Induced Length Metric: Comparison

Proof. i) It is a consequence of the same inequality for W˜_p which in turn is true because it is (by our identification withWˆ_p) a Kantorovich-Wasserstein metric, for which this inequality is but an application of Hölder’s inequality. Indeed, for 1 ≤ p≤q, Hölder’s inequality gives both,P˜_q(Y|X)⊂P˜_p(Y|X)and

W˜p(σ, τ)≤W˜q(σ, τ)

for everyσ, τ ∈P˜_q(Y|X). Hence, it follows that the same is true when taking infima, so that

W_p⁰(µ, ν) = inf

σ,τ∈P˜p(Y|X) σ⁰=µ,τ⁰=ν

W˜_p(σ, τ)≤ inf

σ,τ∈P˜q(Y|X) σ⁰=µ,τ⁰=ν

W˜_q(σ, τ) =W_q⁰(µ, ν).

Now by the same reasoning, we see that alsoW_p^[(µ, ν)≤W_q^[(µ, ν), because

n

X

i=1

W_p⁰(ηi−1, η_i)≤

n

X

i=1

W_q⁰(ηi−1, η_i)

andP_q^sub(Y)⊂ P_p^sub(Y). Finally, the case Wp^] follows again in the same way.

ii) This is again true due to the same result for Kantorovich-Wasserstein metrics.

In the case of bounded spaces, all the spacesP˜_p(Y|X) for every possible p coincide, and so do the P_p^sub(Y)’s, so that we can argue in the same way as in part i), given the starting point

W˜_q≤W˜

p q

p diam(X)

q−p q

for 1≤p≤q.

4.4 Induced Length Metric: Comparison

Also here we continue to assume thatX is a length space.

Just like the identification of W˜p with the Kantorovich-Wasserstein metric Wˆp on the doubled space gave us immediately some properties about the space of charged probabilities, it will now be convenient to compare the previously defined metrics W_p^[ and Wp^] to a Kantorovich-Wasserstein metric on the one-point completion we introduced in Subsection 2.2. We will do so by providing similar representation formulas as the one in Lemma 4.2.6 for W_p⁰. To state them, let us introduce some more auxiliary transport cost functions.

Definition 4.4.1. i) W_p⁰ will denote theL^p-Kantorovich-Wasserstein distance on P_p(Y⁰) induced by the distanced⁰.

ii) Extending each subprobability measure µ∈ P_p^sub(Y)to a probability measure µ⁰∈ P_p(Y⁰)byµ⁰ :=µ+ (1−µ(Y))δ∂induces a bijection betweenP_p^sub(Y)and P_p(Y⁰)(see the lemma below). The induced distance onP_p^sub(Y)will again be denoted byW_p⁰, i.e.

W_p⁰(µ, ν) :=W_p⁰(µ⁰, ν⁰).

iii) For subprobability measures µ, ν with equal mass µ(Y) = ν(Y) we will also make use of the transportation cost

W_p^†(µ, ν)^p := inf

q∈Cpl(µ,ν)

ˆ

Y×Y

d^†(x, y)^pdq(x, y) (4.4.1) induced byd^† (which was defined in (2.2.1)).

iv) For a subprobabilityµ∈ P^sub(Y)define W_p⁰(µ,0)^p :=W_p⁰(µ, δ_∂)^p=

ˆ

Y

d⁰(x, ∂)^pdµ(x), (4.4.2) with 0 denoting the subprobability measure with vanishing total mass.

Lemma 4.4.2. i) The map

P_p^sub(Y)→ P_p(Y⁰), µ7→µ⁰ :=µ+ (1−µ(Y))δ_∂

is a bijection, and it is an isometry when we equip both spaces withW_p⁰. ii) If Y is additionally totally bounded, W_p⁰ metrizes the vague convergence in

P_p^sub(Y).

Proof. i) The map is clearly a bijection with inverse P_p(Y⁰) 3 µ⁰ 7→ µ := µ⁰|_Y ∈ P_p^sub(Y). By definition ofW_p⁰ on P_p^sub(Y) it is an isometry.

ii) Let us show that W_p⁰ metrizes the vague convergence in P_p^sub(Y). Given a vaguely converging sequenceµn→µ∗ inP_p^sub(Y), defineµ⁰_n:=µn+ (1−µn(Y))δ∂ ∈ P_p(Y⁰). This is a sequence of probability measures on a compact space; hence, for every subsequence, Prokhorov’s theorem provides a converging further subsequence.

Since the restriction of all these limits to Y has to coincide with µ∗, the whole sequenceµ⁰_n converges weakly toµ⁰_∗ :=µ∗+ (1−µ∗(Y))δ_∂, so that W_p⁰(µ⁰_n, µ⁰_∗)→0.

Then alsoW_p⁰(µ_n, µ∗)→0.

Assume conversely that W_p⁰(µ_n, µ∗)→0. By definition this means that we have convergence W_p⁰(µ⁰_n, µ⁰_∗) → 0, which in turn assures that µ⁰_n → µ⁰_∗ weakly in Y⁰. Then the restrictions toY converge vaguely.

Remark 4.4.3. Note that without the assumption of total boundedness the vague con-vergence inY wouldnotimply the weak convergence of the corresponding probability measures onY⁰ since they could lose mass at infinity instead of at the boundary.

Remark 4.4.4. One could equally well define W_p⁰⁰(µ, ν) := inf{W_p⁰(ˇµ,ν)ˇ

µ,ˇ νˇ∈ M(Y⁰),µ|ˇ_Y =µ,ν|ˇ_Y =ν}.

For p = 1 the metrics W₁⁰ and W₁⁰⁰ coincide, but for p > 1 this is no longer true.

This is due to the fact that(d(x, ∂) +d(y, ∂))^p =d(x, ∂)^p+d(y, ∂)^p only for p= 1.

Intuitively speaking, for p > 1, it makes a difference if we transport mass through the boundary point, or to it – however, for the latter we need to allow for masses bigger than 1. Take for instance X = R, Y = (−3,3) and µ = δ−2, ν =δ₂. Then µ⁰ = µ and ν⁰ = ν, so that W_p⁰(µ, ν)^p = d⁰(−2,2)^p = 2^p, whereas W_p⁰⁰(µ, ν)^p ≤ W_p⁰(µ+δ_∂, ν+δ_∂)^p =d⁰(−2, ∂)^p+d⁰(2, ∂)^p = 2.

The metricW₂⁰⁰ coincides with Figalli & Gigli’s metricW b₂ [FG10].

4.4 Induced Length Metric: Comparison

We start by characterizing the metric W_p⁰ in terms of L^ptransportation and -annihilation costs.

Lemma 4.4.5. For all µ, ν ∈ P_p^sub(Y) W_p⁰(µ, ν)^p= infn

W_p(µ₁, ν₁)^p+W_p^†(µ₂, ν₂)^p+W_p⁰(µ₀,0)^p+W_p⁰(ν₀,0)^p µ=µ₁+µ₂+µ₀, ν=ν₁+ν₂+ν₀, (µ+ν₀)(Y)≤1,

(ν+µ0)(Y)≤1 o

. (4.4.3) In the casep= 1, contributions from the term W_p^†(µ₂, ν₂)^p can be avoided, in other words, one can always choose µ2 =ν2 = 0.

Proof. The derivation of this formula is straightforward. The transport decomposes into trivial transports within ∂ (which do not appear in the formula), transports betweenY and∂(given byW_p⁰(µ₀,0)^p+W_p⁰(ν₀,0)^p), and transports withinY, and the latter ones into transports usingdandd^†(given byW_p(µ₁, ν₁)^p+W_p^†(µ₂, ν₂)^p). One can construct these decompositions more explicitly like in the proof of Lemma 4.2.5.

The resulting couplings are still optimal between their marginals. The inequalities in the constraints are due to the fact that we compose theprobability measuresµ⁰, ν⁰ instead of the subprobabilitiesµ, νand the trivial transport within∂can be omitted.

For the vanishing of the Wp^†-term note that in the case p = 1 one has [d⁰(x, ∂) + d⁰(x, ∂)]^p = d⁰(x, ∂)^p +d⁰(x, ∂)^p, meaning that the term can be absorbed in the annihilation terms W_p⁰(µ0,0)^p+W_p⁰(ν0,0)^p.

The following lemma discusses the connection between our twoannihilation costs W_p⁰ andW_p^∗.

Lemma 4.4.6. i) For all µ, ν∈ P₁(Y) W₁^∗(µ, ν) = infn

W₁(µ, ξ) +W₁(ξ, ν)

ξ∈ P(∂Y)o . ii) For allp≥1 and all µ∈ P_p(Y)

W_p⁰(µ,0) = inf

Wp(µ, ξ)

ξ∈ P(∂Y) . iii) For allp≥1 and all µ∈ P_p(Y)

2^−1+1/pW_p⁰(µ,0)≤W_p^∗(µ)≤W_p⁰(µ,0).

In particular,W₁^∗(µ) =W₁⁰(µ,0).

Proof. i) By triangle inequality, we have that for everyξ ∈ P(∂Y) W₁^∗(µ, ν)≤W₁(µ, ξ) +W₁(ξ, ν).

Making use of Lemma 4.1.5, we consider the measures as given on the different copies, µ ∈ P(Y⁺) and ν ∈ P(Y⁻). Take now a Wˆ₁-optimal coupling q ∈ Cpl(µ, ν) ⊂ P(Y⁺×Y⁻). Let ε >0 and

Gε(x, y) :={γ ∈C⁰([0,1],X)ˆ

γ0 =x, γ1 =y,|L(γ)−d(x, y)| ≤ε}

be the set of ε-geodesics in Xˆ connecting x and y. Given a curve γ in Xˆ with γ0 ∈ Y⁺ and γ1 ∈ Y⁻, define α(γ) := inf{s > 0|γs 6∈ Y⁺} and z(γ) := γ_α(γ). Thenz(γ)∈∂Y, and given a measurable selection Γε: ˆX×Xˆ →C⁰([0,1],X)ˆ with Γ_ε(x, y)∈G_ε(x, y)(which exists by our measurable selection Lemma 2.5.6), we define the “boundary crossing points” Z := z◦Γ_ε:Y⁺×Y⁻ →∂Y. Using the projection pr₁: ˆX×Xˆ →X,ˆ (x, y)7→x, we get a map

(pr₁,Z) : Y⁺×Y⁻ →Y⁺×∂Y

and define the push-forward measureQ1 := (pr₁,Z)_#q∈ P(Y⁺×∂Y).

Let us check that this is a coupling between µand ξ :=Z_#q ∈ P(∂Y): Given a measurable setA⊂Y⁺,

(pr₁,Z)⁻¹(A×∂Y) ={(x, y)∈Y⁺×Y⁻

(pr₁(x, y),Z(x, y))∈A×∂Y}

={(x, y)∈Y⁺×Y⁻

(x,Z(x, y))∈A×∂Y}

=A×Y⁻,

which yields that Q1(A×∂Y) = q(A×Y⁻) = µ(A). On the other hand, given B ⊂∂Y measurable,

(pr₁,Z)⁻¹(Y⁺×B) ={(x, y)∈Y⁺×Y⁻

(pr₁(x, y),Z(x, y))∈Y⁺×B}

={(x, y)∈Y⁺×Y⁻

(x,Z(x, y))∈Y⁺×B}

=(Y⁺×Y⁻)∩ Z⁻¹(B)

=Z⁻¹(B),

hence in this case we haveQ₁(Y⁺×B) =q(Z⁻¹(B)) =ξ(B). Analogously one sees thatQ₂ := (Z,pr₂)_#q as a coupling betweenξ andν.

Now what is left to prove is that ξ is an “almost-midpoint”. Since fory∈∂Y we haved(x, y) =d^∗(x, y), together with Lemma 2.1.7 we get that

W₁(µ, ξ)≤ ˆ

Y⁺×∂Y

d(x, y) dQ₁(x, y)

= ˆ

Y⁺×Y⁻

d^∗(pr₁(x, y),Z(x, y)) dq(x, y)

= ˆ

Y⁺×Y⁻

d^∗(x,Γε(x, y)t|_{t=α(Γε(x,y))}) dq(x, y)

≤ ˆ

Y⁺×Y⁻

α(Γ_ε(x, y))d^∗(x, y) +εdq(x, y).

4.4 Induced Length Metric: Comparison

Using in the same wayQ₂ as a coupling betweenξ andν, we finally get that W1(µ, ξ) +W1(ξ, ν)≤

ˆ

Y⁺×Y⁻

α(Γ(x, y))d^∗(x, y) +εdq(x, y) +

ˆ

Y⁺×Y⁻

(1−α(Γ(x, y)))d^∗(x, y) +εdq(x, y)

= ˆ

Y⁺×Y⁻

d^∗(x, y) + 2εdq(x, y)

=W₁^∗(µ, ν) + 2ε.

ii) Given an arbitrary ξ ∈ P(∂Y) and a W_p-optimal coupling q ∈ Cpl(µ, ξ), we get

W_p(µ, ξ)^p = ˆ

X×X

d(x, y)^pdq(x, y)≥ ˆ

X×X

d⁰(x, ∂)^pdq(x, y) =W_p⁰(µ,0)^p. For the other inequality, similarly as in part i), we will define a map and use its push-forward measure. Givenx∈Y, let

Gε(x) :=

n z∈∂Y

|d(x, z)−d⁰(x, ∂)| ≤ε o

be the boundary points closest tox. Let us show that the graph of the multivalued mapGε is closed: Take a sequence(xn, zn) withzn∈Gε(x) that converges to(x, z) inX×X. Then z∈∂Y by the closedness of the boundary, and

|d(x, z)−d⁰(x, ∂)|= lim

n→∞|d(x_n, z_n)−d⁰(x_n, ∂)| ≤ε,

hencez∈Gε(x). Thus we can apply the measurable selection Theorem 2.5.4 and get a measurable functionΦ_ε:Y →∂Y such thatΦ_ε(x)∈G_ε(x). Then for the measure ξ_ε:= (Φ_ε)_#µ we see that

Wp(µ, ξε)^p≤ ˆ

X

d(x,Φε(x))^pdµ(x)≤ ˆ

X

d⁰(x, ∂) +εp

dµ(x).

First of all observe that the moment bound of µ implies that also the d⁰-moment

´d⁰(·, ∂)^pdµ is finite. Since µ is a probability measure, constant functions are in-tegrable, thus also the sum d⁰(·, ∂) +ε is in L^p(µ). This sum converges pointwise to d⁰(·, ∂) asε→0, and it is dominated byd⁰(·, ∂) + 1∈L^p(µ). By the dominated convergence theorem we get convergence inL^p(µ)asε→0, i.e.

ε→0limWp(µ, ξε)^p ≤ ˆ

X

d⁰(x, ∂)^pdµ(x) =W_p⁰(µ,0).

iii) The triangle inequality for d^∗ implies that Wp(µ, ξ) +Wp(ξ, µ) ≥ W_p^∗(µ, µ) for all ξ ∈ P(∂Y). ThusW_p⁰(µ,0)≥ ¹₂W_p^∗(µ, µ) =W_p^∗(µ). An estimate in the other direction is obtained as follows

W_p^∗(µ)^p = 2^−pW_p^∗(µ, µ)^p = 2^−p ˆ

X×X

z∈Xinf\Y d(x, z) +d(z, y) p

dq(x, y)

≥2^−p ˆ

X×X

z∈Xinf\Y d(x, z) + inf

w∈X\Y d(w, y) p

dq(x, y)

≥2^1−p ˆ

X×X

z∈X\Yinf d(x, z) p

dq(x, y)

= 2^1−pW_p⁰(µ,0)^p,

whereq denotes any W_p^∗-optimal coupling ofµ andµ.

Remark 4.4.7. In general,W_p^∗(µ)andW_p⁰(µ,0)will not coincide. Our lower bound for W_p^∗(µ)/W_p⁰(µ,0)is sharp. For instance, let Y = (0,2)⊂X =R and µ= ¹₂(δ1+δε) for some ε ∈(0,1). Then W_p⁰(µ,0)^p = ¹₂(d⁰(1, ∂)^p+d⁰(ε, ∂)^p) = ¹₂(1 +ε^p) whereas W_p^∗(µ)^p = ^1+ε₂ p

. Thus W_p^∗(µ)

W_p⁰(µ,0) = 2^−1+1p 1 +ε (1 +ε^p)^1p

−→2^−1+1p asε→0.

Now we are in position to compareW_p^[, Wp^], W_p⁰. It turns out that for p= 1 they all coincide.

Theorem 4.4.8. i) For all µ, ν∈ P₁^sub(Y)

W₁^[(µ, ν) =W₁^](µ, ν) =W₁⁰(µ, ν).

ii) More generally, for all p≥1 and all µ, ν∈ P_p^sub(Y)

W₁⁰(µ, ν)≤W_p^[(µ, ν)≤W_p^](µ, ν)≤W_p⁰(µ, ν).

In particular,W_p^[, Wp^], W_p⁰ do not vanish outside the diagonal.

Proof. i) According to Lemma 4.4.5 and Lemma 4.4.6, W₁⁰(µ, ν) = inf

n

W1(µ1, ν1) +W₁^∗(µ0) +W₁^∗(ν0)

µ=µ1+µ0, ν =ν1+ν0

o (4.4.4) for all subprobability measures µ, ν ∈ P₁^sub(Y). Together with Lemma 4.2.6i) this implies W₁⁰(µ, ν) ≤ W₁⁰(µ, ν). As W₁^[ is the biggest metric below W₁⁰, we have W₁⁰ ≤W₁^[. Using the fact thatW₁⁰ is a length metric, this yields

W₁^](µ, ν) = inf

η:µ ν W₁^[-cont.

sup

0=s0<...<sn=1 n

X

i=1

W₁^[(ηsi−1, ηsi)

≥ inf

η:µ ν W₁^[-cont.

sup

0=s0<...<sn=1 n

X

i=1

W₁⁰(ηsi−1, ηsi)

≥ inf

η:µ ν W₁⁰-cont.

sup

0=s0<...<sn=1 n

X

i=1

W₁⁰(η_si−1, η_si) =W₁⁰(µ, ν).

4.4 Induced Length Metric: Comparison

SinceW₁^] is the length metric induced by W₁^[, one gets W₁^[≤W₁^].

Now we are going to show that W₁^] ≤ W₁⁰. To do so, starting from an almost-geodesic in the representation ofW₁⁰ given by (4.4.4) we define a new curve connecting µandν and estimate itsW₁^[-length by using a clever decomposition in the represen-tation formula forW₁⁰ given by Lemma 4.2.6ii).

Letε >0 and take a decomposition µ=µ₁+µ₀, ν =ν₁+ν₀ in (4.4.4) such that W₁⁰(µ, ν) +ε≥W1(µ1, ν1) +W₁^∗(µ0) +W₁^∗(ν0).

Then we take an ε-W₁-geodesic (η_s,1)_s∈[0,1] connecting µ₁ and ν₁ that is supported onε-geodesics in Y. Define

˜ η⁰_s,0 :=

((1−2s)µ₀+ 2sµ₀(Y⁰)δ_∂, s∈[0,¹₂] (2s−1)ν₀+ 2(1−s)µ₀(Y⁰)δ_∂, s∈(¹₂,1].

This is a curve connecting µ₀ and ν₀. Take the restrictionη˜_s,0 := ˜η_s,0⁰ |_Y and define

˜

η_s:=η_s,1+ ˜η_s,0,

which is a curve connecting µ and ν. To estimate the W₁^[-length of the restricted curveη˜_s it is useful to get a bound onW₁⁰(˜η_s,η˜_t). Consider the case 0≤s≤t≤ ¹₂: We can rewrite

˜

η_s= (η_s,1+ ˜η⁰_t,0)

| {z }

“µ1”in Lemma 4.2.6

+ 2(t−s)µ₀

| {z }

“µ0”

and η˜_t= (η_t,1+ ˜η_t,0⁰ )

| {z }

“ν1”

+ 0

|{z}

“ν0”

.

This is an admissible decomposition in Lemma 4.2.6ii) as for instance

“(µ+ν0)(X)” = (ηs,1+ ˜ηs,0+ 0) (X) = (ηs,1+ (1−2s)µ0)(X)

≤(η0,1+µ0)(X)≤µ(X)≤1,

and similarly for “(ν +µ₀)(X)”. Thus, in this case the representation given by Lemma 4.2.6ii) yields

W₁⁰(˜ηs,η˜t)≤ W1(ηs,1+ ˜ηt,0, ηt,1+ ˜ηt,0) +W₁^∗(2(t−s)µ0)

= W₁(η_s,1, η_t,1) +W₁^∗(2(t−s)µ₀)

= |t−s|W₁(η_0,1, η_1,1) +|t−s|ε+ 2|t−s|W₁^∗(µ₀),

where we made use of the translation invariance of the Kantorovich-Wasserstein metric forp= 1, see (2.5.3).

In the case ¹₂ ≤s≤t≤1 we analogously rewrite

˜

η_s= (η_s,1+ ˜η_s,0)

| {z }

“µ1”

+ 0

|{z}

“µ0”

and η˜_t= (η_t,1+ ˜η_s,0)

| {z }

“ν1”

+ 2(t−s)ν₀

| {z }

“ν0”

,

and end up with

W₁⁰(˜η_s,η˜_t)≤ |t−s|W₁(η_0,1, η_1,1) +|t−s|ε+ 2|t−s|W₁^∗(ν₀).

To compute the length of the curve η˜_s, we enforce the partitions to visit the time step ¹₂, and then use the above estimates for W₁⁰:

L^[₁(˜η) = sup ( _n

X

k=0

W₁^[(˜η_si−1,η˜_si)

n∈N,0 =s₀ <· · ·<_n= 1 )

= sup ( _n

X

k=0

W₁^[(˜η_si−1,η˜_si)

n∈N,0 =s₀ < s_i^∗ = 1

2 <· · ·<_n= 1 )

≤sup ( _n

X

k=0

W₁⁰(˜ηsi−1,η˜si)

n∈N,0 =s0 < si^∗ = 1

2 <· · ·<n= 1 )

≤supn X

si≤¹₂

|s_i−si−1|W₁(η_0,1, η_1,1) +|s_i−si−1|ε+ 2|s_i−si−1|W₁^∗(µ₀)

+ X

si≥¹

2

|s_i−si−1|W₁(η0,1, η1,1) +|s_i−si−s|ε+ 2|s_i−si−1|W₁^∗(ν0) o

= 1

2W1(η0,1, η1,1) +1

2ε+W₁^∗(µ0) +1

2W1(η0,1, η1,1) + 1

2ε+W₁^∗(ν0)

=W₁(µ₁, ν₁) +ε+W₁^∗(µ₀) +W₁^∗(ν₀)

≤W₁⁰(µ, ν) + 2ε.

This finally yields W₁^](µ, ν) ≤ L^[₁(˜η) ≤ W₁⁰(µ, ν) + 2ε. Since ε was arbitrary, this provesW₁^]≤W₁⁰.

By the fact that W₁^[ is the biggest metric below W₁⁰ and we now know that W₁^] = W₁⁰ ≤W₁⁰, we also get W₁^[ ≥W₁^].

ii) Thanks to i) and Lemma 4.3.4 we know that W₁⁰ =W₁^[≤W_p^[. Further, since Wp^]is the length metric induced byW_p^[ we also haveW_p^[≤Wp^]. Hence the only thing left to show is thatWp^]≤W_p⁰.

The idea to do so is that locally (along a geodesic) the contribution of W_p^† is negligible, so that we can compare W_p⁰ and W_p^[ on a small scale and then carry it over to the induced length metrics.

Let subprobabilities µ, ν be given as well as an ε-W_p⁰-geodesic (η_t⁰)_t∈[0,1] connecting the measures µ⁰:=µ+ (1−µ(Y))δ∂ and ν⁰ :=ν+ (1−ν(Y))δ∂. By the continuity of W_p⁰ and W_p^∗ with respect to weak convergence we can assume without loss of generality thatµand ν have compact supports and forα >0small

ηt(Y)≤1−α

for all t∈ (0,1). Again we use the notation that measures without primes are the restrictions to Y. We thus have η_t(∂) = 0, whereas η⁰_t(∂) ≥α. Choose δ > 0 such that η_t(B_δ⁰(∂)) ≤ ^α₂. Let Π be the probability measure on C⁰([0,1], Y⁰) supported on ε-geodesics such that η⁰_t = (et)#Π and denote by L the essential supremum of d⁰(γ₀, γ₁) underΠ (which is finite thanks to the compact supports of µand ν). Let δ⁰:= _L^δ.

4.4 Induced Length Metric: Comparison

We consider η_s and η_t for |s−t| ≤ δ⁰. Using that d^†(x, y)^p ≥ d⁰(x, ∂)^p +d⁰(y, ∂)^p, we see that in the decomposition (4.4.3) it is actually cheaper to annihilate mass at the boundary:

W_p⁰(ηs, ηt)^p = inf n

Wp(ηs,1, ηt,1)^p+W_p^†(ηs,2, ηt,2)^p+W_p⁰(ηs,0,0)^p+W_p⁰(ηt,0,0)^p ηs=ηs,1+ηs,2+ηs,0, ηt=ηt,1+ηt,2+ηt,0,

(η_s+η_t,0)(Y)≤1,(η_t+η_s,0)(Y)≤1o

≥inf n

Wp(ηs,1, ηt,1)^p+W_p⁰(ηs,0+ηs,2,0)^p+W_p⁰(ηt,0+ηt,2,0)^p ηs=ηs,1+ηs,2+ηs,0, ηt=ηt,1+ηt,2+ηt,0,

(ηs+ηt,0)(Y)≤1,(ηt+ηs,0)(Y)≤1 o

. Since W^† only occurs where d^† is smaller than d, its contribution comes from ε-geodesics inB⁰_δ(∂), so that by our choice ofδwe know thatηs,2(Y) =ηs,2(B_δ⁰(∂))≤ ^α₂ and the same forη_t,2. Hence for α small enough we have (η_s+ (η_t,2+η_t,0))(Y)≤1, so that η_s = η_s,1 + ˜η_s,0 with η˜_s,0 := η_s,0 +η_s,2 is an admissible decomposition in (4.4.3). In particular, the above inequality is an equality. Note that we cannot use this trick fors= 0, t= 1because then the constraint might not be satisfied. Thanks to Lemma 4.4.6 we thus have

W_p⁰(η_s, η_t)^p ≥infn

W_p(η_s,1, η_t,1)^p+W_p^∗(˜η_s,0)^p+W_p^∗(˜η_t,0)^p

ηs =ηs,1+ ˜ηs,0, ηt=ηt,1+ ˜ηt,0,(ηs+ ˜ηt,0)(Y)≤1, (ηt+ ˜ηs,0)(Y)≤1

o

≥W_p⁰(η_s, η_t)^p ≥W_p^[(η_s, η_t)^p.

Hence, the W_p⁰-length of the curve (η_t)_t∈[0,1] dominates its W_p^[-length. As this curve is an almost-geodesic forW_p⁰, going to the induced length metrics this finally proves

W_p⁰(µ, ν) +ε≥W_p^](µ, ν).

Sinceεwas arbitrary, the proof is finished.

Remark 4.4.9. As we have seen in the proof of part i), W₁⁰ ≤W₁⁰, and in particular W₁⁰ does not vanish outside the diagonal.

Let us give some simple examples illustrating Theorem 4.4.8.

Example 4.4.10. LetX=R, Y = (−1,1), µ=δx, ν =δy for x, y∈Y. Then W_p⁰(µ, ν) =Wp(µ, ν) =|x−y|,

and for everyp≥1

W_p⁰(µ, ν) =d⁰(x, y) = min{|x−y|,2− |x−y|}. (4.4.5) Hence, by the independence on p on the right-hand side of (4.4.5), Theorem 4.4.8 yields

W_p^[(µ, ν) =W_p^](µ, ν) = min{|x−y|,2− |x−y|}.

Example 4.4.11. LetX =R, Y = (−2,2), µ = _2n+1¹ δ_−1/2, ν = _2n+1¹ δ_+1/2 for n ∈N. Then

W_p⁰(µ, ν)^p =W_p(µ, ν)^p = 1 2n+ 1. Taking

σ := 1 2n+ 1

n

X

k=0

δ 2k 2n+1−¹

2

, 1 2n+ 1

n

X

k=1

δ 2k 2n+1−¹

2

!

and

τ := 1 2n+ 1

n

X

k=0

δ2k+1

2n+1−¹₂, 1 2n+ 1

n−1

X

k=0

δ2k+1 2n+1−¹₂

! , we see that

W_p⁰(µ, ν)^p ≤W˜p(σ, τ)^p= 1

2n+ 1 p

, so that

W_p^[(µ, ν)≤W_p⁰(µ, ν)≤ 1

2n+ 1

<

1 2n+ 1

¹_p

=W_p⁰(µ, ν),

for p > 1, n ≥ 1. In particular, the lower estimate for W_p^[ in assertion ii) of the previous Theorem is sharp.

Lemma 4.4.12. For allµ, ν ∈ P₁^sub(Y) W₁^](µ, ν) = inf

n

W1(µ1, ν1) +W₁^∗(µ0) +W₁^∗(ν0)

µ=µ1+µ0, ν=ν1+ν0

o . Proof. This is a result of the identification of W₁^] with W₁⁰ done in Theorem 4.4.8 together with the characterization ofW₁⁰ shown in Lemma 4.4.5 and the identification of the annihilation costs in Lemma 4.4.6.

Im Dokument Gluing of metric measure spaces and the heat equation with homogeneous Dirichlet boundary values (Seite 83-92)