ) := min { inf y ∈ X ( b.d ( x, y ) − ϕ ∗ ( y )) , a } . As the func-tion X → R , x inf y ∈ X ( b.d ( x, y ) − ϕ ∗ ( y )) is Lipschitz, we have ϕ ∈ C ( X ). For each UALITY AND QUOTIENTS SPACES OF GENERALIZED WASSERSTEIN SPACES 17 x ∈ X , we have ϕ ∗ ( x ) + ϕ ∗ ( y ) ≤ b.d ( x, y ) , ∀ y ∈ X . So ϕ ∗ ( x ) ≤ inf y ∈ X ( b.d ( x, y ) − ϕ ∗ ( y )) , ∀ x ∈ X . Moreover, we also have ϕ ∗ ( x )+ ϕ ∗ ( x ) ≤ b.d ( x, x ) = 0 , ∀ x ∈ X . And from ϕ ∗ i ( x ) ≥ − a, ∀ x ∈ X , i = 1 ,
2, we get that ϕ ∗ i ( x ) ∈ [ − a, a ] , ∀ x ∈ X , i = 1 , . Therefore, ϕ ( x ) ≥ ϕ ∗ ( x ) for every x ∈ X . Besides that, for any x ∈ X one has b.d ( x, y ) − ϕ ∗ ( y ) ≥ − ϕ ∗ ( y ) ≥ − a, ∀ y ∈ X . Thus, ϕ ( x ) ∈ [ − a, a ] for every x ∈ X .Now, we define, for each y ∈ X , ϕ ( y ) := inf x ∈ X ( b.d ( x, y ) − ϕ ( x )) . Then ϕ ∈ C ( X ) and ϕ ( x ) + ϕ ( y ) ≤ b.d ( x, y ) , ∀ x, y ∈ X. By the same arguments as above, we still have ϕ ( y ) ∈ [ − a, a ] , ∀ y ∈ X and ϕ ≥ ϕ ∗ on X .Therefore ( ϕ , ϕ ) ∈ Φ W .Since the function I is nondecreasing, applying (5.4), we obtain X i Z X I ( ϕ i ( x )) dµ i ( x ) = X i Z X \ X I ( ϕ i ( x )) dµ i ( x ) + Z X I ( ϕ i ( x )) dµ ∗ i ( x ) ≥ − a ( µ ( X \ X ) + µ ( X \ X )) + X i Z X I ( ϕ ∗ i ( x )) dµ ∗ i ( x ) ≥ − aε + W a,b ( µ , µ ) − (2 a + 1) ε = W a,b ( µ , µ ) − (4 a + 1) ε. Therefore, sup ( ϕ ,ϕ ) ∈ Φ W P i R X I ( ϕ i ( x )) dµ i ( x ) ≥ W a,b ( µ , µ ) . Applying lemma 5.3, we get that W a,b ( µ , µ ) = sup ( ϕ ,ϕ ) ∈ Φ W X i Z X I ( ϕ i ( x )) dµ i ( x )= sup ( ϕ ,ϕ ) ∈ Φ W X i Z X inf s ≥ ( sϕ i ( x ) + a | − s | ) dµ i ( x ) . (cid:3) Remark 5.5.
In the case X is compact, theorem 1.1 is a special case of [21, Theorem 4.11] although its statement there is slightly different from ours as they consider lower semicon-tinuous functions ϕ , ϕ . For the completeness, we present a proof for this compact case instep 1 and it follows the ideas of the proof of [21, Theorem 4.11] . Let (
X, d ) be a metric space. For a function f : X → R , we denote k f k Lip := sup x,y ∈ X,x = y | f ( x ) − f ( y ) | d ( x, y ) . Now, using the techniques of the proof of [29, Theorem 1.14] and applying theorem 1.1 weare ready to prove theorem 1.2.
Proof of Theorem 1.2.
For every ( ψ, ϕ ) ∈ Φ W , we define ϕ d ( x ) := inf y ∈ X [ b.d ( x, y ) − ϕ ( y )]for every x ∈ X . Then ϕ d is b -Lipschitz function and ϕ d ( x ) ∈ [ − a, a ] for every x ∈ X .Therefore ϕ d ∈ F . Now we define ϕ dd ( y ) := inf x ∈ X [ b.d ( x, y ) − ϕ d ( x )] for every y ∈ X . Then ϕ dd is b -Lipschitz and ϕ d ( x ) + ϕ dd ( y ) ≤ b.d ( x, y ) , for every x, y ∈ X. As − a ≤ ϕ d ( x ) ≤ a we also get that − a ≤ ϕ dd ( y ) ≤ a for every y ∈ X . Therefore we have ϕ dd ∈ F and ( ϕ d , ϕ dd ) ∈ Φ W .On the other hand, as ψ ( x ) + ϕ ( y ) ≤ b.d ( x, y ) for every x, y ∈ X we get that ψ ( x ) ≤ inf y ∈ X [ b.d ( x, y ) − ϕ ( y )] = ϕ d ( x ) for every x ∈ X. Similarly, from the definitions of ϕ dd we also have ϕ dd ( y ) ≥ ϕ ( y ) for every y ∈ Y . Hence Z X I ( ψ ) dµ + Z X I ( ϕ ) dν ≤ Z X I Ä ϕ d ä dµ + Z X I Ä ϕ dd ä dν. Therefore,sup ( ψ,ϕ ) ∈ Φ W ¶ Z X I ( ψ ) dµ + Z X I ( ϕ ) dν © ≤ sup ϕ ∈ C b ( X ) ¶ Z X I Ä ϕ d ä dµ + Z X I Ä ϕ dd ä dν © . As ϕ d is b -Lipschitz we get − ϕ d ( x ) ≤ inf y ∈ X [ b.d ( x, y ) − ϕ d ( y )] . On the other hand, inf y ∈ X [ b.d ( x, y ) − ϕ d ( y )] ≤ − ϕ d ( x ). Hence ϕ dd ( x ) = inf y ∈ X [ b.d ( x, y ) − ϕ d ( y )] = − ϕ d ( x ) . Thus sup ( ψ,ϕ ) ∈ Φ W ¶ Z X I ( ψ ) dµ + Z X I ( ϕ ) dν © ≤ sup ϕ ∈ C b ( X ) ¶ Z X I Ä ϕ d ä dµ + Z X I Ä ϕ dd ä dν © = sup ϕ ∈ C b ( X ) ¶ Z X I Ä ϕ d ä dµ + Z X I Ä − ϕ d ä dν © ≤ sup ϕ ∈ F ¶ Z X I ( ϕ ) dµ + Z X I ( − ϕ ) dν © ≤ sup ( ψ,ϕ ) ∈ Φ W ¶ Z X I ( ψ ) dµ + Z X I ( ϕ ) dν © . So we must have equality everywhere and get the result. (cid:3)
Remark 5.6.
1) Theorem 1.2 has been proved in [25, Theorem 2] for the case a = b = 1 and X = R n by a different method.2) ( [15, 16] ) Let ( X, d ) be a Polish metric space. We denote by M s ( X ) the space of allsigned Borel measures with finite mass on X . Let M ( X ) be the set of all µ ∈ M s ( X ) such UALITY AND QUOTIENTS SPACES OF GENERALIZED WASSERSTEIN SPACES 19 that µ ( X ) = 0 . For every µ ∈ M ( X ) we denote by Ψ µ the set of all nonnegative measures γ ∈ M ( X × X ) such that λ ( X × A ) − λ ( A × X ) = µ ( A ) for every Borel A ⊂ X . Then wedefine for every µ ∈ M ( X ) , k µ k d := inf γ ∈ Ψ µ ¶ Z X × X d ( x, y ) dγ ( x, y ) © . Now, on the vector space M s ( X ) we define an extension Kantorovich-Rubinstein norm asfollowing k µ k d := inf ν ∈ M ( X ) ¶ k ν k d + | µ − ν | ( X ) © , for every µ ∈ M s ( X ) . Then from [15, Theorem 0] (when X is compact) or [16, Theorem 1] (when X is a generalPolish metric space), applying Hahn-Banach theorem we get that k µ k d = sup ¶ Z X f d ( µ − ν ) : f ∈ F © , where F := ¶ f ∈ C b ( X ) , k f k ∞ ≤ , k f k Lip ≤ © . We thank Benedetto Piccoli and FrancescoRossi for pointing [15] out to us, and we have found [16] after that. The corollary 1.3 is from theorem 1.2 as following.
Proof of Corollary 1.3.
It is clear that Ä M ( X ) , W a,b ä is complete, this follows from propo-sition 2.4. For every µ, ν ∈ M ( X ), we define σ := ( µ + ν ) /
2. Then using theorem 1.2 weobtain W a,b ( µ, σ ) = sup f ∈ F Z X f d ( µ − σ )= 12 sup f ∈ F Z X f d ( µ − ν )= 12 W a,b ( µ, ν ) . Similarly, W a,b ( σ, ν ) = W a,b ( µ, ν ). Hence, applying [6, Theorem 2.4.16] or [28, Lemma2.1] we get the result. (cid:3) Using theorem 1.2 we get another proof of [26, Lemma 5].
Corollary 5.7.
For every µ, ν, η ∈ M ( X ) we have W a,b ( µ + η, ν + η ) = W a,b ( µ, ν ) . From theorem 1.2 we also get a similar result in [3, Lemma 1.5].
Corollary 5.8.
Let ( X , d ) and ( X , d ) be two Polish metric spaces. If ψ : ( X , d ) → ( X , d ) is an isometry map then the map ψ ♯ : Ä M ( X ) , W a,b ä → Ä M ( X ) , W a,b ä is also anisometry.Proof. For every µ, ν ∈ M ( X ), it is clear that W a,b ( µ, ν ) ≥ W a,b ( ψ ♯ µ, ψ ♯ ν ) and ψ ♯ issurjective. Hence, we need to show that W a,b ( ψ ♯ µ, ψ ♯ ν ) ≥ W a,b ( µ, ν ). Let F i = { f ∈ C b ( X i ) , k f k ∞ ≤ a, k f k Lip ≤ b } , i = 1 ,
2. By theorem 1.2 one has W a,b ( ψ ♯ µ, ψ ♯ ν ) = sup g ∈ F Z X g d ( ψ ♯ µ − ψ ♯ ν ) = sup g ∈ F Z X g ◦ ψ d ( µ − ν ) . For every f ∈ F and every y ∈ X we define h ( y ) := inf x ∈ X [ b.d ( y, ψ ( x )) + f ( x )]. Then h is b -Lipschitz and h ( y ) ≥ − a for every y ∈ X . Since ψ is surjective, for every y ∈ X , thereexists x ′ ∈ X such that ψ ( x ′ ) = y . Thus, h ( y ) ≤ f ( x ′ ) ≤ a . Therefore, h ∈ F . Moreover,since f is b -Lipschitz, for every x ∈ X one has f ( x ) = inf x ∈ X [ f ( x ) + | f ( x ) − f ( x ) | ] ≤ inf x ∈ X [ f ( x ) + b.d ( x, x )]= inf x ∈ X [ f ( x ) + b.d ( ψ ( x ) , ψ ( x ))] ≤ f ( x ) . Therefore, f ( x ) = h ( ψ ( x )), for all x ∈ X or f = h ◦ ψ . Hence, we get that Z X f d ( µ − ν ) = Z X h ◦ ψ d ( µ − ν ) ≤ sup g ∈ F Z X g ◦ ψ d ( µ − ν ) = W a,b ( ψ ♯ µ, ψ ♯ ν ) . So that W a,b ( µ, ν ) ≤ W a,b ( ψ ♯ µ, ψ ♯ ν ). (cid:3) Now using theorem 1.2 we give another proof of theorem 1.4 for the case p = 1. Corollary 5.9.
Let ( X, d ) be a Polish metric space. Then for every a, b > the pushforward map p ♯ : ( M G ( X ) , W a,b ) → ( M ( X/G ) , W a,b ) is an isometry.Proof. From part 1) of theorem 1.4 we know that W a,b ( p ♯ µ, p ♯ ν ) ≤ W a,b ( µ, ν ) for every µ, ν ∈ M ( X ).Now we prove that for every µ , µ ∈ M G ( X ), we have W a,b ( µ , µ ) ≤ W a,b ( p ♯ µ , p ♯ µ ).Let µ , µ ∈ M G ( X ). Recall that F := { f ∈ C b ( X ) : k f k ∞ ≤ a, k f k Lip ≤ b } and we define F ∗ := { f ∈ C b ( X/G ) : k f k ∞ ≤ a, k f k Lip ≤ b } . For every f ∈ F , the map f : X → R , defined by f ( x ) = R G f ( xg ) dλ ( g ) is well definedand f ( xh ) = f ( x ) for every x ∈ X, h ∈ G and hence we can define the map f ∗ : X/G → R by f ∗ ( p ( x )) = f ( x ) for every x ∈ X . It is clear that f ∗ ∈ C b ( X ) and k f ∗ k ∞ ≤ a . Now wecheck that k f ∗ k Lip ≤ b . For every x ∗ , y ∗ ∈ X/G with x ∗ = y ∗ there exist x ∈ x ∗ , y ∈ y ∗ such that d ( x , y ) = d ∗ ( x ∗ , y ∗ ). As the action is isometry and f is b -Lipschitz, for every x, y ∈ X we have | f ( x ) − f ( y ) | ≤ R G | f ( xg ) − f ( yg ) | dλ ( g ) ≤ b.d ( x, y ) . Therefore, | f ∗ ( x ∗ ) − f ∗ ( y ∗ ) | d ( x ∗ , y ∗ ) = | f ( x ) − f ( y ) | d ( x , y ) ≤ b. Hence k f ∗ k Lip ≤ b and therefore f ∗ ∈ F ∗ . On the other hand, as µ , µ ∈ M G ( X ), one has Z X/G f ∗ ( x ∗ ) d ( p ♯ µ − p ♯ µ )( x ∗ ) = Z X f ( x ) d ( µ − µ )( x ) UALITY AND QUOTIENTS SPACES OF GENERALIZED WASSERSTEIN SPACES 21 = Z X Z G f ( xg ) dλ ( g ) d ( µ − µ )( x )= Z G Z X f ( xg ) d ( µ − µ )( x ) dλ ( g )= Z X f ( x ) d ( µ − µ )( x ) . Therefore, applying theorem 1.2 we get that W a,b ( µ , µ ) ≤ W a,b ( p ♯ µ , p ♯ µ ) . (cid:3) For every µ , µ ∈ M ( X ) we denote by Opt a,b ( µ , µ ) the set of all γ ∈ M ( X × X ) suchthat R X × X d ( x, y ) dγ ( x, y ) < ∞ , γ i ≤ µ i , i = 1 ,
2, and W a,b ( µ , µ ) = a | µ − γ | + a | µ − γ | + b Z X × X d ( x, y ) dγ ( x, y ) , where γ , γ are the marginals of γ . Lemma 5.10.
Let ( X, d ) be a Polish metric space. For every µ , µ ∈ M ( X ) the set Opt a,b ( µ , µ ) is a nonempty, convex and compact subset of M ( X × X ) .Proof. It is clear that Opt a,b ( µ , µ ) is convex.From remark 5.4, we choose a sequence of γ n ∈ M ( X × X ) such that R X × X d ( x, y ) dγ n ( x, y ) < ∞ , ( π i ) ♯ γ n ≤ µ i for every i = 1 , , n ∈ N and a | µ − ( π ) ♯ γ n | + a | µ − ( π ) ♯ γ n | + b Z X × X d ( x, y ) dγ n ( x, y ) → W a,b ( µ , µ ) . As µ , µ ∈ M ( X ) we get that { ( π i ) ♯ γ n } n ∈ N is tight for every i = 1 ,
2. Therefore for every ε > K ε , L ε of X such that( π ) ♯ γ n ( X \ K ε ) < ε and ( π ) ♯ γ n ( X \ L ε ) < ε, for every n ∈ N . And hence γ n ( X × X \ K ε × L ε ) ≤ ( π ) ♯ γ n ( X \ K ε ) + ( π ) ♯ γ n ( X \ L ε ) < ε , for every n ∈ N .Therefore { γ n } n ∈ N is tight. As µ i ∈ M ( X ) and ( π i ) ♯ γ n ≤ µ i for every i = 1 , , n ∈ N we getthat { γ n } n ∈ N is bounded. Hence applying Prokhorov’s theorem, passing to a subsequencewe can assume that γ n → γ as n → ∞ in the weak*-topology for some γ ∈ M ( X × X ).Since the metric function d is nonnegative lower semicontinuous on X × X , we can write d as the pointwise limit of a nondecreasing sequence of nonnegative, continuous functions( c k ) k ∈ N on X × X . Replacing c k by min { c k , k } , we can assume that each c k is bounded. Bymonotone convergence, one has R X × X c k ( x, y ) dγ ( x, y ) → R X × X d ( x, y ) dγ ( x, y ) as k → ∞ . As c k ≤ d for every k ∈ N , we have R X × X d ( x, y ) dγ n ( x, y ) ≥ R X × X c k ( x, y ) dγ n ( x, y ). Moreover,since c k is bounded and continuous, we get thatlim inf n →∞ Z X × X d ( x, y ) dγ n ( x, y ) ≥ lim n →∞ Z X × X c k ( x, y ) dγ n ( x, y ) = Z X × X c k ( x, y ) dγ ( x, y ) . Therefore,lim inf n →∞ Z X × X d ( x, y ) dγ n ( x, y ) ≥ lim k →∞ Z X × X c k ( x, y ) dγ ( x, y ) = Z X × X d ( x, y ) dγ ( x, y ) . As W a,b ( µ , µ ) is finite we get that R X × X d ( x, y ) dγ ( x, y ) < ∞ . As γ n → γ as n → ∞ in theweak*-topology, applying [23, Theorem 6.1 page 40] we also get thatlim sup n →∞ γ n ( X × X ) ≤ γ ( X × X ) ≤ lim inf n →∞ γ n ( X × X ) . Therefore γ ( X × X ) = lim n →∞ γ n ( X × X ).Next we will prove that ( π i ) ♯ γ ≤ µ i for every i = 1 , . Let A be a Borel subset of X .Applying [23, Theorem 6.1 page 40] again we get that γ ( U × X ) ≤ lim inf n →∞ γ n ( U × X ) ≤ µ ( U ) for every U ⊂ X open. Therefore( π ) ♯ γ ( A ) = γ ( A × X )= inf { γ ( W ) : W ⊂ X × X open , A × X ⊂ W }≤ inf { γ ( U × X ) : U ⊂ X open , A ⊂ U }≤ inf { µ ( U ) : U ⊂ X open , A ⊂ U } = µ ( A ) . This means ( π ) ♯ γ ≤ µ . Similarly, we get that ( π ) ♯ γ ≤ µ . Therefore, a | µ i − ( π i ) ♯ γ n | → a | µ i − ( π i ) ♯ γ | as n → ∞ , i = 1 ,
2. Hence, we get that a | µ − ( π ) ♯ γ | + a | µ − ( π ) ♯ γ | + b Z X × X d ( x, y ) dγ ( x, y ) ≤ W a,b ( µ , µ )Therefore, Opt a,b ( µ , µ ) is nonempty.Now we prove that Opt a,b ( µ , µ ) is a compact subset of M ( X × X ). Let { γ n } n ∈ N be asequence in Opt a,b ( µ , µ ). Using the same argument as above we can get a subsequence of { γ n } n ∈ N converging to some γ ∈ Opt a,b ( µ , µ ) in the weak*-topology. (cid:3) Next, we will provide the optimality conditions for generalized Wasserstein distances intheorem 5.11 and theorem 5.14. These results are versions of [21, Theorem 4.14 and Theorem4.15] for generalized Wasserstein distances. Note that as our nonsmooth entropy function F ( s ) = a | − s | is not superlinear and the cost function b.d ( · , · ) does not have compactsublevels when X is a general Polish metric space, they do not satisfy coercive conditionsas in [21, Theorem 4.14 and Theorem 4.15]. Theorem 5.11.
Let ( X, d ) be a Polish metric space and let a, b > . For every µ , µ ∈ M ( X ) , there exists ( ϕ , ϕ ) ∈ Φ W such that W a,b ( µ , µ ) = X i Z X I ( ϕ i ( x )) dµ i ( x ) . Before giving the proof of theorem 5.11, we prove the following elementary lemma.
Lemma 5.12.
Let ( X, d ) be a separable metric space and let a, b > . If a sequence { ϕ n } in C b ( X ) satisfies that ϕ n is b -Lipschitz and | ϕ n ( x ) | ≤ a for every x ∈ X and every n ∈ N then { ϕ n } has a pointwise convergent subsequence on X .Proof. Let S = { s , s , . . . } be a countable dense subset of X . As | ϕ n ( s ) | ≤ a for every n ∈ N , s ∈ S , using a standard diagonal argument there exists a subsequence of { ϕ n } whichwe still denote by { ϕ n } such that ϕ n ( s ) converges as n → ∞ for every s ∈ S . As S is UALITY AND QUOTIENTS SPACES OF GENERALIZED WASSERSTEIN SPACES 23 dense in X , for every x ∈ X and every ε > s ∈ S such that d ( x, s ) < ε/b .Since { ϕ m ( s ) } converges, there exists N > m , m > N we have | ϕ m ( s ) − ϕ m ( s ) | < ε . Furthermore, ϕ m and ϕ m are b -Lipschitz on X . Therefore, | ϕ m ( x ) − ϕ m ( x ) | ≤ | ϕ m ( x ) − ϕ m ( s ) | + | ϕ m ( x ) − ϕ m ( s ) | + | ϕ m ( s ) − ϕ m ( s ) | < b.d ( x, s ) + ε< ε. Hence, we get the result. (cid:3)
Proof of Theorem 5.11.
For every n ∈ N , n ≥
1, using theorem 1.1, we get that there exists Ä ϕ ,n , ϕ ,n ä ∈ Φ W such that X i Z X I Ä ϕ i,n ( x ) ä dµ i ( x ) ≥ W a,b ( µ , µ ) − n . For every x ∈ X , we define e ϕ ,n ( x ) := inf y ∈ X î b.d ( x, y ) − ϕ ,n ( y ) ó and for every y ∈ X wedefine e ϕ ,n ( x ) := inf x ∈ X [ b.d ( x, y ) − e ϕ ,n ( x )]. Then ( e ϕ ,n , e ϕ ,n ) ∈ Φ W and e ϕ i,n is b -Lipschitzon X, i = 1 ,
2. Moreover, e ϕ i,n ( x ) ≥ ϕ i,n ( x ) for every x ∈ X, i = 1 ,
2. Thus, X i Z X I ( e ϕ i,n ( x )) dµ i ( x ) ≥ X i Z X I Ä ϕ i,n ( x ) ä dµ i ( x ) . Applying lemma 5.12, there exists a subsequence { e ϕ ,n k } k pointwise convergent to e ϕ on X . Then e ϕ ( x ) ∈ [ − a, a ] for every x ∈ X . We now consider the subsequence { e ϕ ,n k } k , it isclear that e ϕ ,n k is b -Lipschitz and e ϕ ,n k ( y ) ∈ [ − a, a ] for every y ∈ X . Thus, using lemma5.12 again we obtain that there exists a subsequence ¶ e ϕ ,n kl © l pointwise convergent to e ϕ on X . Then e ϕ ( y ) ∈ [ − a, a ] for every y ∈ X . For every x, y ∈ X , from e ϕ ,n kl ( x ) + e ϕ ,n kl ( y ) ≤ b.d ( x, y ) we also have e ϕ ( x )+ e ϕ ( y ) ≤ b.d ( x, y ). Since the Lebesgue’s dominated convergencetheorem and I ( ϕ ) = ϕ for every ϕ ∈ [ − a, a ], we obtain X i Z X I ( e ϕ i ( x )) dµ i ( x ) = lim l →∞ X i Z X I Ä e ϕ i,n kl ( x ) ä dµ i ( x ) ≥ lim l →∞ X i Z X I (cid:16) ϕ i,n kl ( x ) (cid:17) dµ i ( x ) ≥ W a,b ( µ , µ ) . Now, for every x ∈ X , we define ϕ ( x ) := inf y ∈ X [ b.d ( x, y ) − e ϕ ( y )] and for every y ∈ X , wedefine ϕ ( y ) := inf x ∈ X [ b.d ( x, y ) − ϕ ( x )]. Then ( ϕ , ϕ ) ∈ Φ W and ϕ i ( x ) ≥ e ϕ i ( x ) for every x ∈ X, i = 1 ,
2. Hence, we get that X i Z X I ( ϕ i ( x )) dµ i ( x ) ≥ X i Z X I ( e ϕ i ( x )) dµ i ( x ) ≥ W a,b ( µ , µ ) . Therefore, P i R X I ( ϕ i ( x )) dµ i ( x ) = W a,b ( µ , µ ). (cid:3) We say that a pair ( ϕ , ϕ ) ∈ Φ W is a dual optimal for W a,b ( µ , µ ) if W a,b ( µ , µ ) = P i R X I ( ϕ i ( x )) dµ i ( x ). Corollary 5.13.
Let a compact group G act on the right of a locally compact Polish metricspace ( X, d X ) by isometries. Let p : X → X/G be the natural quotient map and let any µ , µ ∈ M G ( X ) . If a pair ( ϕ , ϕ ) is a dual optimal for W a,b ( p ♯ µ , p ♯ µ ) then ( ϕ , ϕ ) isalso a dual optimal for W a,b ( µ , µ ) , where ϕ i is defined by ϕ i := ϕ i ◦ p, i = 1 , .Proof. Since ϕ i ( x ∗ ) ∈ [ − a, a ] for every x ∗ ∈ X/G and ϕ i ∈ C b ( X/G ), we get that ϕ i ( x ) ∈ [ − a, a ] for every x ∈ X and ϕ i ∈ C b ( X ), i = 1 ,
2, since p is continuous. Moreover, forevery x, y ∈ X one has ϕ ( x ) + ϕ ( y ) = ϕ ( x ∗ ) + ϕ ( y ∗ ) ≤ b.d X/G ( x ∗ , y ∗ ) ≤ b.d X ( x, y ).Therefore, ( ϕ , ϕ ) ∈ Φ W . Since ( ϕ , ϕ ) is a dual optimal for W a,b ( p ♯ µ , p ♯ µ ) and usingtheorem 1.4 we get that W a,b ( µ , µ ) = W a,b ( p ♯ µ , p ♯ µ )= X i Z X/G I ( ϕ i ( x ∗ )) dp ♯ µ i ( x ∗ )= X i Z X I ( ϕ i ( p ( x ))) dµ i ( x )= X i Z X I ( ϕ i ( x )) dµ i ( x ) . Hence, ( ϕ , ϕ ) is a dual optimal for W a,b ( µ , µ ). (cid:3) The above result has been proved for Wasserstein distances in [14, Corollary 3.4].Next, we provide the conditions between a optimal plan γ ∈ Opt a,b ( µ , µ ) and a dualoptimal ( ϕ , ϕ ). Theorem 5.14.
Let ( X, d ) be a Polish metric space and let µ , µ ∈ M ( X ) , γ ∈ M ≤ ( µ , µ ) .Then for every a, b > the plan γ ∈ Opt a,b ( µ , µ ) if and only if there exist a pair ( ϕ , ϕ ) ∈ Φ W and two Borel subsets A , A of X that satisfy the following conditions(i) γ i ( X \ A i ) = µ ⊥ i ( A i ) = 0 , i = 1 , , where γ i is the marginal of γ and µ i = g i γ i + µ ⊥ i isthe Lebegues decomposition of µ i with respect to γ i .(ii) ϕ ( x ) + ϕ ( y ) = b.d ( x, y ) γ -a.e in X × X .(iii) ( a − ϕ i ( x )) (1 − f i ( x )) = 0 µ i -a.e in A i , i = 1 , , where f i : X → [0 , is the Boreldensity of γ i with respect to µ i .(iv) ϕ i ( x ) = a µ ⊥ i -a.e in X \ A i , i = 1 , .Proof. Let γ ∈ Opt a,b ( µ , µ ). For each i ∈ { , } , since µ ⊥ i ⊥ γ i , there exists a Borelsubset A i of X such that γ i ( X \ A i ) = µ ⊥ i ( A i ) = 0. By theorem 5.11, let ( ϕ , ϕ ) ∈ Φ W be a dual optimal for W a,b ( µ , µ ). Since ϕ i ( x ) ∈ [ − a, a ] for every x ∈ X , I ( ϕ i ( x )) =inf s ≥ ( sϕ i ( x ) + a | − s | ) = ϕ i ( x ) , i = 1 ,
2. Hence, we get that X i Z X ϕ i ( x ) dµ i ( x ) = W a,b ( µ , µ ) UALITY AND QUOTIENTS SPACES OF GENERALIZED WASSERSTEIN SPACES 25 = a ( | µ − γ | + | µ − γ | ) + b Z X × X d ( x, y ) dγ ( x, y ) ≥ X i Z X a (1 − f i ( x )) dµ i ( x ) + Z X ϕ ( x ) dγ ( x, y ) + Z X ϕ ( y ) dγ ( x, y )= X i î Z A i a (1 − f i ( x )) dµ i ( x ) + Z X \ A i a (1 − f i ( x )) dµ i ( x ) + Z A i ϕ i ( x ) dγ i ( x ) ó = X i î Z A i [ a (1 − f i ( x )) + f i ( x ) ϕ i ( x )] dµ i ( x ) + Z X \ A i a (1 − f i ( x )) dµ i ( x ) ó . Since γ i ( X \ A i ) = 0 , i = 1 , ϕ i ( x ) ≤ a for every x ∈ X , one has Z X \ A i a (1 − f i ( x )) dµ i ( x ) = a Z X \ A i dµ ⊥ i ( x ) − aγ i ( X \ A i ) ≥ Z X \ A i ϕ i ( x ) dµ ⊥ i ( x ) . Moreover, from f i ( x ) ∈ [0 , , ϕ i ( x ) ∈ [ − a, a ] for every x ∈ X , we get that ( a − ϕ i ( x )) (1 − f i ( x )) ≥ a (1 − f i ( x )) + f i ( x ) ϕ i ( x ) ≥ ϕ i ( x ) for every x ∈ X . Therefore, X i Z X ϕ i ( x ) dµ i ( x ) ≥ X i Ä Z A i ϕ i ( x ) dµ i ( x ) + Z X \ A i ϕ i ( x ) dµ ⊥ i ( x ) ä = X i Z X ϕ i ( x ) dµ i ( x ) . Hence, we must have equality everywhere and we get the conditions ( i ) − ( iv ).Conversely, assume that there exist ( ϕ , ϕ ) ∈ Φ W and two Borel subsets A , A of X that satisfy four conditions ( i ) − ( iv ). Since the conditions ( i ) and ( ii ) we obtain a ( | µ − γ | + | µ − γ | ) + b Z X × X d ( x, y ) dγ ( x, y ) = X i Z X a (1 − f i ( x )) dµ i ( x ) + Z X ϕ i ( x ) dγ i ( x )= X i Z A i [ a (1 − f i ( x )) + f i ( x ) ϕ i ( x )] dµ i ( x )+ Z X \ A i a (1 − f i ( x )) dµ i ( x ) . On the other hand, from the conditions ( i ) and ( iv ), for each i ∈ { , } we get that Z X \ A i a (1 − f i ( x )) dµ i ( x ) = a Z X \ A i dµ ⊥ i ( x ) − aγ i ( X \ A i ) = Z X \ A i ϕ i ( x ) dµ ⊥ i ( x ) . Furthermore, the condition ( iii ) implies that Z A i [ a (1 − f i ( x )) + f i ( x ) ϕ i ( x )] dµ i ( x ) = Z A i ϕ i ( x ) dµ i ( x ) , i = 1 , . Hence, we get that a ( | µ − γ | + | µ − γ | ) + b Z X × X d ( x, y ) dγ ( x, y ) = X i Z A i ϕ i ( x ) dµ i ( x ) + Z X \ A i ϕ i ( x ) dµ ⊥ i ( x )= X i Z X I ( ϕ i ( x )) dµ i ( x ) ≤ W a,b ( µ , µ ) . However, we always have the opposite inequality. Therefore, γ ∈ Opt a,b ( µ , µ ). (cid:3) References [1] Luigi Ambrosio, Nicola Gigli, and Giuseppe Savar´e,
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Nhan-Phu Chung, Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Korea. Tel: +82 031-299-4819
E-mail address : [email protected];[email protected] Thanh-Son Trinh, Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Korea.
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