Lecture Notes on Optimal
Transport Problems ∗
Luigi Ambrosio, [email protected]
†
Contents
1 Some elementary examples
3
2 Optimal transport plans: existence and regularity
5
3 The one dimensional case
14
4 The ODE version of the optimal transport problem
17
5 The PDE version of the optimal transport problem and the plaplacian approximation
31
6 Existence of optimal transport maps
35
7 Regularity and uniqueness of the transport density
40
8 The Bouchitté–Buttazzo mass optimization problem
44
9 Appendix: some measure theoretic results
46
Introduction
These notes are devoted to the Monge–Kantorovich optimal transport problem.
This problem, in the original formulation of Monge, can be stated as follows: given
∗
Lectures given in Madeira (PT), Euro Summer School “Mathematical aspects of evolving
interfaces”, 2–9 July 2000
†
Scuola Normale Superiore, Pisa; work partially supported by the GNAFA project “Problemi
di Monge–Kantorovich e strutture deboli”
1
two distributions with equal masses of a given material g0 (x), g1 (x) (corresponding
for instance to an embankment and an excavation), find a transport map ψ which
carries the first distribution into the second and minimizes the transport cost
Z
C(ψ) :=
|x − ψ(x)|g0 (x) dx.
X
The condition that the first distribution of mass is carried into the second can be
written as
Z
Z
g0 (x) dx =
g1 (y) dy
∀B ⊂ X Borel
(1)
ψ −1 (B)
B
or, by the change of variables formula, as
g1 (ψ(x)) |det∇ψ(x)| = g0 (x)
for Ln -a.e. x ∈ B
if ψ is one to one and sufficiently regular.
More generally one can replace the functions g0 , g1 by positive measures f0 , f1
with equal mass, so that (1) reads f1 = ψ# f0 , and replace the euclidean distance by
a generic cost function c(x, y), studying the problem
Z
min
c(x, ψ(x)) df0 (x).
(2)
ψ# f0 =f1
X
The infimum of the transport problem leads also to a c-dependent distance between
measures with equal mass, known as Kantorovich–Wasserstein distance.
The optimal transport problem and the Kantorovich–Wasserstein distance have
a very broad range of applications: Fluid Mechanics [9], [10]; Partial Differential
Equations [32, 29]; Optimization [13], [14] to quote just a few examples. Moreover,
the 1-Wasserstein distance (corresponding to c(x, y) = |x − y| in (2)) is related to
the so-called flat distance in Geometric Measure Theory, which plays an important
role in its development (see [6], [24], [30], [27], [44]). However, rather than showing
specific applications (for which we mainly refer to the Evans survey [21] or to the
introduction of [9]), the main aim of the notes is to present the different formulations
of the optimal transport problem and to compare them, focussing mainly on the
linear case c(x, y) = |x − y|. The main sources for the preparation of the notes have
been the papers by Bouchitté–Buttazzo [13, 14], Caffarelli–Feldman–McCann [15],
Evans–Gangbo [22], Gangbo–McCann [26], Sudakov [42] and Evans [21].
The notes are organized as follows. In Section 1 we discuss some basic examples
and in Section 2 we discuss Kantorovich’s generalized solutions, i.e. the transport
plans, pointing out the connection between them and the transport maps. Section 3
is entirely devoted to the one dimensional case: in this situation the order structure
plays an important role and considerably simplifies the theory. Sections 4 and 5 are
2
devoted to the ODE and PDE based formulations of the optimal transport problem
(respectively due to Brenier and Evans–Gangbo); we discuss in particular the role of
the so-called transport density and the equivalence of its different representations.
R1
Namely, we prove that any transport density µ can be represented as 0 πt# (|y −
R1
x|γ) dt, where γ is an optimal planning, as 0 |Et | dt, where Et is the “velocity field”
in the ODE formulation, or as the solution of the PDE div(∇µ uµ) = f1 − f0 , with
no regularity assumption on f1 , f0 . Moreover, in the same generality we prove
convergence of the p-laplacian approximation.
In Section 6 we discuss the existence of the optimal transport map, following
essentially the original Sudakov approach and filling a gap in his original proof
(see also [15, 43]). Section 7 deals with recent results, related to those obtained
in [25], on the regularity and the uniqueness of the transport density. Section 8 is
devoted to the connection between the optimal transport problem and the so-called
mass optimization problem. Finally, Section 9 contains a self contained list of the
measure theoretic results needed in the development of the theory.
Main notation
X
B(X)
Ln
Hk
Lip(X)
Lip1 (X)
Σu
πt
So (X)
Sc (X)
M(X)
M+ (X)
M1 (X)
|µ|
µ+ , µ −
f# µ
1
a compact convex subset of an Euclidean space Rn
Borel σ-algebra of X
Lebesgue measure in Rn
Hausdorff k-dimensional measure in Rn
real valued Lipschitz functions defined on X
functions in Lip(X) with Lipschitz constant not greater than 1
the set of points where u is not differentiable
projections (x, y) 7→ x + t(y − x), t ∈ [0, 1]
open segments ]]x, y[[ with x, y ∈ X
closed segments [[x, y]] with x, y ∈ X, x 6= y
signed Radon measures with finite total variation in X
positive and finite Radon measures in X
probability measures in X
total variation of µ ∈ [M(X)]n
positive and negative part of µ ∈ M(X)
push forward of µ by f
Some elementary examples
In this section we discuss some elementary examples that illustrate the kind of
phenomena (non existence, non uniqueness) which can occur. The first one shows
that optimal transport maps need not exist if the first measure f0 has atoms.
3
Example 1.1 (Non existence) Let f0 = δ0 and f1 = (δ1 + δ−1 )/2. In this case
the optimal transport problem has no solution simply because there is no map ψ
such that ψ# f0 = f1 .
The following two examples deal with the case when the cost function c in X ×X
is |x − y|, i.e. the euclidean distance between x and y. In this case we will use as
a test for optimality the fact that the infimum of the transport problem is always
greater than
Z
u d(f1 − f0 ) : u ∈ Lip1 (X) .
sup
(3)
X
Indeed,
Z
Z
u d(f1 − f0 ) =
X
Z
u(ψ(x)) − u(x) df0 (x) ≤
X
|ψ(x) − x| df0 (x)
X
for any admissible transport ψ. Actually we will prove this lower bound is sharp if
f0 has no atom (see (6) and (13)).
Our second example shows that in general the solution of the optimal transport
problem is not unique. In the one-dimensional case we will obtain (see Theorem 3.1),
uniqueness (and existence) in the class of nondecreasing maps.
Example 1.2 (Book shifting) Let n ≥ 1 be an integer and f0 = χ[0,n] L1 and
f1 = χ[1,n+1] L1 . Then the map ψ(t) = t + 1 is optimal. Indeed, the cost relative
to ψ is n and, choosing the 1-Lipschitz function u(t) = t in (3), we obtain that the
supremum is at least n, whence the optimality of ψ follows. But since the minimal
cost is n, if n > 1 another optimal map ψ is given by
(
t + n on [0, 1]
ψ(t) =
t
on [1, n].
In the previous example the two transport maps coincide when n = 1; however
in this case there is one more (and actually infinitely many) optimal transport map.
Example 1.3 Let f0 = χ[0,1] L1 and f1 = χ[1,2] L1 (i.e. n = 1 in the previous
example). We have already seen that ψ(t) = t + 1 is optimal. But in this case also
the map ψ(t) = 2 − t is optimal as well.
In all the previous examples the optimal transport maps ψ satisfy the condition
ψ(t) ≥ t. However it is easy to find examples where this does not happen.
Example 1.4 Let f0 = χ[−1,1] L1 and f1 = δ−1 + δ1 . In this case the optimal
transport map ψ is unique (modulo L1 -negligible sets); it is identically equal to −1
on [−1, 0) and identically equal to 1 on (0, 1]. The verification is left to the reader
as an exercise.
4
We conclude this section with some two dimensional examples.
Example 1.5 Assume that 2f0 is the sum of the unit Dirac masses at (1, 1) and
(0, 0), and that 2f1 is the sum of the unit Dirac masses at (1, 0) and (0, 1). Then
the “horizontal” transport and the “vertical” transport are both optimal. Indeed,
the cost of these transports is 1 and choosing
(p
if x1 + x2 ≤ 1
x21 + x22
p
u(x1 , x2 ) =
(1 − x1 )2 + (1 − x2 )2 if x1 + x2 > 1
in (3) we obtain that the infimum of the transport problem is at least 1.
Example 1.6 Assume that f1 is the sum of two Dirac masses at A, B ∈ R2 and
assume that f0 is supported on the middle axis between them. Then
Z
Z
|x − ψ(x)| df0 (x) =
|x − A| df0 (x)
X
X
whenever ψ(x) ∈ {A, B}, hence any admissible transport is optimal.
2
Optimal transport plans: existence and regularity
In this section we discuss Kantorovich’s approach to the optimal transport problem.
His idea has been to look for optimal transport “plans” , i.e. probability measures γ
in the product space X × X, rather than optimal transport maps. We will see that
this more general viewpoint can be used in several situations to prove that actually
optimal transport maps exist (this intermediate passage through a weak formulation
of the problems is quite common in PDE and Calculus of Variations).
(MK) Let f0 , f1 ∈ M1 (X). We say that a probability measure γ in M1 (X × X) is
admissible if its marginals are f0 and f1 , i.e.
π0# γ = f0 ,
π1# γ = f1 .
Then, given a Borel cost function c : X × X → [0, ∞], we minimize
Z
I(γ) :=
c(x, y) dγ(x, y)
X×X
5
among all admissible γ and we denote by Fc (f0 , f1 ) the value of the infimum.
We also call an admissible γ a transport plan. Notice also that in Kantorovich’s
setting no restriction on f0 or f1 is necessary to produce admissible transport plans:
the product measure f0 × f1 is always admissible. In particular the following definition is well posed and produces a family of distances in M1 (X).
Definition 2.1 (Kantorovich–Wasserstein distances) Let p ≥ 1 and f0 , f1 ∈
M1 (X). We define the p-Wasserstein distance between f0 and f1 by
1/p
Z
p
Fp (f0 , f1 ) :=
min
|x − y| dγ
.
(4)
π0# γ=f0 , π1# γ=f1
X×X
The difference between transport maps and transport plans can be better understood with the following proposition.
Proposition 2.1 (Transport plans versus transport maps) Any Borel transport map ψ : X → X induces a transport plan γψ defined by
γψ := (Id × ψ)# f0 .
(5)
Conversely, a transport plan γ is induced by a transport map if γ is concentrated on
a γ-measurable graph Γ.
Proof. Let ψ be a transport map. Since π0 ◦ (Id × ψ) = Id and π1 ◦ (Id × ψ) = ψ
we obtain immediately that π0# γψ = f0 and π1# γψ = ψ# f0 = f1 . Notice also that,
by Lusin’s theorem, the graph of ψ is γψ -measurable.
Conversely, let Γ ⊂ X × X be a γ-measurable graph on which γ is concentrated
and write
Γ = {(x, φ(x)) : x ∈ π0 (Γ)}
for some function φ : π0 (Γ) → X. Let (Kh ) be an increasing sequence of compact
subsets of Γ such that γ(Γ \ Kh ) → 0 and notice that
f0 (π0 (Kh )) = γ π0−1 (π0 (Kh )) ≥ γ(Kh ) → 1.
Hence, π0 (Γ) ⊃ ∪h π0 (Kh ) is f0 -measurable and with full measure in X. Moreover,
representing γ as γx ⊗ f0 as in (58) we get
Z
γx ({y : (x, y) ∈
/ ∪h Kh }) df0 (x)
0 = lim γ(X × X \ Kh ) =
h→∞
X
Z ∗
≥
γx (X \ {φ(x)}) df0 (x)
π0 (Γ)
6
R∗
(here
denotes the outer integral). Hence γx is the unit Dirac mass at φ(x) for
f0 -a.e. x ∈ X. Since
Z
x 7→ ψ(x) :=
y dγx (y)
X
is a Borel map coinciding with φ f0 -a.e., we obtain that γx is the unit Dirac mass
at ψ(x) for f0 -a.e. x. For A, B ∈ B(X) we get
Z
γ(A × B) =
γx (B) df0 (x) = f0 ({x : (x, ψ(x)) ∈ A × B}) = γψ (A × B)
A
and therefore γ = γψ . The existence of optimal transport plans is a straightforward consequence of the
w -compactness of probability measures and of the lower semicontinuity of I.
∗
Theorem 2.1 (Existence of optimal plans) Assume that c is lower semicontinuous in X × X. Then there exists γ ∈ M1 (X × X) solving (MK). Moreover, if c is
continuous and real valued we have
Z
min (MK) = inf
c(x, ψ(x)) df0 (x)
(6)
ψ# f0 =f1
X
provided f0 has no atom.
Proof. Clearly the set of admissible γ’s is closed, bounded and w∗ -compact for the
w∗ -convergence of measures (i.e. in the duality with continuous functions in X ×X).
Hence, it suffices to prove that
I(γ) ≤ lim inf I(γh )
h→∞
whenever γh w∗ -converge to γ. This lower semicontinuity property follows by the
fact that c can be approximated from below by an increasing sequence of continuous
and real valued functions ch (this is a well known fact: see for instance Lemma 1.61
in [4]). The functionals Ih induced by ch converge monotonically to I, whence the
lower semicontinuity of I follows.
In order to prove the last part we need to show the existence, for any γ ∈
M+ (X × X) with π0# γ = f0 and π1# γ = f1 , of Borel maps ψh : X → X such that
ψh# f0 = f1 and δψh (x) ⊗ f0 weakly converge to γ in M(X × X). The approximation
Theorem 9.3 provides us, on the other hand, with a Borel map ϕ : X → X such that
ϕ# f0 has no atom, is arbitrarily close to f1 and δϕ(x) ⊗ f0 is arbitrarily close (with
respect to the weak topology) to γ. We will build ψh by an iterated application of
this result.
7
By a standard approximation argument we can assume that L = Lip(c) is finite
and that
δ|x − y| ≤ c(x, y)
∀x, y ∈ X
for some δ > 0. Possibly replacing c by c/δ we assume δ = 1.
Fix now an integer h, set f00 = f0 and choose ϕ0 : X → X such that ϕ0# f00 has
no atom and
Z
0
−h
F1 (f1 , ϕ0# f0 ) < 2
and
c(ϕ0 (x), x) df00 (x) < Fc (f1 , f00 ) + 2−h .
X
Then, we set f01 = ϕ0# f00 , find ϕ1 : X → X such that ϕ1# f01 has no atom and
Z
1
−1−h
F1 (f1 , ϕ1# f0 ) < 2
c(ϕ1 (x), x) df01 (x) < Fc (f1 , f01 ) + 2−1−h .
and
X
Proceeding inductively and setting f0k = ϕ(k−1)# f0k−1 we find ϕk such that
Z
k
−k−h
F1 (f1 , ϕk# f0 ) < 2
and
c(ϕk (x), x) df0k (x) < Fc (f1 , f0k ) + 2−k−h
X
and ϕk# f0k has no atom. Then, we set φ0 (x) = x and φk = ϕk−1 ◦· · ·◦ϕ0 for k ≥ 1, so
that f0k = φk# f0 . We claim that (φk ) is a Cauchy sequence in L1 (X, f0 ; X). Indeed,
∞ Z
X
k=0
|φk+1 (x) − φk (x)| df0 (x) =
X
∞ Z
X
k=0
1−h
≤ 2
|ϕk (y) − y| df0k (y)
X
+
∞
X
F1 (f1 , f0k ) < ∞.
k=0
Denoting by ψh the limit of φk , passing to the limit as k → ∞ we obtain ψh# f0 = f1 ;
moreover, we have
Z
Z
c(φk (x), x) df0 (x) ≤
X
c(ϕ0 (x), x) df0 (x) + L
X
≤ Fc (f1 , f0 ) + 2
−h
+L
k Z
X
i=1
k Z
X
i=1
|φi (x) − φi−1 (x))| df0 (x)
X
|ϕi (y) − y| df0i (y) ≤ Fc (f1 , f0 ) + 2−h (1 + 2L).
X
Passing to the limit as k → ∞ we obtain
Z
c(ψh (x), x) df0 (x) ≤ Fc (f1 , f0 ) + 2−h (1 + 2L)
X
8
and the proof is achieved. For instance in the case of Example 1.1 (where transport maps do not exist at
all) it is easy to check that the unique optimal transport plan is given by
1
1
δ0 × δ−1 + δ0 × δ1 .
2
2
In general, however, uniqueness fails because of the linearity of I and of the convexity
of the class of admissible plans γ. In Example 1.2, for instance, any measure
t(Id × ψ1 )# f0 + (1 − t)(Id × ψ2 )# f0
is optimal, with t ∈ [0, 1] and ψ1 , ψ2 optimal transport maps.
In order to understand the regularity properties of optimal plans γ we introduce,
following [26, 36], the concept of cyclical monotonicity.
Definition 2.2 (Cyclical monotonicity) Let Γ ⊂ X × X. We say that Γ is cciclically monotone if
n
n
X
X
c(xi+1 , yi ) ≥
c(xi , yi )
(7)
i=1
i=1
whenever n ≥ 2 and (xi , yi ) ∈ Γ for 1 ≤ 1 ≤ n, with xn+1 = x1 .
The cyclical monotonicity property can also be stated in a (apparently) stronger
form:
n
n
X
X
c(xσ(i) , yi ) ≥
c(xi , yi )
(8)
i=1
i=1
for any permutation σ : {1, . . . , n} → {1, . . . , n}. The equivalence can be proved
either directly (reducing to the case when σ has no nontrivial invariant set) or
verifying, as we will soon do, that any cyclically monotone set is contained in the csuperdifferential of a c-concave function and then checking that the superdifferential
fulfils (8).
Theorem 2.2 (Regularity of optimal plans) Assume that c is continuous and
real valued. Then, for any optimal γ the set sptγ is c-ciclically monotone. Moreover,
the union of sptγ as γ range among all optimal plans is c-ciclically monotone.
Proof. Assume by contradiction that there exist an integer n ≥ 2 and points
(xi , yi ) ∈ sptγ, i = 1, . . . , n, such that
f ((xi ), (yi )) :=
n
X
c(xi+1 , yi ) − c(xi , yi ) < 0
i=1
9
with xn+1 = x1 . For 1 ≤ i ≤ n, let Ui , Vi be compact neighbourhoods of xi and yi
respectively such that γ(Ui × Vi ) > 0 and f ((ui ), (vi )) < 0 whenever ui ∈ Ui and
vi ∈ Vi .
Set now λ = mini γ(Ui × Vi ) and denote by γi ∈ M1 (Ui × Vi ) the normalized
restriction of γ to Ui × Vi . We can find a compact space Y , a probability measure σ
in Y and Borel maps ηi = ui × vi : X → Ui × Vi such that γi = ηi# σ for i = 1, . . . , n
(it suffices for instance to define Y as the product of Ui × Vi , so that ηi are the
projections on the i-coordinate) and define
n
γ 0 := γ +
λX
(ui+1 × vi )# σ − (ui × vi )# σ.
n i=1
Since ληi# σ = λγi ≤ γ we obtain that γ 0 ∈ M+ (X × X); moreover, it is easy to
check that π0# γ 0 = f0 and π1# γ 0 = f1 . This leads to a contradiction because
Z X
n
λ
0
c(ui+1 , vi ) − c(ui , vi ) dσ < 0.
I(γ ) − I(γ) =
n Y i=1
In order to show the last part of the statement we notice that the collection of
optimal transport plans is w∗ -closed and compact. If (γh )h≥1 is a countable dense
set, then
!
h
h
[
X
1
sptγi = spt
γi
h
i=1
i=1
is c-ciclically monotone for any h ≥ 1. Passing to the limit as h → ∞ we obtain
that the closure of the union of sptγh is c-ciclically monotone. By the density of
(γh ), this closure contains sptγ for any optimal plan γ. Next, we relate the c-cyclical monotonicity to suitable concepts (adapted to c)
of concavity and superdifferential.
Definition 2.3 (c-concavity) We say that a function u : X → R is c-concave if
it can be represented as the infimum of a family (ui ) of functions given by
ui (x) := c(x, yi ) + ti
for suitable yi ∈ X and ti ∈ R.
Remark 2.1 [Linear and quadratic case] In the case when c(x, y) is a symmetric
function satisfying the triangle inequality, the notion of c-concavity is equivalent to
1-Lipschitz continuity with respect to the metric dc induced by c. Indeed, given
u ∈ Lip1 (X, dc ), the family of functions whose infimum is u is simply
{c(x, y) + u(y) : y ∈ X} .
10
In the quadratic case c(x, y) = |x − y|2 /2 a function u is c-concave if and only if
u − |x|2 /2 is concave. Indeed, u = inf i c(·, yi ) + ti implies
1
1
u(x) − |x|2 = inf hx, −yi i + |yi |2 + ti
i
2
2
and therefore the concavity of u − |x|2 /2. Conversely, if v = u − |x|2 /2 is concave,
from the well known formula
v(x) =
inf
y, p∈∂ + v(y)
v(y) + hp, x − yi
(here ∂ + v is the superdifferential of v in the sense of convex analysis) we obtain
u(x) =
inf+
y, −p∈∂
1
|p − x|2 + c(p, y).
v(y) 2
Definition 2.4 (c-superdifferential) Let u : X → R be a function. The csuperdifferential ∂c u(x) of u at x ∈ X is defined by
∂c u(x) := {y : u(z) ≤ u(x) + c(z, y) − c(x, y) ∀z ∈ X} .
(9)
The following theorem ([40], [41], [36]) shows that the graphs of superdifferentials
of c-concave functions are maximal (with respect to set inclusion) c-cyclically monotone sets. It may considered as the extension of the well known result of Rockafellar
to this setting.
Theorem 2.3 Any c-ciclically monotone set Γ is contained in the graph of the
c-superdifferential of a c-concave function. Conversely, the graph of the csuperdifferential of a c-concave function is c-ciclically monotone.
Proof. This proof is taken from [36]. We fix (x0 , y0 ) ∈ Γ and define
u(x) := inf c(x, yn ) − c(xn , yn ) + · · · + c(x1 , y0 ) − c(x0 , y0 )
∀x ∈ X
where the infimum runs among all collections (xi , yi ) ∈ Γ with 1 ≤ i ≤ n and
n ≥ 1. Then u is c-concave by construction and the cyclical monotonicity of Γ gives
u(x0 ) = 0 (the minimum is achieved with n = 1 and (x1 , y1 ) = (x0 , y0 )).
We will prove the inequality
u(x) ≤ u(x0 ) + c(x, y 0 ) − c(x0 , y 0 )
11
(10)
for any x ∈ X and (x0 , y 0 ) ∈ Γ. In particular (choosing x = x0 ) this implies that
u(x0 ) > −∞ and that y 0 ∈ ∂c u(x0 ). In order to prove (10) we fix λ > u(x0 ) and find
(xi , yi ) ∈ Γ, 1 ≤ i ≤ n, such that
c(x0 , yn ) − c(xn , yn ) + · · · + c(x1 , y0 ) − c(x0 , y0 ) < λ.
Then, setting (xn+1 , yn+1 ) = (x0 , y 0 ) we find
u(x) ≤ c(x, yn+1 ) − c(xn+1 , yn+1 ) + c(xn+1 , yn ) − c(xn , yn )
+ · · · + c(x1 , y0 ) − c(x0 , y0 ) ≤ c(x, y 0 ) − c(x0 , y 0 ) + λ.
Since λ is arbitrary (10) follows.
Finally, if v is c-concave, yi ∈ ∂c v(xi ) for 1 ≤ i ≤ n and σ is a permutation we
can add the inequalities
v(xσ(i) ) − v(xi ) ≤ c(xσ(i) , yi ) − c(xi , yi )
to obtain (8). In the following corollary we assume that the cost function is symmetric, continuous and satisfies the triangle inequality, so that c-concavity reduces to 1-Lipschitz
continuity with respect to the distance induced by c.
Corollary 2.1 (Linear case) Let γ ∈ M1 (X ×X) with π0# γ = f0 and π1# γ = f1 .
Then γ is optimal for (MK) if and only if there exists u : X → R such that
|u(x) − u(y)| ≤ c(x, y)
∀(x, y) ∈ X × X
(11)
u(x) − u(y) = c(x, y)
for (x, y) ∈ sptγ.
(12)
In addition, there exists u satisfying (11) such that (12) holds for any optimal planning γ. We will call any function u with these properties a maximal Kantorovich
potential.
Proof. (Sufficiency) Let γ 0 be any admissible transport plan; by applying (11) first
and then (12) we get
Z
Z
Z
0
0
I(γ ) ≥
u(x) − u(y) dγ =
u df0 −
u df1
X×X
X
X
Z
=
u(x) − u(y) dγ = I(γ).
X
(Necessity) Let Γ be the closure of the union of sptγ 0 as γ 0 varies among all optimal
plans for (MK). Then we know that Γ is c-cyclically monotone, hence there exists a
12
c-concave function u such that Γ ⊂ ∂c u. Then, (11) follows by the c-concavity of u,
while the inclusion Γ ⊂ ∂c u implies
u(y) − u(x) ≤ c(y, y) − c(x, y) = −c(x, y)
for any (x, y) ∈ sptγ ⊂ Γ. This, taking into account (11), proves (12). A direct consequence of the proof of sufficiency is the identity
Z
min (MK) = max
ud(f0 − f1 ) : u ∈ Lip1 (X, dc )
(13)
(where dc is the distance in X induced by c) and the maximum on the right is
achieved precisely whenever u satisfies (12).
In the following corollary, instead, we consider the case when c(x, y) = |x−y|2 /2.
The result below, taken from [26], was proved first by Brenier in [7, 8] under more restrictive assumptions on f0 , f1 (see also [26] for the general case c(x, y) = h(|x−y|)).
Before stating the result we recall that the set Σv of points of nondifferentiability of
a real valued concave function v (i.e. the set of points x such that ∂ + v(x) is not a
singleton) is countably (CC) regular (see [45] and also [1]). This means that Σv can
be covered with a countable family of (CC) hypersurface, i.e. graphs of differences
of convex functions of n − 1 variables. This property is stronger than the canonical Hn−1 -rectifiability: it implies that Hn−1 -almost all of Σv can be covered by a
sequence of C 2 hypersurfaces.
Corollary 2.2 (Quadratic case) Assume that any (CC) hypersurface is f0 negligible. Then the optimal planning γ is unique and is induced by an optimal
transport map ψ. Moreover ψ is the gradient of a convex function.
Proof. Let u : X → R be a c-concave function such that the graph Γ of its superdifferential contains the support of any optimal planning γ. As c(x, y) = |x − y|2 /2,
an elementary computation shows that (x0 , y0 ) ∈ Γ if and only if
−y0 ∈ ∂ + v(x0 )
where v(x) = u(x)−|x|2 /2 is the concave function already considered in Remark 2.1.
Then, by the above mentioned results on differentiability of concave functions, the
set of points where v is not differentiable is f0 -negligible, hence for f0 -a.e. x ∈ X
there is a unique y0 = −∇v(x0 ) ∈ X such that (x0 , y0 ) ∈ Γ. As sptγ ⊂ Γ, by
Proposition 2.1 we infer that
γ = (Id × ψ)# f0
for any optimal planning γ, with ψ = −∇v. 13
Example 2.1 (Brenier polar factorization theorem) A remarkable consequence of Corollary 2.2 is the following result, known as polar factorization theorem.
A vector field r : X → X such that
Ln (r−1 (B)) = 0
whenever
Ln (B) = 0
(14)
can be written as ∇u ◦ η, with u convex and η measure preserving.
It suffices to apply the Corollary to the measure f0 = r# (Ln X) (absolutely
continuous, due to (14)) and f1 = Ln X; we have then
Ln X = (∇v)# (r# (Ln X)) = (∇v ◦ r)# (Ln X)
for a suitable convex function v, hence η = (∇v) ◦ r is measure preserving. The
desired representation follows with u = v ∗ , since ∇u = (∇v)−1 .
Let us assume that f0 = Ln X and let f1 ∈ M+ (X) be any other measure
such that f1 (X) = Ln (X); in general the problem of mapping f0 into f1 through a
Lipschitz map has no solution (it suffices to consider, for instance, the case when X =
B 1 and f1 = n1 H1 ∂B1 ). However, another remarkable consequence of Corollary 2.2
is that the problem has solution if we require the transport map to be only a function
of bounded variation: indeed, bounded monotone functions (in particular gradients
of Lipschitz convex functions) are functions of bounded variation (see for instance
Proposition 5.1 of [2]). Moreover, we can give a sharp quantitative estimate of the
error made in the approximation by Lipschitz transport maps.
Theorem 2.4 There exists a constant C = C(n, X) with the following property:
for any µ ∈ M+ (X) with µ(X) = Ln (X) and any M > 0 there exist a Lipschitz
function φ : X → X and B ∈ B(X) such that Lip(φ) ≤ M , Ln (X \ B) ≤ C/M and
µ = φ# (Ln B) + µs
with µs ∈ M+ (X) and µs (X) ≤ Ln (X \ B).
Proof. By Corollary 2.2 we can represent µ = ψ# (Ln X) with ψ : X → X equal
to the gradient of a convex function. Let Ω be the interior of X; by applying
Proposition 5.1 of [2] (valid, more generally, for monotone operators) we obtain that
the total variation |Dψ|(Ω) can be estimated with a suitable constant C depending
only on n and X. Therefore, by applying Theorem 5.34 of [4] we can find a Borel
set B ⊂ X (it is a suitable sublevel set of the maximal function of |Dψ|) such that
Ln (X \B) ≤ c(n)|Dψ|(Ω)/M and the restriction of ψ to B is a M -Lipschitz function,
i.e. with Lipschitz constant not greater than M . By Kirszbraun theorem (see for
14
instance [24]) we can extend ψ|B to a M -Lipschitz function φ : X → X. Setting
B c = X \ B we have then
µ = ψ# (Ln B) + ψ# (Ln B c ) = φ# (Ln B) + ψ# (Ln B c )
and setting µs = ψ# (Ln B c ) the proof is achieved. 3
The one dimensional case
In this section we assume that X = I is a closed interval of the real line; we also
assume for simplicity that the transport cost is c(x, y) = |x − y|p for some p ≥ 1.
Theorem 3.1 (Existence and uniqueness) Assume that f0 is a diffuse measure,
i.e. f0 ({t}) = 0 for any t ∈ I. Then
(i) there exists a unique (modulo countable sets) nondecreasing function ψ :
sptf0 → X such that ψ# f0 = f1 ;
(ii) the function ψ in (i) is an optimal transport, and if p > 1 is the unique optimal
transport.
In the one-dimensional case these results are sharp: we have already seen that
transport maps need not exist if f0 has atoms (Example 1.1) and that, without the
monotonicity constraint, are not necessarily unique when p = 1.
Proof. (i) Let m = min I and define
ψ(s) := sup {t ∈ I : f1 ([m, t]) ≤ f0 ([m, s])} .
(15)
It is easy to check that the following properties hold:
(a) ψ is non decreasing;
(b) ψ(I) ⊃ sptf1 ;
(c) if ψ(s) is not an atom of f1 we have
f1 ([m, ψ(s)]) = f0 ([m, s]).
(16)
Let T be the at most countable set made by the atoms of f1 and by the points
t ∈ I such that ψ −1 (t) contains more than one point; then ψ −1 is well defined on
15
ψ(I) \ T and ψ −1 ([t, t0 ]) = [ψ −1 (t), ψ −1 (t0 )] whenever t, t0 ∈ ψ(I) \ T with t < t0 .
Then (c) gives
f1 ([t, t0 ]) = f1 ([m, t0 ]) − f1 ([m, t]) = f0 ([m, ψ −1 (t0 )]) − f0 ([m, ψ −1 (t)])
= f0 ([ψ −1 (t), ψ −1 (t0 )]) = f0 (ψ −1 ([t, t0 ]))
(notice that only here we use the fact that f0 is diffuse). By (b) the closed intervals
whose endpoints belong to ψ(I) \ T generate the Borel σ-algebra of sptf1 , and this
proves that ψ# f0 = f1 .
Let φ be any nondecreasing function such that φ# f0 = f1 and assume, possibly
modifying φ on a countable set, that φ is right continuous. Let
T := {s ∈ sptf0 : (s, s0 ) ∩ sptf0 = ∅ for some s0 > s } .
and notice that T is at most countable (since we can index with T a family of
pairwise disjoint open intervals). We claim that φ ≥ ψ on sptf0 \ T ; indeed, for
s ∈ sptf0 \ T and s0 > s we have the inequalities
f1 ([m, φ(s0 )]) = f0 (φ−1 ([m, φ(s0 )])) ≥ f0 ([m, s0 ]) > f0 ([m, s])
and, by the definition of ψ, the inequality follows letting s0 ↓ s. In particular
Z
Z
Z
Z
Z
φ − ψ df0 = φ df0 − ψ df0 = 1 df1 − 1 df1 = 0
I
I
I
I
I
whence φ = ψ f0 -a.e. in I. It follows that φ(s) = ψ(s) at any continuity point
s ∈ sptf0 of φ and ψ.
(ii) By a continuity argument it suffices to prove that ψ is the unique solution of the
transport problem for any p > 1 (see also [26, 15]). Let γ be an optimal planning
and notice that the cyclical monotonicity proved in Theorem 2.2 gives
|x − y 0 |p + |x0 − y|p ≥ |x − y|p + |x0 − y 0 |p
whenever (x, y), (x0 , y 0 ) ∈ sptγ. If x < x0 , this condition implies that y ≤ y 0 (this
is a simple analytic calculation that we omit, and here the fact that p > 1 plays a
role). This means that the set
T := {x ∈ sptf0 : card({y : (x, y) ∈ sptγ}) > 1}
is at most countable (since we can index with T a family of pairwise disjoint open
intervals) hence f0 -negligible. Therefore for f0 -a.e. x ∈ I there exists a unique
y = ψ̃(x) ∈ I such that (x, y) ∈ sptγ (the existence of at least one y follows by the
16
fact that the projection of sptγ on the first factor is sptf0 ). Notice also that ψ̃ is
nondecreasing in its domain.
Arguing as in Proposition 2.1 we obtain that
γ = (Id × ψ̃)# f0 .
In particular
f1 = π1# γ = π1# (Id × ψ̃)# f0 = ψ̃# f0
and since ψ̃ is non decreasing it follows that ψ̃ = ψ (up to countable sets) on sptf0 .
4
The ODE version of the optimal transport
problem
In this and in the next section we rephrase the optimal transport problem in differential terms. In the following we consider a fixed auxiliary open set Ω containing
X; we assume that Ω is sufficiently large, namely that the open r-neighbourhood
of X is contained in Ω, with r > diam(X) (the necessity of this condition will be
discussed later on).
The first idea, due to Brenier ([10, 9] and also [11]) is to look for all the paths
ft in M+ (X) connecting f0 to f1 . In the simplest case when ft = δx(t) , it turns out
that the velocity field Et = ẋ(t)δx(t) is related to ft by the equation
f˙t + ∇ · Et = 0
in (0, 1) × Ω
(17)
in the distribution sense. Indeed, given ϕ ∈ Cc∞ (0, 1) and φ ∈ Cc∞ (Ω), it suffices
to take ϕ(t)φ(x) as test function in (17) and to use the definitions of ft and Et to
obtain
Z 1
˙
(ft + ∇ · Et )(ϕφ) = −
ϕ̇(t)φ(x(t)) + ϕ(t)h∇φ(x(t)), ẋ(t)i dt
0
Z 1
d
[ϕ(t)φ(x(t))] dt = 0.
= −
0 dt
More generally, regardless of any assumption on (ft , Et ) ∈ M+ (X) × [M(Ω)]n ,
it is easy to check that (17) holds in the distribution sense if and only if
f˙t (φ) = ∇φ · Et
in (0,1)
∀φ ∈ Cc∞ (Ω)
in the distribution sense. We will use both interpretations in the following.
17
(18)
One more interpretation of (17) is given in the following proposition. Recall that
a map f defined in (0, 1) with values in a metric space (E, d) is said to be absolutely
continuous if for any ε > 0 there exists δ > 0 such that
X
X
(yi − xi ) < δ
=⇒
d (f (yi ), f (xi )) < ε
i
i
for any family of pairwise disjoint intervals (xi , yi ) ⊂ (0, 1).
Proposition 4.1 If some
R 1family (ft ) ⊂ M1 (X) fulfils (17) for suitable measures
n
Et ∈ [M(Ω)] satisfying 0 |Et |(Ω) dt < ∞ then f is an absolutely continuous map
between (0, 1) and M1 (X), endowed with the 1-Wasserstein distance (4) and
F1 (ft+h , ft )
≤ |Et |(Ω)
h→0
|h|
lim
for L1 -a.e. t ∈ (0, 1).
(19)
Conversely, if ft is an absolutely continuous map we can choose Et so that equality
holds in (19).
Proof. In (13) we can obviosuly restrict to test functions u such that |u| ≤ r =
diam(X) on X; by our assumption on Ω any of these function can be extended
to Rn in such a way that the Lipschitz constant is still less than 1 and u ≡ 0 in a
neighbourhood of Rn \Ω. In particular, choosing an optimal u and setting uε = u∗ρε
we get
Z
Z t
uε d(ft − fs ) = lim+
∇ · Eτ (uε ) dτ
F1 (fs , ft ) = lim+
ε→0
ε→0
X
s
Z t
Z t
|Eτ |(Ω) dτ
(20)
= − lim+
Eτ · ∇uε dτ ≤
ε→0
s
s
whenever 0 ≤ s ≤ t ≤ 1 and this easily leads to (19).
In the proof of the converse implication we can assume with no loss of generality
(up to a reparameterization by arclength) that f is a Lipschitz map. We can consider
M1 (X) as a subset of the dual Y = G∗ , where
G := φ ∈ C 1 (Rn ) ∩ Lip(Rn ) : φ ≡ 0 on Rn \ Ω
endowed with the norm kφk = Lip(φ). By using convolutions and (13) it is easy
to check that F1 (µ, ν) = kµ − νkY , so that M1 (X) is isometrically embedded in
Y . Moreover, using Hahn–Banach theorem it is easy to check that any y ∈ Y is
representable as the divergence of a measure E ∈ [M(Ω)]n with kyk = |E|(Ω) (E is
not unique, of course).
18
By a general result proved in [5], valid also for more than one independent
variable, any Lipschitz map f from (0, 1) into the dual Y of a separable Banach
space is weakly∗ -differentiable for L1 -almost every t, i.e.
∃w∗ − lim
h→0
ft+h − ft
=: f˙(t)
h
Rt
and ft − fs = s f˙(τ ) dτ for s, t ∈ (0, 1). In addition, the map is also metrically
differentiable for L1 -almost every t, i.e.
kft+h − ft k
=: mf˙(t)
h→0
|h|
∃ lim
Although f˙ is only a w∗ -limit of the difference quotients, it turns out (see [5]) that
the metric derivative mf˙ is L1 -a.e. equal to kf˙k.
Putting together these informations the conclusion follows. According to Brenier, we can formulate the optimal transport problem as follows.
(ODE) Let f0 , f1 ∈ M1 (X) be given probability measures. Minimize
Z 1
J(E) :=
|Et |(Ω) dt
(21)
0
among all Borel maps ft : [0, 1] → M+ (X) and Et : [0, 1] → [M(Ω)]n such that (17)
holds.
Example 4.1 In the case considered in Example 1.3 the measures
ft = χ[t,t+1] L1 ,
Et = χ[t,t+1] L1
provide an admissible and optimal flow, obviously related to the optimal transport
map x 7→ x + 1. But, quite surprisingly, we can also define an optimal flow by

1
1

 1−2t χ[2t,1] L if 0 ≤ t < 1/2
f t = δ1
if t = 1/2

 1
1
χ
L if 1/2 < t ≤ 1
2t−1 [1,2t]
whose “velocity field” is
( 2(1−x)
Et =
χ
L1
(1−2t)2 [2t,1]
2(x−1)
χ
L1
(2t−1)2 [1,2t]
19
if 0 ≤ t < 1/2
if 1/2 < t ≤ 1.
It is easy to check that f˙t = ∇·Et = 0 and that |Et | are probability measures for any
t 6= 1/2, hence J(E) = 1. The relation of this new flow with the optimal transport
map x 7→ 2 − x will be seen in the following (see Remark 4.1(3)).
In order to relate solutions of (ODE) to solutions of (MK) we will need an
uniqueness theorem, under regularity assumptions in the space variable, for the
ODE f˙t + ∇ · (gt ft ) = 0. If ft , gt are smooth (say C 2 ) with respect to both the
space and time variables, uniqueness is a consequence of the classical method of
characteristics (see for instance §3.2 of [20]), which provides the representation
Z t
ft (xt ) = f0 (x) exp −
cs (xs ) ds
0
where ct = ∇ · gt and xt solves the ODE
ẋt = gt (xt ),
t ∈ (0, 1).
x0 = x,
See also [33] for more general uniqueness and representation results in a weak setting.
Theorem 4.1 Assume that
f˙t + ∇ · (gt ft ) = 0
in (0, 1) × Rn
(22)
R1
where 0 |ft |(Rn ) dt < ∞ and |gt |+Lip(gt ) ≤ C, with C independent of t and f0 = 0.
Then ft = 0 for any t ∈ (0, 1).
Proof. Let gtε be obtained from gt by a mollification with respect to the space and
time variables and define X ε (s, t, x) as the solution of the ODE ẋ = gsε (x) (with s
as independent variable) such that X ε (t, t, x) = x. Define, for ψ ∈ Cc∞ ((0, 1) × Rn )
fixed,
Z
1
ϕε (t, x) := −
ψ (s, X ε (s, t, x)) ds.
t
ε
ε
ε
Since X (s, t, X (t, 0, x)) = X (s, 0, x) we have
Z 1
ε
ε
ϕ (t, X (t, 0, x)) = −
ψ (s, X ε (s, 0, x)) ds
t
and, differentiating both sides, we infer
ε
∂ϕ
ε
ε
+ gt · ∇ϕ (t, X ε (t, 0, x)) = ψ (t, X ε (t, 0, x))
∂t
whence ψ = (∂ϕε /∂t + gtε · ∇ϕε ) in (0, 1) × Rn .
20
Insert now the test function ϕε in (22) and take into account that f0 = 0 to
obtain
Z 1Z
Z 1Z
∂ϕε
ε
ft ψ + ft (gt − gtε ) · ∇ϕε dxdt.
+ gt · ∇ϕ ) dxdt =
0=
ft (
∂t
0
Rn
0
Rn
The proof is finished letting ε → 0+ and noticing that |gtε | + |∇ϕε | ≤ C, with C
independent of ε. In the following theorem we show that (MK) and (ODE) are basically equivalent.
Here the assumption that Ω is large enough plays a role: indeed, if for instance
X = [[x0 , x1 ]], f0 = δx0 and f1 = δx1 , the infimum of (ODE) is easily seen to be less
than
dist(x0 , ∂Ω) + dist(x1 , ∂Ω).
Therefore, in the general case when f0 , f1 are arbitrary measures in X, we require
that dist(∂Ω, X) > diam(X).
Theorem 4.2 ((MK) versus (ODE)) The problem (ODE) has at least one solution and min (ODE) = min (MK). Moreover, for any optimal planning γ ∈
M1 (X × X) for (MK) the measures
ft := πt# γ,
Et := πt# ((y − x)γ)
t ∈ [0, 1]
(23)
with πt (x, t) = x + t(y − x) solve (ODE).
Proof. Let (ft , Et ) be defined by (23). For any φ ∈ C ∞ (Rn ) we compute
Z
Z
d
d
φ dft =
φ (x + t(y − x)) dγ
dt X
dt X×X
Z
=
∇φ (x + t(y − x)) · (y − x) dγ
X×X
=
n Z
X
i=1
∇i φ dEt,i = −∇ · Et (φ)
X
hence the ODE (17) is satisfied. Then, we simply evaluate the energy J(E) in (21)
by
Z 1
Z 1
J(E) =
|πt# ((y − x)γ)|(X) dt ≤
πt# (|y − x|γ)(X) dt
(24)
0
0
Z 1Z
=
|x − y| dγdt = I(γ).
0
X×X
21
This shows that inf (ODE) ≤ min (MK).
In order to prove the opposite inequality we first use Proposition 4.1 and then
we present a different strategy, which provides more geometric informations.
By Proposition 4.1 we have
Z 1
Z 1
d
F1 (ft , f0 ) dt ≤
|Et |(Ω) dt
F1 (f0 , f1 ) ≤
0 dt
0
for any admissible flow (ft , Et ). This proves that min (MK) ≤ inf (ODE).
For given (and admissible) (ft , Et ), and assuming that
Z 1
|Et | dt ⊂⊂ Ω
spt
0
we exhibit an optimal planning γ with I(γ) ≤ J(E).
∞
To this
R 1 aim we fix a cut-off function θ ∈ Cc (Ω) with 0 ≤ θ ≤ 1 and θ ≡ 1 on
X ∪ spt 0 |Et | dt; then, we define
ftε := ft ∗ ρε + εθ
and Etε := Et ∗ ρε
where ρ is any convolution kernel with compact support and ρε (x) = ε−n ρ(x/ε).
Notice that ftε are strictly positive on sptEt ; moreover (17) still holds, sptEtε ⊂ Ω
for ε small enough and L1 -a.e. t and by (53) we have
Z
|Etε |(y) dy ≤ |Et |(Ω)
∀t ∈ [0, 1].
(25)
Rn
We define gtε = Etε /ftε and denote by ψtε (x) the semigroup in [0, 1] associated to the
ODE
ψ̇tε (x) = gtε (ψtε (x)) ,
ψ0ε (x) = x.
(26)
Notice that this flow for ε small enough maps Ω into itself and leaves Rn \ Ω fixed.
ε
Now, we claim that ftε = ψt#
(f0ε Ln ) for any t ∈ [0, 1]. Indeed, since by definition
f˙tε + ∇ · (gtε ftε ) = f˙tε + ∇ · Etε = 0
by Theorem 4.1 and the linearity of the equation we need only to check that also
ε
νtε = ψt#
(f0ε Ln ) satisfies the ODE ν̇tε + ∇ · (g ε νtε ) = 0. This is a straightforward
computation based on (26).
Using this representation, we can view ψ1ε as approximate solutions of the optimal
transport problem and define
γ ε := (Id × ψ1ε )# (f0ε Ln ),
22
i.e.
Z
ε
Z
ϕ(x, y) dγ (x) =
Rn
ϕ (x, ψ1ε (x)) f0ε (x) dx
for any bounded Borel function ϕ in Rn × Rn .
In order to evaluate I(γ ε ), we notice that
Z 1
ε
|gtε |(ψtε (x)) dt
Length(ψt (x)) =
0
f0ε
hence, by multiplying by
and integrating we get
Z
Z 1Z
ε
ε
Length(ψt (x))f0 (x) dx =
|gtε (y)|ftε (y) dydt
Rn
0
Rn
Z 1Z
Z 1
ε
|Et |(y) dydt ≤
=
|Et |(Ω) dt.
0
As
ε
Z
I(γ ) =
Rn
|ψ1ε (x)
−
x|f0ε (x) dx
Rn
(27)
0
Z
≤
Rn
Length(ψtε (x)) df0 (x),
this proves that I(γ ε ) ≤ J(E). Passing to the limit as ε ↓ 0 and noticing that
π0# γ ε = f0ε Ln ,
π1# γ ε = f1ε Ln
we obtain, possibly passing to a subsequence, an admissible planning γ ∈ M1 (Rn ×
Rn ) for f0 and f1 such that I(γ) ≤ J(E). We can turn γ in a planning supported
in X × X simply replacing γ by (π × π)# γ, where π : Rn → X is the orthogonal
projection. Remark 4.1 (1) Starting from (ft , Et ), the (possibly multivalued) operation leading to γ and then to the new flow
f˜t := πt# γ,
Ẽt := πt# ((y − x)γ)
can be understood as a sort of “arclength reparameterization” of (ft , Et ). However,
since this operation is local in space (consider for instance the case of two line paths,
one parameterized by arclength, one not), in general there is no function ϕ(t) such
that
(f˜t , Ẽt ) = (fϕ(t) , Eϕ(t) ).
(2) The solutions (ft , Et ) in Example 4.1 are built as πt# γ and πt# ((y − x)γ) where
γ = (Id × ψ)# f0 and ψ(x) = x + 1, ψ(x) = 2 − x. In particular, in the second
example, f1/2 = δ1 because 1 is the midpoint of any transport ray.
(3) By making a regularization first in space and then in time of (ft , Et ) one can
avoid the use of Theorem 4.1, using only the classical representation of solutions of
(22) with characteristics. 23
Remark
4.2 (Optimality conditions) (1) If (ft , Et ) is optimal, then
R1
spt 0 |Et | dt ⊂⊂ Ω. Indeed, we can find Ω0 ⊂⊂ Ω which still has the property
that the open r-neighbourhood of X is contained in Ω0 , with r = diam(X), hence
Z 1
Z 1
|Et |(Ω) dt = min (MK) =
|Et |(Ω0 ) dt.
0
0
(2) Since the two sides in the chain of inequalities (24) are equal when γ is optimal,
we infer
|πt# ((y − x)γ)| = πt# (|y − x|γ)
for L1 -a.e. t ∈ (0, 1).
(28)
Analogously, we have
Z
0
1
Z
Et dt =
1
|Et | dt
(29)
0
whenever (ft , Et ) is optimal. If the strict inequality < holds, then
Z 1
fˆt = f0 + t(f1 − f0 ) and Êt =
Eτ dτ
0
provide an admissible pair for (ODE) with strictly less energy. Remark 4.3 (Nonlinear cost) When the cost function c(x, y) is |x − y|p for some
p > 1 the corresponding problem (MK) is still equivalent to (ODE), provided we
minimize, instead of J, the energy
Z 1
Jp (f, E) :=
Φp (ft , Et ) dt
0
where
Z p
ν 

 σ dσ
X
Φp (σ, ν) =



+∞
if |ν| << σ;
otherwise.
The proof of the equivalence is quite similar to the one given in Theorem 4.2. Again
the essential ingredient is the inequality
Φp (ν ∗ ρε , σ ∗ ρε ) ≤ Φp (ν, σ)
The latter follows by Jensen’s inequality and the convexity of the map (z, t) 7→
|z|p t1−p in Rn × (0, ∞), which provide the pointwise estimate
p ν ∗ ρ ε p
σ ∗ ρε ≤ ν σ ∗ ρε .
σ ∗ ρε σ
24
In this case, under the same assumptions on f0 made in Corollary 2.2, due to the
uniqueness of the optimal planning γ = (Id × ψ# )f0 , we obtain that, for optimal
(ft , Et ), γ ε = (Id × ψ1ε )# f0ε converge to γ as ε → 0+ . As a byproduct, the maps ψ1ε
converge to ψ as Young measures. If f0 << Ln it follows that ψh converge to ψ in
[Lr (f0 )]n for any r ∈ [1, ∞) (see Lemma 9.1 and the remark following it).
See §2.6 of [4] for a systematic analysis of the continuity and semicontinuity
properties of the functional (σ, ν) 7→ Φp (σ, ν) with respect to weak convergence of
measures. In the previous proof I don’t know whether it is actually possible to show full
convergence of γ ε as ε → 0+ . However, using a more refined estimate and a geometric
lemma (see Lemma 4.1 below) we will prove that any limit measure γ satisfies the
condition
Z 1
Z 1
|Et | dt.
(30)
πt# (|y − x|γ) dt =
0
0
We call transport density any of the measures in (30). This double representation
will be relevant in Section 7, where under suitable assumptions we will obtain the
uniqueness of the transport density.
Corollary 4.1 Let (ft , Et ) be optimal for (ODE).
R 1 Then there exists an optimal
planning γ such that (30) holds. In particular spt 0 |Et | dt ⊂ X.
Proof. Let
R 1 m = min (MK) = min (ODE) and recall that, by Remark 4.2(1), the
measure 0 |Et | dt has compact support in Ω. It suffices to build an optimal planning
γ such that
Z
Z
1
1
πt# ((y − x)γ) dt =
Et dt.
(31)
0
0
Indeed, by (29) we get
Z
1
Z
πt# (|y − x|γ) ≥
0
1
|Et | dt
0
and the two measures coincide, having both mass equal to m.
Keeping the same notation used in the final part of the proof of Theorem 4.2,
we define γtε as (Id × ψtε )# f0ε and we compute
Z 1Z
Z 1
d
ψtε (x) − x
ε
ε
I(γ ) =
I(γt ) dt =
· ψ̇tε (x)f0ε (x) dxdt.
ε
0 dt
0
BR |ψt (x) − x|
Since I(γ ε ) → m as ε → 0+ and since (by (25))
Z 1Z
Z
Z 1Z
ε
ε
ε
|ψ̇t (x)|f0 (x) dxdt =
|Et | dxdt ≤
0
BR
0
BR
25
0
1
|Et |(Ω) dt = m
we infer
Z
Z
1
lim
ε→0+
BR
0
2
ψ̇ ε (x)
ε
(x)
−
x
ψ
t
− tε
|ψ̇ ε (x)|f0ε (x) dtdx = 0.
ε
|ψ̇t (x)| |ψt (x) − x| t
(32)
Now using Lemma 4.1 and the Young inequality 2ab ≤ δa2 + b2 /δ, for any φ ∈
Cc∞ (Rn ) we obtain
Z 1
[πt# ((y − x)γ ε ) − Etε ](φ) dt
Z0 Z 1
ε
ε
ε
ε
ε
= [(ψ1 (x) − x)φ(t(ψ1 (x) − x)) − ψ̇t (x)φ(ψt (x))]f0 (x) dtdx
X 0
Z
≤ RLip(φ)δ
Length(ψtε )f0ε (x) dx
X
2
Z Z 1 ε
RLip(φ)
ψtε (x) − x ψ̇t (x)
+
− ε
ε
|ψ̇ ε (x)|f0ε (x) dtdx.
δ
|ψt (x) − x| t
X 0 |ψ̇t (x)|
with R = diam(Ω). Passing to the limit first as ε → 0+ , taking (32) into account
and then passing to the limit as δ → 0+ we obtain
Z 1
Z 1
Et (φ) dt.
πt# ((y − x)γ)(φ) dt =
0
0
Remark 4.4 [Decomposition of vectorfields] Let E be a vectorfield with ∇·E =
f in Ω and assume that |E|(Ω) = min (MK) with f0 = f + and f1 = f − . Then,
choosing Et = E and ft = f0 +t(f1 −f0 ) in (31), we obtain that E can be represented
as the superposition of elementary vectorfields
Z
E=
Exy dγ(x, y)
with
Exy := (y − x) H1 [[x, y]]
Ω×Ω
without cancellations, i.e. in such a way that
Z
Z
|E| =
|Exy | dγ(x, y),
|∇ · E| =
Ω×Ω
|∇ · Ex,y | dγ(x, y).
Ω×Ω
The possibility to represent a generic vectorfield whose divergence is a measure
as a superposition without cancellations of elementary vectorfields associated to
rectifiable curves is discussed in [39]. It turns out that this property is not true
in general, at least if n ≥ 3; hence the optimality condition |E|(Ω) = min (MK) is
essential. 26
Lemma 4.1 Let ψ ∈ Lip([0, 1], Rn ) with ψ(0) = 0 and φ ∈ Lip(Rn ). Then
Z 1
Z 1 ψ(t)
ψ̇(t)
≤ LR
−
ψ(1)φ(tψ(1))
−
ψ̇(t)φ(ψ(t))
dt
|ψ̇(t)| dt
|ψ(t)|
|ψ̇(t)|
0
0
with L = Lip(φ), R = sup |ψ|.
Proof. We start from the elementary identity
d
d
[φ(sψ(t))ψi (t)] −
sφ(sψ(t))ψ̇i (t)
dt
ds
n
X
∂φ
(sψ(t)) ψi (t)ψ̇j (t) − ψj (t)ψ̇i (t)
= s
∂xj
j=1
(33)
and integrate both sides in [0, 1] × [0, 1]. Then, the left side becomes exactly the
integral that we need to estimate from above. The right side, up to the multiplicative
constant Lip(φ), can be estimated with
!
Z 1
Z 1
ψ(t)
ψ̇(t)
|ψ(t) ∧ ψ̇(t)| dt =
|ψ(t)| ψ̇(t) ∧
−
dt
|ψ̇(t)| |ψ(t)| 0
0
Z 1 ψ̇(t)
ψ(t)
≤ sup |ψ|
−
|ψ̇(t)| dt.
|ψ(t)|
|ψ̇(t)|
0
Remark 4.5 The geometric meaning of the proof above is the following: the integral to be estimated is the action on the 1-form φdxi of the closed and rectifiable
current T associated to the closed path starting at 0, arriving at ψ(1) following
the curve ψ(t), and then going back to 0 through the segment [[0, ψ(1)]]; for any
2-dimensional current G such that ∂G = T , as
T (φdxi ) = G(dφ ∧ dxi ),
(34)
this action can be estimated by the mass of G times Lip(φ). The cone construction
provides a current G with ∂G = T , whose mass can be estimated by
Z 1Z 1
s|ψ(t) ∧ ψ̇(t)| dsdt.
0
0
With this choice of G the identity (34) corresponds to (33). 27
In order to relate also (ft , Et ) to the Kantorovich potential, we define the transport
rays and the transport set and we prove the differentiability of the potential on the
transport set.
Definition 4.1 (Transport rays and transport set) Let u ∈ Lip1 (X). We say
that a segment ]]x, y[[⊂ X is a transport ray if it is a maximal open oriented segment
whose endpoints x, y satisfy the condition
u(x) − u(y) = |x − y|.
(35)
The transport set Tu is defined as the union of all transport rays. We also define
Tue as the union of the closures of all transport rays.
Denoting by F the compact collection of pairs (x, y) with x 6= y such that (35)
holds, the transport set is also given by
[ [
Tu =
{x + t(y − x)}
t∈(0,1) (x,y)∈F
and therefore is a Borel set (precisely a countable union of closed sets).
We can now easily prove that the u is differentiable at any point in Tu .
Proposition 4.2 (Differentiability of the potential) Let u ∈ Lip1 (X). Then
u is differentiable at any point z ∈ Tu . Moreover −∇u(z) is the unit vector parallel
to the transport ray containing z.
Proof. Let x, y ∈ X be such that (35) holds. By the triangle inequality and the
1-Lipschitz continuity of u we get
u(x) − u(x + t(y − x)) = t|y − x|
∀t ∈ [0, 1].
This implies that, setting ν = (y − x)/|y − x|, the partial derivative of u along
ν is equal to −1 for any internal point z of the segment. For any unit vector ξ
perpendicular to ν we have
p
p
u(z + hξ) − u(z) = u(z + hξ) − u(z + |h|ν) + u(z + |h|ν) − u(z)
p
p
≤
|h|2 + |h| − |h| = O(|h|3/2 ) = o(|h|).
A similar argument also proves that u(z + hξ) − u(z) ≥ o(|h|). This proves the
differentiability of u at z and the identity ∇u(z) = −ν. In Section 6 we will need a mild Lipschitz property of the potential ∇u on Tue
(see also [15, 43] for the case of general strictly convex norms). We will use this
property to prove in Corollary 6.1 that Tue \ Tu is Lebesgue negligible. This property
can also be used to prove that ∇u is approximately differentiable Ln -a.e. on Tu (see
for instance Theorem 3.1.9 of [24]).
28
Theorem 4.3 (Countable Lipschitz property) Let u ∈ Lip1 (X). There exists
a sequence of Borel sets Th covering Ln -almost all of Tue such that ∇u restricted to
Th is a Lipschitz function.
Proof. Given a direction ν ∈ Sn−1 and a ∈ R, let R be the union of the half closed
transport rays [[x, y[[ with hy − x, νi ≥ 0 and hy, νi ≥ a. It suffices to prove that the
restriction of ∇u to
T := R ∩ {x : x · ν < a} \ Σu
has the countable Lipschitz property stated in the theorem. To this aim, since BVloc
funtions have this property (see for instance Theorem 5.34 of [4] or [24]), it suffices
to prove that ∇u coincides Ln -a.e. in T with a suitable function w ∈ [BVloc (Sa )]n ,
where Sa = {x : x · ν < a}. To this aim we define
ũ(x) := min {u(y) + |x − y| : y ∈ Ya }
where Ya is the collection of all right endpoints of transport rays with y · ν ≥ a. By
construction ũ ≥ u and equality holds on R ⊃ T .
We claim that, for b < a, ũ − C|x|2 is concave in Sb for C = C(b) large enough.
Indeed, since |x − y| ≥ a − b > 0 for any y ∈ Ya and any x ∈ Sb , the functions
u(y) + |x − y| − C|x|2 ,
y ∈ Ya
are all concave in H for C large enough depending on a − b. In particular, as
gradients of real valued concave functions are BVloc (see for instance [2]), we obtain
that
w := ∇ũ = ∇(u − C|x|2 ) + 2Cx
is a BVloc function in Sa . Since ∇u = w Ln -a.e. in T the proof is achieved.
By a similar argument, using semi-convexity in place of semi-concavity, one can
take into account the right extreme points of the transport rays. As a byproduct of the equivalence between (MK) and (ODE) we can prove that
(ODE) has “regular” solutions, related to the transport set and to the gradient of
any maximal Kantorovich potential u.
Theorem 4.4 (Regularity of (ft , Et )) For any solution (ft , Et ) of (ODE) representable as in Theorem 4.2 for a suitable optimal planning γ there exists a 1-Lipschitz
function u such that
(i) ft is concentrated on the transport set Tu and |Et | ≤ Cft for L1 -a.e. t ∈ (0, 1),
with C = diam(X);
(ii) Et = −∇u|Et | for L1 -a.e. t ∈ (0, 1);
29
(iii) for any convolution kernel ρ we have
Z 1Z
lim+
|∇u − ∇u ∗ ρε |2 d|Et | dt = 0;
ε→0
(iv) |Et |(X) =
R1
0
0
(36)
X
|Eτ |(X) dτ for L1 -a.e. t ∈ (0, 1).
Proof. (i) Let u be any maximal Kantorovich potential and let T = Tu be the
associated transport set. For any t ∈ (0, 1) we have
Z
ft (X \ T ) =
χ{(x,y): x+t(y−x)∈T
/ } dγ = 0
X×X
because, by (12), any segment ]]x, y[[ with (x, y) ∈ sptγ is contained in a transport
ray, hence contained in T . The inequality |Et | ≤ Cft simply follows by the fact that
|x − y| ≤ C on sptγ.
(ii) Choosing t satisfying (28) and taking into account Proposition 4.2, for any
φ ∈ C(X) we obtain
Z
Z
φ(πt )∇u(πt )|y − x| dγ
φ∇u d|Et | =
X×X
X
Z
= −
φ(πt )(y − x) dγ = −Et (φ).
X×X
(iii) Let m = J(E) = min (MK). By (13) we get
Z
Z
m =
u d(f0 − f1 ) = lim+
u ∗ ρε d(f1 − f0 )
ε→0
X
X
Z Z 1
Z 1
∇u ∗ ρε · Et dt
= lim+
u ∗ ρε df˙t dt = lim+
ε→0
ε→0
X 0
0
Z 1
≤
|Et |(X) dt = m.
0
This proves that
Z
Z
|Et |(X) −
lim
ε→0+
1
0
h∇u ∗ ρε , ∇ui d|Et | dt = 0
X
whence, taking into account that |∇u| = 1 and |∇u ∗ ρε | ≤ 1, (36) follows.
(iv) The inequality |Et |(X) ≤ I(γ) = J(E) has been estabilished during the proof
of Theorem 4.2. By minimality equality holds for L1 -a.e. t ∈ (0, 1). 30
Remark 4.6 (1) It is easy to produce examples of optimal flows (ft , Et ) satisfying
(i), (ii), (iii), (iv) which are not representable as in Theorem 4.2 (again it suffices
to consider the sum of two paths, with constant sum of velocities). It is not clear
which conditions must be added in order to obtain this representation property.
(2) Notice that condition (i) actually implies that f is a Lipschitz map from [0, 1]
into M1 (X) endowed with the 1-Wasserstein metric. 5
The PDE version of the optimal transport problem and the p-laplacian approximation
In this section we see how, in the case when the cost function is linear, the optimal
transport problem can be rephrased using a PDE. As in the previous section we
consider an auxiliary bounded open set Ω such that the open r-neighbourhood of
X is contained in Ω, with r > diam(X); we assume also that Ω has a Lipschitz
boundary.
Specifically, we are going to consider the following problem and its connections
with (MK) and (ODE).
(PDE) Let f ∈ M(X) be a measure with f (X) = 0. Find µ ∈ M+ (Ω) and
u ∈ Lip1 (Ω) such that:
(i) there exist smooth functions uh uniformly converging to u on X, equal to 0 on
∂Ω and such that the functions ∇uh converge in [L2 (µ)]n to a function ∇µ u
satisfying
|∇µ u| = 1
µ-a.e. in Ω;
(37)
(ii) the following PDE is satisfied in the sense of distributions
−∇ · (∇µ u µ) = f
in Ω.
(38)
Given (µ, u) admissible for (PDE), we call µ the transport density (the reason
for that soon will be clear) and ∇µ u the tangential gradient of u.
Remark 5.1 (1) Choosing uh as test function in (38) gives
Z
Z
2
µ(Ω) =
|∇µ u| dµ = lim
h∇µ u, ∇uh i dµ
h→∞
Ω
=
lim f (uh ) = f (u),
h→∞
31
Ω
(39)
so that the total mass µ(Ω) depends only on f and u. We will prove in Theorem 5.1
that actually µ(Ω) depends only on f and that µ is concentrated on X.
(2) It is easy to prove that, given (µ, u), there is at most one tangential gradient
˜ µ u we have
∇µ u: indeed if ∇uh → ∇µ u and ∇ũh → ∇
Z
Z
˜
∇µ u · ∇µ u dµ = lim
∇µ u · ∇ũh dµ = lim f (ũh ) = f (u) = µ(Ω)
Ω
h→∞
h→∞
Ω
˜ µ u µ-a.e. On the other hand, Example 5.1 below shows that to
whence ∇µ u = ∇
some u there could correspond more than one measures µ solving (PDE). We will
obtain uniqueness in Section 7 under absolute continuity assumptions on f + or f − .
We will need the following lemma, in which the assumption that Ω is “large
enough” plays a role.
Lemma 5.1 If (µ, u) solves (PDE) then spt µ ⊂⊂ Ω.
Proof. Let Ω0 ⊂⊂ Ω00 ⊂⊂ Ω be such that |x − y| > r = diam(X) for any x ∈ X
and any y ∈ Rn \ Ω0 . We choose a minimum point x0 for the restriction of u to X
and define
w(x) := min [u(x) − u(x0 )]+ , dist(x, Rn \ Ω0 ) .
Then, it is easy to check that w(x) = u(x) − u(x0 ) on X and that w ≡ 0 on Rn \ Ω0 .
Since
µ(Ω) = f (u) = f (u − u(x0 )) = lim+ f (w ∗ ρε )
→0
Z
= lim+ ∇µ u · ∇w ∗ ρε dx ≤ µ(Ω00 )
→0
Ω
we conclude that µ(Ω \ Ω00 ) = 0. In the following theorem we build a solution of (PDE) choosing µ as a transport
density of (MK) with f0 = f + and f1 = f − . As a byproduct, by Corollary 4.1 we
obtain that the support of µ is contained in X and that µ is representable as the
right side in (30).
Theorem
5.1 ((ODE) versus (PDE)) Assume
that
(ft , Et )
with
R1
spt 0 |Et | dt ⊂⊂ Ω solves (ODE) and let u be a maximal Kantorovich potenR1
tial relative to f0 , f1 . Then, setting f = f0 − f1 , the pair (µ, u) with µ = 0 |Et | dt
solves (PDE) with ∇µ u = ∇u and, in particular, sptµ ⊂ X.
32
Conversely, if (µ, u) solves (PDE), setting f0 = f + and f1 = f − , the measures
(ft , Et ) defined by
ft := f0 + t(f1 − f0 ),
Et := −∇µ u µ
solve (ODE) and spt µ ⊂ X. In particular
µ(X) = min (ODE) = min (MK).
Proof. By Corollary 4.1 we can assume that µ is representable as in (30) for a
suitable optimal planning γ. Then, condition (i) in (PDE) with ∇µ u = ∇u follows
by (36). By (17) and condition (ii) of Theorem 4.4 we get
∇ · (∇µ u|Ft |) = ∇ · (∇u|Ft |) = f˙t
with Ft = πt# ((y − x)γ). Integrating in time and taking into account the definition
of µ, we obtain (38). Notice also that µ(X) = J(E).
If (ft , Et ) are defined as above, we get
f˙t + ∇ · Et = f1 − f0 − ∇ · (∇µ uµ) = f1 − f0 + f = 0.
Also in this case J(E) = µ(X). Example 5.1 (Non uniqueness of µ) In general, given f , the measure which
solves (PDE) for a given u ∈ Lip1 (X) is not unique. As an example one can consider
the situation in Example 1.5, where µ can be concentrated either on the horizontal
sides of the square or on the vertical sides of the square.
As shown in [22], an alternative construction can be obtained by solving the
problem “−∆∞ u = f ”, i.e. studying the following problems
(
−∇ · (|∇up |p−2 ∇up ) = f in Ω
(40)
up = 0
on ∂Ω
as p → ∞. In this case µ is the limit, up to subsequences, of |∇up |p−1 Ln . Under
suitable regularity assumptions, Evans and Gangbo prove that µ = aLn for some
a ∈ L1 (Ω) and use (a, u) to build an optimal transport ψ; their construction is
based on a careful regularization corresponding to the one used in Theorem 4.2 in
the special case Et = aLn and ft = f0 + t(f1 − f0 ). However, since the goal in
Theorem 4.2 is to build only an optimal planning, and not an optimal transport,
our proof is much simpler and works in a much greater generality (i.e. no special
assumption on f ).
Now we also prove that the limiting procedure of Evans and Gangbo leads to
solutions of (PDE), regardless of any assumption on the data f0 , f1 .
33
Theorem 5.2 Let up be solutions of (40) with p ≥ n + 1. Then
(i) the measures Hp = |∇up |p−2 ∇up Ln Ω are equi-bounded in Ω;
(ii) (up ) are equibounded and equicontinuous in Ω;
(iii) If (H, u) = limj (Hpj , upj ) with pj → ∞, then (µ, u) solves (PDE) with µ = |H|.
Proof. (i) Using up as test function and the Sobolev embedding theorem we get
Z
1/p0
Z
p
p0
|∇up0 | dx
|∇up | dx = f (up ) ≤ |f |(X)kup k∞ ≤ C
Ω
Ω
with p0 = n + 1. Using Hölder inequality we infer
Z
p−p0
p
|∇up |p dx ≤ C p−1 Ln (Ω) p0 (p−1) .
Ω
(ii) Follows by (i) and the Sobolev embedding theorem.
(iii) Clearly −∇ · H = f in Ω. By (i) and Hölder inequality we infer
Z
sup
|∇u|q dx < ∞
q≥n+1
Ω
whence |∇u| ≤ 1 Ln -a.e. in Ω.
We notice first that the functional
2
Z ν
φ d|ν|
ν 7→
−
w
|ν|
Ω
is lower semicontinuous with respect to the weak convergence of measures for any
nonnegative φ ∈ Cc (Ω), w ∈ [C(Ω)]n . The verification of this fact is straightforward:
it suffices to expand the squares. Second, we notice that
2
Z Hp
lim+ lim sup − ∇uε d|Hp | = 0
→0
p→∞
Ω |Hp |
whenever uε ∈ Lip1 (Rn ) are smooth functions uniformly converging to u in X and
equal to 0 on ∂Ω. Indeed, as |∇uε | ≤ 1, we have
2
Z Z
Hp
∇u
·
∇u
ε
p
p−1
− ∇uε d|Hp | ≤ 2 |∇up |
1−
dx
|∇up |
Ω |Hp |
Ω
Z
≤ 2 |∇up |p−2 (|∇up |2 − ∇uε · ∇up ) dx + ωp
Ω
= 2f (up ) − 2f (uε ) + ωp
34
where ωp = supt≥0 tp−1 − tp tends to 0 as p → ∞.
Taking into account these two remarks, setting p = pj and passing to the limit
as j → ∞ we obtain
2
Z H
− ∇uε φ d|H| = 0.
lim+ →0
|H|
Ω
for any nonnegative φ ∈ Cc (Ω). This implies that ∇uε converge in L2loc (|H|) to the
Radon–Nikodým derivative H/|H|. Since |∇uε | ≤ 1 we have also convergence in
[L2 (|H|)]n .
We now set µ = |H| and ∇µ u = lim ∇u , so that H = ∇µ uµ and (38) is satisfied.
6
Existence of optimal transport maps
In this section we prove the existence of optimal transport maps in the case when
c(x, y) = |x − y| and f0 is absolutely continuous with respect to Ln , following
essentially the original Sudakov approach and filling a gap in his original proof
(see the comments after Theorem 6.1). Using a maximal Kantorovich potential we
decompose almost all of X in transport rays and we build an optimal transport maps
by gluing the 1-dimensional transport maps obtained in each ray. The assumption
f0 << Ln is used to prove that the conditional measures f0C within any transport
ray C are non atomic (and even absolutely continuous with respect to H1 C), so
that Theorem 3.1 is applicable. Therefore the proof depends on the following two
results whose proof is based on the countable Lipschitz property of ∇u stated in
Theorem 4.3.
Theorem 6.1 Let B ∈ B(X) and let π : B → Sc (X) be a Borel map satisfying the
conditions
(i) π(x) ∩ π(x0 ) = ∅ whenever π(x) 6= π(x0 );
(ii) x ∈ π(x) for any x ∈ B;
(iii) the direction ν(x) of π(x) is a Sn−1 -valued countably Lipschitz map on B.
Then, for any measure λ ∈ M+ (X) absolutely continuous with respect to Ln B,
setting µ = π# λ ∈ M+ (Sc (X)), the measures λC of Theorem 9.1 are absolutely
continuous with respect to H1 C for µ-a.e. C ∈ Sc (X).
Proof. Being the property stated stable under countable disjoint unions we may
assume that
35
(a) there exists a unit vector ν such that ν(x) · ν ≥
1
2
for any x ∈ B;
(b) ν(x) is a Lipschitz map on B;
(c) B is contained in a strip
{x : a − b ≤ x · ν ≤ a}
with b > 0 sufficiently small (depending only on the Lipschitz constant of ν)
and π(x) intersects the hyperplane {x : x · ν = a}.
Assuming with no loss of generality ν = en and a = 0, we write x = (y, z) with
y ∈ Rn−1 and z < 0. Under assumption (a), the map T : π(B) → Rn−1 which
associates to any segment π(x) the vector y ∈ Rn−1 such that (y, 0) ∈ π(x) is well
defined. Moreover, by condition (i), T is one to one. Hence, setting f = T ◦ π :
B → Rn−1 ,
ν = T# µ = f# λ,
C(y) = T −1 (y) ⊃ f −1 (y)
and representing λ = ηy ⊗ ν with ηy = λC(y) ∈ M1 (f −1 (y)), we need only to prove
that ηy << H1 C(y) for ν-a.e. y.
To this aim we examine the Jacobian, in the y variables, of the map f (y, t).
Writing ν = (νy , νt ), we have
f (y, t) = y + τ (y, t)νy (y, t)
with
τ (y, t) = −
t
.
νt (y, t)
Since νt ≥ 1/2 and τ ≤ 2b on B we have
t
det (∇y f (y, t)) = det Id + τ ∇y νy + 2 ∇y νt ⊗ νy > 0
νt
if b is small enough, depending only on Lip(ν).
Therefore, the coarea factor
sX
det2 A
Cf :=
A
(where the sum runs on all (n − 1) × (n − 1) minors A of ∇f ) of f is strictly positive
on B and, writing λ = gLn with g = 0 out of B, Federer’s coarea formula (see for
instance [4], [38]) gives
λ=
g
g
1
−1
n−1
Cf Ln =
= ηy0 ⊗ ν 0
H f (y) ⊗ L
Cf
Cf
36
with L = {y ∈ Rn−1 : H1 (f −1 (y)) > 0} and
ηy0
g
Cf
H1 f −1 (y)
:= R
,
g/Cf d H1
f −1 (y)
0
Z
ν =
f −1 (y)
g
1
d H Ln−1 L.
Cf
By Theorem 9.2 we obtain ν = ν 0 and ηy = ηy0 for ν-a.e. y, and this concludes the
proof. Remark 6.1 (1) In [42] V.N.Sudakov stated the theorem above (see Proposition 78
therein) for maps π : B → So (X) without the countable Lipschitz assumption (iii)
and also in a greater generality, i.e. for a generic “Borel decomposition” of the
space in open affine regions, even of different dimensions. However, it turns out that
the assumption (iii) is essential even if we restrict to 1-dimensional decompositions.
Indeed, G.Alberti, B.Kirchheim and D.Preiss [3] have recently found an example of a
compact family of open and pairwise disjoint segments in R3 such that the collection
B of the midpoints of the segments has strictly positive Lebesgue measure. In this
case, of course, if λ = Ln B the conditional measures λC are unit Dirac masses
concentrated at the midpoint of C, so that the conclusion of Theorem 6.1 fails.
If n = 2 it is not hard to prove that actually condition (iii) follows by (ii), so
that the counterexample mentioned above is optimal.
(2) Since any open segment can be approximated from inside by closed segments,
by a simple approximation argument one can prove that Theorem 6.1 still holds as
well for maps π : B → So (X) or for maps with values into half open [[x, y[[ segments.
Also the assumption (i) that the segments do not intersect can be relaxed. For
instance we can assume that the segments can intersect only at their extreme points,
provided the collection of these extreme points is Lebesgue negligible. This is precisely the content of the next corollary; again, a counterexample in [28] shows that
this property need not be true for a generic compact family of disjoint line segments
in Rn , n ≥ 3. Corollary 6.1 (Negligible extreme points) Let u ∈ Lip1 (X). Then the collection of the extreme points of the transport rays is Lebesgue negligible.
Proof. We prove that the collection L of all left extreme points is negligible, the
proof for the right ones being similar. Let B = L \ Σu and set
π(x) := [[x, x −
r(x)
∇u(x)]]
2
where r(x) is the length of the transport ray emanating from x (this ray is unique
due to the differentiability of u at x). By Theorem 4.3 the map π has the countable
37
Lipschitz property on B and, by construction, π(x) ∩ π(x0 ) = ∅ whenever x 6= x0 .
By Theorem 6.1 we obtain
Z
λ=
λC dν
Sc (X)
with λ = Ln B, ν = π# λ and λC << H1 C probability measures concentrated
on π −1 (C) for ν-a.e. C. Since π −1 (C) contains only one point for any C ∈ π(B) we
obtain λC = 0 for ν-a.e. C, whence λ(B) = ν(Sc (X)) = 0. Theorem 6.2 (Sudakov) Let f0 , f1 ∈ M1 (X) and assume that f0 << Ln . Then
there exists an optimal transport ψ mapping f0 to f1 . Moreover, if f1 << Ln we can
−1
choose ψ so that ψ −1 is well defined f1 -a.e. and ψ#
f1 = f0 .
Proof. Let γ be an optimal planning. In the first two steps we assume that
for f0 -a.e. x there exists y 6= x such that (x, y) ∈ sptγ.
(41)
This condition holds for instance if f0 ∧ f1 = 0 because in this case any optimal
planning γ does not charge the diagonal ∆ of X × X, i.e. γ(∆) = 0 (otherwise
h = π0# (χ∆ γ) = π1# (χ∆ γ) would be a nonzero measure less than f0 and f1 ).
Let u ∈ Lip1 (X) be a maximal Kantorovich potential given by Corollary 2.1, i.e.
a function satisfying
u(x) − u(y) = |x − y|
γ-a.e. in X × X
(42)
for any optimal planning γ and let Tu be the transport set relative to u (see Definition 4.1). In the following we set
e = {x ∈ X \ Σu : ∃y 6= x s.t. (x, y) ∈ sptγ} .
X
By (41) and the absolute continuity assumption we know that f0 is concentrated
e Notice also that for any x ∈ X
e there exists a unique closed transport ray
on X.
e is a differentiability point of
containing x: this follows by the fact that any x ∈ X
u and by Proposition 4.2.
Step 1. We define r : X × X → Sc (X) as the map which associates to any pair
(x, y) the closed transport ray containing [[x, y]]. By (42) the map r is well defined
e r is also well
γ-a.e. out of the diagonal ∆; moreover, being f0 concentrated on X,
defined γ-a.e. on ∆. Hence, according to Theorem 9.1 we can represent
γ = γC ⊗ ν
with
ν := r# γ
and (42) gives
u(x) − u(y) = |x − y|
38
γC -a.e. in X × X
(43)
for ν-a.e. C ∈ Sc (X). By the sufficiency part in Corollary 2.1 we infer that γC is an
optimal planning relative to the probability measures
f0C := π0# γC ,
f1C := π1# γC
for ν-a.e. C ∈ Sc (X). By (56) we infer
Z
πt# γC (B) dν(C)
πt# γ(B) =
∀t ∈ [0, 1], B ∈ B(X).
(44)
Sc (X)
Notice also that
Z
F1 (f0 , f1 ) =
F1 (f0C , f1C ) dν(C).
(45)
Sc (X)
Indeed,
Z
Z
Z
Z
|x − y| dγ =
I(γ) =
|x − y| dγC dν(C) =
Sc (X)
X×X
I(γC ) dν(C)
Sc (X)
X×X
and we know by (43) that γC is optimal for ν-a.e. C ∈ Sc (X).
e → Sc (X) the natural map, so that π(x) is the closed
Step 2. We denote by π : X
e × X we obtain
transport ray containing x. Since r = π ◦ π0 on X
ν = r# γ = π# (π0# γ) = π# f0 .
Moreover (44) gives
Z
f0 (B) =
f0C (B) dν(C)
∀B ∈ B(X).
(46)
Sc (X)
e can intersect only at their right extreme
Notice that the segments π(x), x ∈ X,
point and that, by Corollary 6.1, the collection of these extreme points is Lebesgue
negligible. As a consequence of (46) with ν = π# f0 and Remark 6.1(2), the measures
f0C are absolutely continuous with respect to H1 C for µ-a.e. C ∈ Sc (X). Hence,
by Theorem 3.1, for µ-a.e. C ∈ Sc (X) we can find a nondecreasing map ψC : C → C
(this notion makes sense, since C is oriented) such that ψC# f0C = f1C . Notice also
that ψ −1 is well defined f1C -a.e., if also f1C << H1 C.
e are pairwise disjoint
Taking into account that the closed transport rays in π(X)
e we can glue all the maps ψC to produce a single Borel map ψ : X
e → X. The
in X,
map ψ is Borel because we have been able to exhibit the one dimensional transport
map constructively and because of the Borel property of the maps C 7→ fiC (see
(15) and (54)); the simple but boring details are left to the reader.
39
Since ψ# f0C = ψC# f0C = f1C for µ-a.e. C ∈ Sc (X), taking (44) into account we
infer
Z
Z
ψ# f0C dν(C) =
f1C dν(C) = f1 .
ψ# f0 =
Sc (X)
Sc (X)
Finally ψ is an optimal transport because (45) holds and any ψC is an optimal
transport.
Step 3. In this step we show how the assumption (41) can be removed. We define
X 0 := {x ∈ X : (x, x) ∈ sptγ and (x, y) ∈
/ sptγ ∀y 6= x}
and L = {(x, x) : x ∈ X 0 }. Then, we set f00 = f0 X 0 and f000 = f0 X 00 , with
X 00 = X \ X 0 , and
f100 := f1 − f10 .
f10 := π1# (γ L),
Since L ⊂ ∆, we have f10 = π0# (γ L) = f00 , hence we can choose on X 0 the map
ψ1 = Id as transport map to obtain (ψ1 )# f00 = f10 . Since f000 is concentrated on X 00
the condition (41) is satisfied with f000 in place of f0 and we can find an optimal
transport map ψ2 : X 00 → X such that (ψ2 )# f000 = f100 . Gluing these two transport
maps we obtain a transport map ψ such that ψ# f0 = f1 ; since, by construction,
u(x) − u(ψ(x)) = |x − ψ(x)|
f0 -a.e. in X
we infer that ψ is optimal. This proof strongly depends on the strict convexity of the euclidean distance,
which provides the first and second order differentiability properties of the potential
u on the transport set Tu . Notice also that if c(x, y) = kx − yk and the norm k · k
is not strictly convex, then the “transport rays” need not be one-dimensional and,
to our knowledge, the existence of an optimal transport map is an open problem
in this situation. Indeed, this existence result is stated by Sudakov in [42] but his
proof is faulty, for the reasons outlined in Remark 6.1(1).
Under special assumptions on the data f0 , f1 (absolute continuity, separated
supports, Lipschitz densities) Evans and Gangbo provided in [22] a proof based on
differential methods of the existence of optimal transport maps. For stricly convex
norms, the first fully rigorous proofs of the existence of an optimal transport map
under the only assumption that f0 << Ln have been given in [15] and [43]. As in
Theorem 6.2 the proof is strongly based on the differentiability of the directions of
transport rays.
40
7
Regularity and uniqueness of the transport
density
In this section we investigate the regularity and the uniqueness properties of the
transport density µ arising in (PDE). Recall
that, as Theorem 5.1 shows, any such
R1
measure µ can also be represented as 0 |Et | dt for a suitable optimal pair (ft , Et )
for (ODE) (even
R 1 with Et independent of t). In turn, by Corollary 4.1, any optimal
measure µ = 0 |Et | dt can be represented as
Z
1
πt# (|y − x|γ) dt
µ=
(47)
0
for a suitable optimal planning γ. For this reason, in the following we restrict our
attention to the representation (47), valid for any optimal measure µ for (PDE).
A more manageable formula for µ, first considered by G.Bouchitté and
G.Buttazzo in [14], is given in the following elementary lemma.
Lemma 7.1 Let µ be as in (47). Then
Z
Length (]]x, y[[∩B) dγ(x, y)
µ(B) =
∀B ∈ B(X).
(48)
X×X
Proof. For any Borel set B we have
Z 1Z
Z
Z 1
µ(B) =
|y − x| dγ(x, y) dt =
|y − x|
χπt−1 (B) dt dγ(x, y)
0
πt−1 (B)
X×X
0
Z
=
Length (]]x, y[[∩B) dγ(x, y).
X×X
A direct consequence of (48) is that µ is concentrated on the transport set (since
γ-a.e. segment ]]x, y[[ is contained in a transport ray); another consequence is the
density estimate
µ(Br (x))
≤ f0 (X)f1 (X)
(49)
2r
because Length(]]x, y[[∩Br (x)) ≤ 2r for any ball Br (x) and any segment ]]x, y[[. The
first results on µ that we state, proved by A.Pratelli in [34] (see also [19]), relate
the dimensions of f0 and f1 to the dimension of µ and show that necessarily µ is
absolutely continuous with respect to the Lebesgue measure if f0 (or, by symmetry,
f1 ) has this property.
41
Theorem 7.1 Assume that for some k > 0 we have
f0 (Br (x))
<∞
rk
r∈(0,1)
sup
f0 -a.e. in X.
Then µ has the same property. In particular µ << Ln if f0 << Ln .
The proof of Theorem 7.1 is based on (48); the density estimate on µ is achieved
by a careful analysis of the transport rays crossing a generic ball Br (x). A similar
analysis proved the following summability estimate (see [19]).
Theorem 7.2 Assume that f0 = g0 Ln and f1 = g1 Ln with g0 , g1 ∈ Lp (X), p > 1.
Then µ = hLn with h ∈ L∞ (X) if p = ∞, h ∈ Lq (X) for any q < p if p < ∞.
It is not known whether g0 , g1 ∈ Lp implies h ∈ Lp for p < ∞.
Definition 7.1 (Hausdorff dimension of a measure) Let µ ∈ M+ (X). The
Hausdorff dimension H-dim(µ) is the supremum of all k ≥ 0 such that µ << Hk .
In other words µ(B) = 0 whenever Hk (B) = 0 for some k < H-dim(µ) and for
any k > H-dim(µ) there exists a Borel set B with µ(B) > 0 and Hk (B) = 0. Notice
that if µ is made of pieces of different dimensions, then H-dim(µ) is the smallest of
these dimensions.
Using the density estimates and the implications (see for instance Theorem 2.56
of [4] or [38]; here t > 0 and k > 0)
lim sup
r→0+
lim sup
r→0+
µ(Br (x))
≥ t ∀x ∈ B
ωk rk
µ(Br (x))
≤ t ∀x ∈ B
ωk r k
=⇒
=⇒
µ(B) ≥ t Hk (B)
(50)
µ(B) ≤ 2k t Hk (B)
(51)
we can prove a natural lower bound on the Hausdorff dimension of µ.
Corollary 7.1 Let µ be a transport density. Then
H-dim(µ) ≥ max {1, H-dim(f0 ), H-dim(f1 )} .
Proof. By (49) we infer that µ has finite (and even bounded) 1-dimensional density
at any point. In particular (51) gives µ(B) = 0 whenever H1 (B) = 0, so that
H-dim(µ) ≥ 1.
Let k = H-dim(f0 ) and k 0 < k; then f0 has finite k 0 -dimensional density f0 -a.e.,
0
otherwise by (50) the set where the density is not finite would be Hk -negligible
and with strictly positive f0 -measure. By Theorem 7.1 we infer that µ has finite
42
k 0 -dimensional density µ-a.e. By the same argument used before with k 0 = 1 we
obtain that H-dim(µ) ≥ k 0 and therefore, since k 0 < k is arbitrary, H-dim(µ) ≥ k.
A symmetric argument proves that H-dim(µ) ≥ H-dim(f1 ). We conclude this section proving the uniqueness of the transport density, under
the assumption that either f0 << Ln or f1 << Ln (see Example 5.1 for a nonuniqueness example if neither f0 nor f1 are absolutely continuous). Similar results have
been first announced by Feldman and McCann (see [25]).
We first deal with the one dimensional case, where this absolute continuity assumption is not needed.
Lemma 7.2 Let µ be a transport density in X ⊂ R and assume that the interior
(a, b) of X is a transport ray. Then µ = hL1 in (a, b) with
h(t) = σf ((a, t)) + c L1 -a.e. in (a, b)
for some constant c, where f = f1 − f0 and σ = 1 if u0 = 1 in (a, b), σ = −1 if
u0 = −1 in (a, b). Moreover
R
F1 (f0 , f1 ) − σ (a,b) (b − t) df (t)
.
c=
b−a
Proof. The proof of the first statement is based on the equation µ0 = σf and on a
smoothing argument. By integrating both sides we get
Z b
Z bZ
c(b − a) = µ(X) − σ
f ((a, t)) dt = µ(X) − σ
1 df (τ ) dt
a
Z
a
Z
= µ(X) − σ
b
(a,t)
Z
1 dt df (τ ) = µ(X) − σ
(a,b)
τ
(b − τ ) df (τ ).
(a,b)
The conclusion is achieved taking into account that µ(X) = min (MK) = F1 (f0 , f1 ).
Theorem 7.3 (Uniqueness) Assume that either f0 or f1 are absolutely continuous with respect to Ln . Then the transport density is absolutely continuous and
unique.
Proof. We already know from Theorem 7.1 that any transport density is absolutely
continuous and we can assume that f0 << Ln . By Corollary 4.1 we know that
the class of transport densities relative to (f0 , f1 ) is equal to the class of transport
densities relative to (f0 − h, f1 − h) where h is any measure in M+ (X) such that
43
h ≤ f0 ∧ f1 (indeed, this subtraction does not change the velocity field Et ). Hence,
it is not restrictive to assume that f0 ∧ f1 = 0.
We can assume that µ is representable as in (30) for a suitable optimal planning
γ. Adopting the same notation of the proof of Theorem 6.1, we have
γ = γC ⊗ ν
where ν = π# f0 does not depend on γ (and here the absolute continuity assumption
on f0 plays a crucial role) and γC is an optimal planning relative to the measures
f0C = π0# γC , f1C = π1# γC for ν-a.e. C. In particular
Z
1
Z
πt# (|y − x|γC ⊗ ν) dt =
µ=
0
1
πt# (|y − x|γC ) dt ⊗ ν.
0
Hence, it suffices to show that the measures
Z 1
πt# (|y − x|γC ) dt
µC :=
0
do not depend on γ (up to ν-negligible sets of course). To this aim, taking into
account Lemma 7.2, it suffices to show that f0C and f1C do not depend on γ.
Indeed, we already know from (46) and Theorem 9.2 that f0C do not depend on
γ.
The argument for f1C is more involved. First, since γ(∆) = 0 (due to the
assumption f0 ∧ f1 = 0), we have |x − y| > 0 γ-a.e., and therefore |x − y| > 0
γC -a.e. for ν-a.e. C ∈ Sc (X). As a consequence, setting C = [[xC , yC ]], we obtain
0
that f1C (xC ) = 0 for ν-a.e. C. Second, we examine the restriction f1C
of f1C to the
relative interior of C noticing that (44) gives
Z
Z
0
f1C Tu dν(C) =
f1C
dν(C)
f1 T u =
Sc (X)
Sc (X)
because Tu ∩ C is the relative interior of C for any C ∈ π(X̃). By Theorem 9.2 we
0
obtain that f1C
depend only on f1 , Tu and ν.
In conclusion, since
0
0
f1C = f1C
+ (1 − f1C
(X)) δyC
we obtain that f1C do not depend on γ as well. 44
8
The Bouchitté–Buttazzo mass optimization
problem
In [13, 14] Bouchitté and Buttazzo consider the following problem. Given f ∈ M(X)
with f (X) = 0, they define
Z
1
2
∞
|∇v| dµ − f (v) : v ∈ C (X)
E(µ) := inf
X 2
for any µ ∈ M+ (X). Then, they raised the following mass optimization problem.
(BB) Given m > 0, maximize E(µ) among all measures µ ∈ M+ (X) with µ(X) =
m.
A possible physical interpretation of this problem is the following: we may imagine that µ represents the conductivity of some material, thinking that the conductivity (i.e. the inverse resistivity) is zero out of sptµ; accordingly we may imagine
that f = f + − f − is a balanced density of positive and negative charges. Then
−E(µ) represents the heating corresponding to the given conductivity, so that there
is an obvious interest in maximizing E(µ) and, for a minimizer u, the (formal) first
variation of the energy
−∇ · (∇u µ) = f + − f −
corresponds to Ohm’s law.
More generally, problems of this sort appear in Shape Optimization and Linear
Elasticity. In these cases u is no longer real valued and general energies of the form
Z
F(u) =
f (x, ∇u(x)) dµ(x)
(52)
X
must be considered in order to take into account light structures, corresponding to
mass distributions not absolutely continuous with respect to Ln . This was the main
motivation for Bouchitté, Buttazzo and Seppecher [12] in their development of a
general differential calculus with measures. This calculus, based on a suitable concept of tangent space to a measure, enables the study of the relaxation of functionals
(52) and provides an explicit formula for their lower semicontinuous envelope.
Coming back to the scalar problem (BB), a remarkable fact discovered in [13, 14]
is its connection with the Monge–Kantorovich optimal transport problem, in the
case when c(x, y) = |x − y|. It turns out that the solutions of (BB) are in one to
one correspondence with constant multiples of transport densities for (PDE) (or,
equivalently, for (MK) with f0 = f + and f1 = f − ).
45
As a byproduct we have that µ is unique and absolutely continuous with respect
to Ln if either f0 or f1 are absolutely continuous with respect to Ln .
Theorem 8.1 ((PDE) versus (BB)) Let (µ, u) be a solution of (PDE) and set
m̃ = µ(X). Then µ̃ = m
µ solves (BB) and any solution of (BB) is representable in
m̃
this way. Moreover
1 m̃2
.
max(BB) = −
2m
Proof. Setting v = λuh (with uh as in the definition of (PDE)), for any ν ∈ M+ (X)
with ν(X) = m we estimate
E(ν) ≤
m 2
λ − λf (uh )
2
so that, letting h → ∞ and taking into account (39), we obtain
E(ν) ≤
m 2
λ − m̃λ.
2
By minimizing with respect to λ we obtain E(ν) ≤ −m̃2 /(2m).
On the other hand, using Young inequality and choosing λ so that λm̃ = m we
can estimate
Z
Z
1
m
2
h∇v, λ∇µ ui dµ̃ − λ2 − f (v)
|∇v| dµ̃ − f (v) ≥
2
X
X 2
1 m̃2
= −∇ · (∇µ u µ)(v) − f (v) −
2m
1 m̃2
= −
2m
for any v ∈ C ∞ (X). This proves that µ̃ is optimal for (BB).
Conversely, it has been proved in [13, 14] that for any solution σ of (BB) there
exists v ∈ Lip(X) such that |∇σ v| = m/m̃ σ-a.e. and
−∇ · (∇σ vσ) = f,
where ∇σ v is understood in the Bouchitté–Buttazzo sense. But since (see [14] again)
Z
Z
2
2
∞
|∇σ v| dσ = inf lim inf
|∇vh | dσ : vh → v uniformly, vh ∈ C (X)
X
h→∞
X
we obtain a sequence of smooth functions vh uniformly converging to v such that
m̃
∇vh → ∇σ v in [L2 (σ)]n . Setting u = m
v and µ = m
σ, this proves that (σ, u) solves
m̃
(PDE). 46
9
Appendix: some measure theoretic results
In this section we list all the measure theoretic results used in the previous section;
reference books for the content of this section are [37], [18] and [4].
Let us begin with some terminology.
• (Measures) Let X be a locally compact and separable metric space. We denote
by [M(X)]m the space of Radon measures with values in Rm and with finite total
variation in X. We recall that the total variation measure of µ = (µ1 , . . . , µm ) ∈
[M(X)]m is defined by
(∞
)
∞
X
G
|µ|(B) := sup
|µ(Bi )| : B =
Bi , Bi ∈ B(X)
i=1
i=1
and belongs to M+ (X). By Riesz theorem the space [M(X)]m endowed with the
norm kµk = |µ|(X) is isometric to the dual of [C(X)]m . The duality is given by the
integral, i.e.
m Z
X
hµ, ui :=
ui dµi .
i=1
X
Recall also that, for µ ∈ M+ (X) and f ∈ [L1 (X, µ)]m , the measure f µ ∈ [M(X)]m
is defined by
Z
f µ(B) :=
∀B ∈ B(X)
f dµ
B
and |f µ| = |f |µ.
• (Push forward of measures) Let µ ∈ [M(X)]m , let Y be another metric space
and let f : X → Y be a Borel map. Then the push forward measure f# µ ∈ [M(Y )]m
is defined by
f# µ(B) := µ f −1 (B)
∀B ∈ B(Y )
and satisfies the more general property
Z
Z
u df# µ =
u ◦ f dµ
for any bounded Borel function u : Y → R.
Y
X
It is easy to check that |f# µ| ≤ f# |µ|.
• (Support) We say that µ ∈ [M(X)]m is concentrated on a Borel set B if |µ|(X \
B) = 0 and we denote by sptµ the smallest closed set on which µ is concentrated
(the existence of a smallest set follows by the separability of X, precisely by the
Lindelöf property). The support is also given by the formula (sometimes taken as
the definition)
sptµ = {x ∈ X : |µ|(B% (x)) > 0 ∀% > 0} .
47
• (Convolution) If X ⊂ Rn , µ ∈ [M(X)]m and ρ ∈ Cc∞ (Rn ) (for instance a
convolution kernel); we define
Z
µ ∗ ρ(x) :=
ρ(x − y) dµ(y)
x ∈ Rn .
X
Notice that µ ∗ ρ ∈ [C ∞ (Rn )]m because Dα (µ ∗ ρ) = (Dα ρ) ∗ µ for any multiindex α
and
sup |Dα (µ ∗ ρ)| ≤ sup |Dα ρ|kµk.
Moreover, using Jensen’s inequality it is easy to check (see for instance Theorem 2.2(ii) of [4]) that
Z
|µ ∗ ρ| dx ≤ |µ|(Cρ ) where Cρ := {x : dist(x, C) ≤ diam(sptρ)}
(53)
C
for any closed set C ⊂ Rn .
• (Weak convergence) Assume that X is a compact metric space. We say that a
family of measures (µh ) ⊂ M(X) weakly converges to µ ∈ M(X) if
Z
Z
lim
u dµh =
u dµ
∀u ∈ C(X)
h→∞
X
X
i.e., if µh weakly∗ converge to µ as elements of the dual of C(X). In the case X
is locally compact and separable, the concept is analogous, simply replacing C(X)
the the subspace Cc (X) of functions with compact support. It is easy to check that
µ ∗ ρε Ln weakly converge to µ in Rn whenever ρ is a convolution kernel.
We mention also the following criterion for the weak convergence of positive
measures (see for instance Proposition 1.80 of [4]): if µh , µ ∈ M+ (X) satisfy
lim inf µh (A) ≥ µ(A) ∀A ⊂ X open
h→∞
and
lim sup µh (X) ≤ µ(X)
h→∞
R
R
then X φ dµh → X φ dµ for any bounded continuous function φ : X → R.
• (Measure valued maps) Let X, Y be locally compact and separable metric
spaces. Let y 7→ λy be a map which assigns to any y ∈ Y a measure λy ∈ [M(X)]m .
We say that λy is a Borel map if y 7→ λy (A) is a real valued Borel map for any open
set A ⊂ X. By a monotone class argument it can be proved that y 7→ λy is a Borel
map if and only if
y 7→ λy ({x : (y, x) ∈ B})
is a Borel map for any B ∈ B(Y × X).
48
(54)
Moreover y 7→ |λy | is a Borel map whenever y 7→ λy is Borel (detailed proofs are
in §2.5 of [4]). If µ ∈ M+ (Y ), analogous statements hold for B(Y )µ -measurable
measure valued maps, where B(Y )µ is the σ-algebra of µ-measurable sets (in this
case one has to replace B(Y × X) by B(Y )µ ⊗ B(X) in (54)).
• (Decomposition of measures) The following result plays a fundamental role in
these notes; it is also known as disintegration theorem.
Theorem 9.1 (Decomposition of measures) Let X, Y be locally compact and
separable metric spaces and let π : X → Y be a Borel map. Let λ ∈ [M(X)]m and
set µ = π# |λ| ∈ M+ (Y ). Then there exist measures λy ∈ [M(X)]m such that
(i) y 7→ λy is a Borel map and |λy | is a probability measure in X for µ-a.e. y ∈ Y ;
(ii) λ = λy ⊗ µ, i.e.
Z
λ(A) =
λy (A) dµ(y)
∀A ∈ B(X);
(55)
Y
(iii) |λy | (X \ π −1 (y)) = 0 for µ-a.e. y ∈ Y .
The representation provided by Theorem 9.1 of λ can be used sometimes to
compute the push forward of λ. Indeed,
f# (λy ⊗ µ) = f# λy ⊗ µ
(56)
for any Borel map f : X → Z, where Z is any other compact metric space. Notice
also that
|λ| = |λy | ⊗ µ.
(57)
Indeed, the inequality ≤ is trivial and the opposite one follows by evaluating both
measures at B = X, using the fact that |λy |(X) = 1 for µ-a.e. y.
In the case when m = 1 and λ ∈ M+ (X) the proof of Theorem 9.1 is available
in many textbooks of measure theory or probability (in this case λy are the the socalled conditional probabilities induced by the random variable π, see for instance
[18]); in the vector valued case one can argue component by component, but the
fact that |λy | are probability measures is not straightforward.
In Theorem 2.28 of [4] the decomposition theorem is proved in the case when
X = Y × Z is a product space and π(y, z) = y is the projection on the first variable;
in this situation, since λy are concentrated on π −1 (y) = {y} × Z, it is sometimes
convenient to consider them as measures on Z, rather than measures on X, writing
(55) in the form
Z
λ(B) =
λy ({z : (y, z) ∈ B}) dµ(y)
∀B ∈ B(X).
(58)
Y
49
Once the decomposition theorem is known in the special case X = Y × Z and
π(y, z) = z the general case can be easily recovered: it suffices to embed X into the
product Y × X through the map f (x) = (π(x), x) and to apply the decomposition
theorem to λ̃ = f# λ.
Now we discuss the uniqueness of λx and µ in the representation λ = λx ⊗ µ.
For simplicity we discuss only the case of positive measures.
Theorem 9.2 Let X, Y and π be as in Theorem 9.1; let λ ∈ M+ (X), µ ∈ M+ (Y )
and let y 7→ ηy be a Borel M+ (X)-valued map defined on Y such that
R
(i) λ = ηy ⊗ µ, i.e. λ(A) = Y ηy (A) dµ(y) for any A ∈ B(X);
(ii) ηy (X \ π −1 (y)) = 0 for µ-a.e. y ∈ Y .
Then ηy are uniquely determined µ-a.e. in Y by (i), (ii) and, setting B =
{y : ηy (X) > 0}, the measure µ B is absolutely continuous with respect to π# λ.
In particular
µ B
ηy = λy
for π# λ-a.e. y ∈ Y
(59)
π# λ
where λy are as in Theorem 9.1.
Proof. Let ηy , ηy0 be satisfying (i), (ii). We have to show that ηy = ηy0 for µ-a.e. y.
Let (An ) be a sequence of open sets stable by finite intersection which generates the
Borel σ-algebra of X. Choosing A = An ∩ π −1 (B), with B ∈ B(Y ), in (i) gives
Z
Z
ηy (An ) dµ(y) =
ηy0 (An ) dµ(y).
B
B
Being B arbitrary, we infer that ηy (An ) = ηy0 (An ) for µ-a.e. y, and therefore there
exists a µ-negligible set N such that ηy (An ) = ηy0 (An ) for any n ∈ N and any
y ∈ Y \ N . By Proposition 1.8 of [4] we obtain that ηy = ηy0 for any y ∈ Y \ N .
Let B 0 ⊂ B be any π# λ-negligible set; then π −1 (B 0 ) is λ-negligible and therefore
(ii) gives
Z
Z
−1
0
0=
ηy π (B ) dµ(y) =
ηy (X) dµ(y).
B0
Y
0
0
As ηy (X) > 0 on B ⊃ B this implies that µ(B ) = 0. Writing µ B = hπ# λ we
obtain λ = hηy ⊗ π# λ and λ = λy ⊗ π# λ. As a consequence (59) holds. • (Young measures) These measures, introduced by L.C.Young, arise in a natural
way in the study of oscillatory phenomena and in the analysis of weak limits of
nonlinear quantities (see [31] for a comprehensive introduction to this wide topic).
50
Specifically, assume that we are given compact metric spaces X, Y and a sequence of Borel maps ψh : X → Y ; in order to understand the limit behaviour of
ψh we associate to them the measures
Z
γψh = (Id × ψh )# µ = δψh (x) dµ(x)
and we study their limit in M(X ×Y ). Assuming, possibly passing to a subsequence,
that γψh → γ, due to the fact that π0# (γψh ) = µ for any h we obtain that π0# γ = µ,
hence according to Theorem 9.1 we can represent γ as
γ = γx ⊗ µ
for suitable probability measures γx in Y , with x 7→ γx Borel. The family of measures
γx is called Young limit of the sequence (ψh ); once the Young limit is known, we can
compute the w∗ -limit of γ(ψh ) in L∞ (X, µ) for any γ ∈ C(Y ): indeed, using test
function of the form φ(x)γ(y), we easily obtain that the limit (in the dual of C(X)
and therefore in L∞ (X, µ)) is given by
Z
Lγ (x) :=
γ(y) dγx (y).
Y
We will use the following two well known results of the theory of Young measures.
Theorem 9.3 (Approximation theorem) Let γ ∈ M+ (X × Y ) and write γ =
γx ⊗ µ with µ = π0# γ. Then, if µ has no atom, there exists a sequence of Borel
maps ψh : X → Y such that
γx ⊗ µ = lim δψh (x) ⊗ µ.
h→∞
Moreover, we can choose ψh in such a way that the measures ψh# µ have no atom
as well.
Proof. (Sketch) We assume first that γx = γ is independent of x. By approximation
we can also assume that
p
X
γ=
p i δy i
i=1
for suitable yi ∈ Y and pi ∈ [0, 1]. Let Qh , h ∈ Zn , be a partition of Rn in cubes with
side length 1/h; since µ has no atom, by Lyapunov theorem we can find a partition
Xh1 , . . . , Xhp of X ∩ Qh such that µ(Xhi ) = pi µ(X ∩ Qh ). Then we define
ψ h = pi
51
on Xhi .
For any φ ∈ C(X) and ϕ ∈ C(Y ) we have
p
XX
Z
φϕ d(δψh ⊗ µ) =
X×Y
Z
ϕ(yi )
XX
∼
φ dµ
Xhi
h∈Zn i=1
p
Z
Z
φϕ dµ × γ
φ dµ =
pi ϕ(yi )
X∩Qh
h∈Zn i=1
X×Y
and this proves that
µ × γ = lim δψh ⊗ µ.
h→∞
If we want ψh# µ to be non atomic, the above construction needs to be modified only
slightly: it suffices to take small balls Bi centered at yi and to define ψh equal to φi
on Xhi , where φi : Rn → Bi is any Borel and one to one map.
If γx is piecewise constant (say in a canonical subdivision of X induced by a
partition in cubes) then we can repeat the local construction above in each region
where γx is constant. Moreover we can approximate Lipschitz functions γx (with
respect to the 1-Wasserstein distance) with piecewise constant ones. Finally, any
Borel map γx can be approximated by Lipschitz ones through a convolution. We will also need the following result.
Lemma 9.1 Let ψh , ψ : X → Y be Borel maps and µ ∈ M1 (X). Then ψh → ψ
µ-a.e. if and only if
γh := δψh (x) ⊗ µ(x) → γ := δψ(x) ⊗ µ(x)
weakly in M(X × Y ).
Proof. Assume that ψh → ψ µ-a.e. Since
Z
Z
ϕ dγϕ =
ϕ(x, φ(x)) dµ(x)
X×Y
∀ϕ ∈ C(X × Y )
X
the dominated convergence theorem gives that γh → γ. To prove the opposite
implication, fix ε > 0 and a compact set K such that ψ|K is continuous and µ(X \
K) < ε. We use as test function
ϕ(x, y) = χK (x)γ (y − ψ(x))
with γ(t) = 1 ∧ |t|/ε (by approximation, although not continuous in X × Y , this is
an admissible test function) to obtain
lim µ ({x ∈ K : |ψh (x) − ψ(x)| ≥ }) = 0.
h→∞
52
The conclusion follows letting ε → 0+ . With a similar proof one can obtain a slightly more general result, namely
γh := δψh (x) ⊗ µh → γ := δψ(x) ⊗ µ.
implies ψh → ψ µ-a.e. provided |µh − µ|(X) → 0.
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