On interpolation and curvature via Wasserstein geodesics

Definition A.1 (Orlicz–Wasserstein space)

Let μi be two probability measures on M and define

with the convention inf⁡∅=∞.

Lemma A.2 ([35, Proposition 3.1])

For every μi∈PL⁢(M) there is an optimal coupling πopt of (μ0,μ1) such that

Let Φ be a convex increasing function with Φ⁢(1)=1. Then

This just uses Sturm’s idea to show the same inequality for Luxemburg norm of Orlicz spaces. Compare this also to [37, Remark 6.6].

Proof.

This follows easily from Jensen’s inequality. Let μ0,μ1 be two measures and λ>0 and π be a coupling such that ∫(Φ∘L)λ⁢(d⁢(x,y))⁢𝑑π⁢(x,y)≤1. Then since (Φ∘L)λ=Φ∘Lλ,

Since Φ⁢(1)≤1 and Φ is increasing, we see that

which implies wL⁢(μ0,μ1)≤wΦ∘L⁢(μ0,μ1).∎

Assume for all λ>0,

If μn,μ∞∈PL⁢(M) and μn converges weakly to μ∞, then

for all 0<λ<λ0.

This generalizes [36, Theorem 7.12]. The other equivalences in Villani’s theorem can be proven similarly. However, only the stated condition is needed below.

Proof.

Fix some x0∈M. It is not difficult to see that for any λ>0 and any μ′∈𝒫L⁢(M),

First assume wL⁢(μn,μ∞)→0 and let πn be the optimal plans with ln=wL⁢(μn,μ∞) and

For n large, for any λ>0 choose a sequence rn≤12 such that ln=rn⁢λ. Then using the triangle inequality and convexity of L we get

since L(1-rn)⁢λ≤L12⁢λ. Therefore,

Now assume that

for any 0<λ<λ0 and μn converges weakly to μ∞. This bound ensures that μ∞ is in 𝒫L⁢(M).

Take any λ>0 and an optimal coupling πn of (μn,μ∞) with respect to Lλ. For R>0 and A∧B=min⁡{A,B} we have

and thus by convexity of L and L⁢(0)=0

Thus integrating over πn we get

We first take the lim sup with n→∞ and then R→∞ and conclude that the last two terms converges to zero by our assumption and since Lλ3⁢(d⁢(x,y)∧R) is a bounded continuous function and πn converges weakly to the trivial coupling (Id×id)*⁢μ∞, the first term converges to zero as well. In particular, for n≥N⁢(λ) we have

and thus

Since λ was arbitrary, we conclude wL⁢(μn,μ∞)→0. ∎

Assume M is a proper metric space and Φ is convex, increasing, Φ⁢(1)=1 and rΦ⁢(r)→0 as r→∞. In addition, assume for all λ>0,

Suppose A is closed subset of PL⁢(M) such that wL~ is bounded, where L~=Φ∘L. Then A is precompact in PL⁢(M).

Compare this to [17, Theorem 6] for the case L⁢(t)=tp, Φ⁢(t)=tr for p≥1 and r>1.

Proof.

It suffices to show that each wL~-ball is compact in 𝒫L⁢(M). So for some r>0 and μ0∈𝒫L~⁢(M)⊂𝒫L⁢(M) let

and let (μn)n∈ℕ be a sequence in B~. Then there are (optimal) couplings πn such that

Using the proposition above, we see

Because of the stability of optimal couplings and lower semicontinuity of the cost [37, Theorem 5.20, Lemma 4.3], we only need to show that (μn)n∈ℕ is weakly precompact and

i.e. it is precompact in 𝒫L⁢(M) by the lemma above.

Since B~ is bounded with respect to wL~, we can assume that for some R>0,

for some λ0>0. For c⁢λ=λ0 and c∈(0,1) we have

for some C>0 depending only on λ0,c and L. Hence by the fact that L⁢(R),Φ⁢(R)→∞ as R→∞ we conclude

In order to show weak precompactness notice that L⁢(R)≥1 for R≥r0=r0⁢(L) implies tightness, which is equivalent to precompactness by the classical Prokhorov theorem. Indeed, BR⁢(x0) is compact and for r0≤R→∞,

uniformly in n. ∎

Assume M is a geodesic space. Let πopt be the optimal coupling of (μ0,μ1). Then there is a Π supported on the geodesics such that

Furthermore, let μt=(et)*⁢Π. Then

In particular, PL⁢(M) is a geodesic space.

Proof.

The first part follows from using the measurable selection theorem for

similar to [19] in case of p-Wasserstein spaces.

For the second part note for λmin=wL⁢(μ0,μ1) that

Hence

So t↦μt is absolutely continuous in 𝒫L⁢(M) and |μ˙t|≤λmin. But we also have

Therefore, |μt|=λmin and

If x,y∈M and z∈Zt⁢(x,y) for some t∈[0,1], then for all m∈M,

Furthermore, choosing x=m, this becomes an equality.

This extends Lemma 2.7.

Proof.

Since L is convex and increasing,

Dividing by t we get the inequality, and choosing x=m we see that all inequalities are actually equalities. ∎

Let η:[0,1]→M be a geodesic between two distinct points x and y. For t∈(0,1] define

Then for some fixed t∈[0,1] the function h⁢(m):=ft⁢(m)-t-1⁢f1⁢(m) has a minimum at x.

Proof.

Using Lemma 2.7 above for t∈(0,1) we have for z=ηt∈Zt⁢(x,y),

Let X,Y be compact subsets of M and let t∈(0,1]. If ϕ∈IcL⁢(X,Y), then t-1⁢ϕ∈Ict⁢(X,Zt⁢(X,Y)).

Proof.

For t=1 there is nothing to prove. For the rest we follow the strategy of [9, Lemma 5.1].

Set Ly⁢(x):=L⁢(d⁢(x,y)), let t∈(0,1] and y∈Y, and define ϕ⁢(x):=cL⁢(x,y)=Ly⁢(x). We claim that the following representation holds:

Indeed, by Lemma A.7 the left-hand side is less than or equal to the right-hand side for any z∈Zt⁢(X,y). Furthermore, choosing x=m we get an equality and thus showing the representation.

Now note that the claim implies that t-1⁢ϕ is the c¯t-transform of the function

and therefore t-1⁢ϕ is ct-concave relative to (X,Zt⁢(X,y)). Since ℐct⁢(X,Zt⁢(X,y))⊂ℐct⁢(X,Zt⁢(X,Y)), we see that each t-1⁢Ly is in ℐct⁢(X,Zt⁢(X,Y)).

It remains to show that for an arbitrary cL-concave function ϕ and t∈(0,1] the function t-1⁢ϕ is ct-concave relative to (X,Zt⁢(X,Y)). Since ϕ=ϕcL⁢c¯L, we have

But each function

is ct-concave relative to (X,Zt⁢(X,Y)) and ϕ is proper, thus also the infimum is ct-concave relative to (X,Zt⁢(X,Y)), i.e. t-1⁢ϕ∈ℐct⁢(X,Zt⁢(X,Y)). ∎