The connection of branched transport and transport problem as discussed in Section 5.1 was first pointed out by Q. Xia in [81]. An equivalent model was proposed by F. Maddalena, J.-M. Morel and S. Solimini in [61]. In [81], [60] and [15] the existence of an optimal branched transport (Theorem 5.2) was also provided. Later, this result has been extended in several directions, see for instance the works A. Brancolini, G. Buttazzo and F. Santambrogio ([16]) and Bianchini-Brancolini [15]. The interior regularity result (Theorem 5.3) has been proved By Q. Xia in [82] and M. Bernot, V. Caselles and J.-M. Morel in [14]. Also, we remark that L. Brasco, G. Buttazzo and F. Santambrogio proved a kind of Benamou-Brenier formula for branched transport in [17].
The content of Section 5.2 comes from J. Dolbeault, B. Nazaret and G. Savaré [33] and [26] of J.
Carrillo, S. Lisini, G. Savaré and D. Slepcev.
Section 5.3 is taken from a work of the second author and A. Figalli [37].
6 More on the structure of ( P
2(M ), W
2)
The aim of this Chapter is to give a comprehensive description of the structure of the ‘Riemannian manifold’(P2(Rd), W2), thus the content of this part of the work is the natural continuation of what we discussed in Subsection 2.3.2. For the sake of simplicity, we are going to stick to the Wasserstein space onRd, but the reader should keep in mind that the discussions here can be generalized with only little effort to the Wasserstein space built over a Riemannian manifold.
6.1 “Duality” between the Wasserstein and the Arnold Manifolds
The content of this section is purely formal and directly comes from the seminal paper of Otto [67].
We won’t even try to provide a rigorous background for the discussion we will do here, as we believe that dealing with the technical problems would lead the reader far from the geometric intuition. Also, we will not use the “results” presented here later on: we just think that these concepts are worth of mention. Thus for the purpose of this section just think that ‘each measure is absolutely continuous with smooth density’, that ‘eachL2function isC∞’, and so on.
Let us recall the definition of Riemannian submersion. LetM, Nbe Riemannian manifolds and letf :M →Na smooth map.f is a submersion provided the map:
df: Ker⊥ df(x)
→Tf(x)N,
is a surjective isometry for anyx∈M. A trivial example of submersion is given in the caseM = N×L(for some Riemannian manifoldL, withM endowed with the product metric) andf :M →N is the natural projection. More generally, iff is a Riemannian submersion, for eachy ∈N, the set f−1(y)⊂Mis a smooth Riemannian submanifold.
The “duality” between the Wasserstein and the Arnold Manifolds consists in the fact that there exists a Big ManifoldBMwhich is flat and a natural Riemannian submersion fromBMtoP2(Rd) whose fibers are precisely the Arnold Manifolds.
Let us define the objects we are dealing with. Fix once and for all a reference measureρ ∈ P2(Rd)(recall that we are “assuming” that all the measures are absolutely continuous with smooth densities - so that we will use the same notation for both the measure and its density).
• The Big Manifold BMis the space L2(ρ)of maps from Rd toRd which areL2 w.r.t. the reference measureρ. The tangent space at some mapT ∈BMis naturally given by the set of vector fields belonging toL2(ρ), where the perturbation ofTin the direction of the vector field uis given byt7→T+tu.
• The target manifold of the submersion is the Wasserstein “manifold”P2(Rd). We recall that the tangent spaceTanρ(P2(Rd))at the measureρis the set
Tanρ(P2(Rd)) :=n
∇ϕ : ϕ∈Cc∞(Rd)o ,
endowed with the scalar product ofL2(ρ)(we neglect to take the closure inL2(ρ)because we want to keep the discussion at a formal level). The perturbation of a measureρin the direction of a tangent vector∇ϕis given byt7→(Id+t∇ϕ)#ρ.
• The Arnold ManifoldArn(ρ)associated to a certain measureρ∈P2(Rd)is the set of maps S:Rd →Rdwhich preserveρ:
Arn(ρ) :=n
S:Rd→Rd : S#ρ=ρ}.
We endowArn(ρ)with theL2distance calculated w.r.t. ρ. To understand who is the tangent space atArn(ρ)at a certain mapS, pick a vector fieldvonRdand consider the perturbation t 7→ S+tvofS in the direction ofv. Thenvis a tangent vector if and only if dtd|t=0(S+ tv)#ρ= 0. Observing that
d
dt|t=0(S+tv)#ρ= d
dt|t=0(Id+tv◦S−1)#(S#ρ) = d
dt|t=0(Id+tv◦S−1)#ρ=∇·(v◦S−1ρ), we deduce
TanSArn(ρ) =n
vector fieldsvonRdsuch that∇ ·(v◦S−1ρ) = 0o , which is naturally endowed with the scalar product inL2(ρ).
We are calling the manifoldArn(ρ)an Arnold Manifold, because ifρis the Lebesgue measure restricted to some open, smooth and bounded setΩ, this definition reduces to the well known definition of Arnold manifold in fluid mechanics: the geodesic equation in such space is - formally - the Euler equation for the motion of an incompressible and inviscid fluid inΩ.
• Finally, the “Riemannian submersion”PffromBMtoP2(Rd)is the push forward map:
Pf : BM → P2(Rd), T 7→ T#ρ,
We claim thatPfis a Riemannian submersion and that the fiberPf−1(ρ)is isometric to the manifold Arn(ρ).
We start considering the fibers. Fixρ∈P2(Rd). Observe that Pf−1(ρ) =n
T ∈BM : T#ρ=ρo ,
and that the tangent spaceTanTPf−1(ρ)is the set of vector fieldsusuch thatdtd|t=0(T+tu)#ρ= 0, so that from
d
dt|t=0(T+tu)#ρ= d
dt|t=0(Id+tu◦T−1)#(T#ρ) = d
dt|t=0(Id+tu◦T−1)#ρ=∇ ·(u◦T−1ρ), we have
TanTPf−1(ρ) =n
vector fieldsuonRdsuch that∇ ·(u◦T−1ρ) = 0o ,
and the scalar product between two vector fields inTanTPf−1(ρ)is the one inherited by the one in BM, i.e. is the scalar product inL2(ρ).
Now choose a distinguished mapTρ ∈Pf−1(ρ)and notice that the right composition withTρ provides a natural bijective map fromArn(ρ)intoPf−1(ρ), because
S#ρ=ρ ⇔ (S◦Tρ)#ρ=ρ.
We claim that this right composition also provides an isometry between the “Riemannian manifolds”
Arn(ρ) andPf−1(ρ): indeed, if v ∈ TanSArn(ρ), then the perturbed mapsS +tvare sent to S◦Tρ+tv◦Tρ, which means that the perturbationvofSis sent to the perturbationu:=v◦Tρ ofS◦Tρby the differential of the right composition. The conclusion follows from the change of variable formula, which gives
Z
|v|2dρ= Z
|u|2dρ.
Clearly, the kernel of the differentialdPfofPfatT is given byTanTPf−1 Pf(T)
, thus it remains to prove that its orthogonal is sent isometrically ontoTanPf(T)(P2(Rd))bydPf. FixT ∈BM, let ρ:= Pf(T) =T#ρand observe that
Tan⊥T Pf−1 ρ
=n
vector fieldsw : Z
hw, uidρ= 0, ∀us.t.∇ ·(u◦T−1ρ) = 0o
=n
vector fieldsw : Z
w◦T−1, u◦T−1
dρ= 0, ∀us.t.∇ ·(u◦T−1ρ) = 0o
=n
vector fieldsw : w◦T−1=∇ϕfor someϕ∈Cc∞(Rd)o . Now pickw∈Tan⊥T Pf−1 ρ
, letϕ∈Cc∞(Rd)be such thatw◦T−1=∇ϕand observe that d
dt|t=0Pf(T+tw) = d
dt|t=0(T+tw)#ρ= d
dt|t=0(Id+tw◦T−1)#(T#ρ) = d
dt|t=0(Id+t∇ϕ)#ρ, which means, by definition ofTanρ(P2(Rd))and the action of tangent vectors, that the differential dPf(T)(w)ofPfcalculated atTalong the directionwis given by∇ϕ. The fact that this map is an isometry follows once again by the change of variable formula
Z
|w|2dρ= Z
|w◦T−1|2dρ= Z
|∇ϕ|2dρ.