Weak optimal entropy transport problems and applications doctor of philosophy major mathematics

Ph D Dissertation Weak Optimal Entropy Transport Problems and Applications THANH SON TRINH The Graduate School Sungkyunkwan University Department of Mathematics Ph D Dissertation Weak Optimal Entropy[.]

Ph.D Dissertation

Weak Optimal Entropy Transport Problems and Applications

THANH SON TRINH

The Graduate School Sungkyunkwan University Department of Mathematics

Ph.D Dissertation

Weak Optimal Entropy Transport Problems and Applications

THANH SON TRINH

The Graduate School Sungkyunkwan University Department of Mathematics

Weak Optimal Entropy Transport Problems and Applications

THANH SON TRINH

A Ph.D Dissertation Submitted to the Department of Mathematics

and the Graduate School of Sungkyunkwan University

in partial fulfillment of the requirements for the degree of Ph.D in Mathematics

October 2021

Supervised by Nhan Phu Chung Major Advisor

This certifies that the dissertation

of THANH SON TRINH is approved.

3 Weak Optimal Entropy Transport Problems and Martingale Optimal

3.1 Weak Optimal Entropy Transport Problems 193.2 Martingale Optimal Entropy Transport Problems 393.3 Examples 48

4 Unbalanced Optimal Total Variation Transport Problems and

Weak Optimal Entropy Transport

Problems and Applications

In the thesis, we introduce weak optimal entropy transport problems that cover bothoptimal entropy transport problems and weak optimal transport problems introduced byLiero, Mielke, and Savar´e [34]; and Gozlan, Roberto, Samson and Tetali [25], respectively.Under some mild assumptions of entropy functionals, we establish a Kantorovich typeduality for our weak optimal entropy transport problem We also introduce martingaleoptimal entropy transport problems, and express them in terms of duality, homogeneousmarginal perspective functionals and homogeneous constraints

On the other hand, we also establish a Kantorovich duality for unbalanced optimal tal variation transport problem which is a special case of weak optimal entropy transportproblems As consequences, we recover a version of duality formula for partial optimaltransports established by Caffarelli and McCann [12]; and we also get another proof ofKantorovich-Rubinstein Theorem for generalized Wasserstein distance W1a,bproved before

to-xi

by Piccoli and Rossi [38] Then we apply our duality formula to study generalized stein barycenters We show the existence of these barycenters for measures with compactsupports Finally, we prove the consistency of our barycenters.

Wasser-Keywords: Weak optimal entropy transport , Kantorovich Duality, Martingale straint, Generalized Wasserstein space, Barycenter

con-Chapter 1

Introduction

After pioneering works of Kantorovich in 1940s [30, 29], the theory of classical Kantorovich optimal transport problems has been developed by many authors It has manyapplications in other fields such as economics, geometry of nonsmooth metric spaces, imageprocessing, PDEs, logarithmic Sobolev inequalities, probability and statistics, We refers

Monge-to the monographs [4, 20, 37, 43, 46, 45] for a more detailed presentation and referencestherein The primal Monge-Kantorovich problem is written in the form

where μ1, μ2 are given probability measures on Polish metric spaces X1 and X2, c : X1×

X2→ (−∞, +∞] is a cost function, and Π(μ1, μ2) is the set of all probability measuresγ

on X1× X2 with marginals μ1and μ2.

Recently, in a seminal paper [34], Liero, Mielke and Savar´e introduced theory of mal Entropy Transport problems between nonnegative and finite Borel measures in Polishspaces which may have different masses Since then it has been investigated further in [13,

Opti-14, 17, 19, 32, 33, 35] They relaxed the marginal constraints γ i := π  i γ = μ i via adding

1

[0, ∞) → [0, ∞] are given convex, lower semi-continuous entropy functions with their recession constants (F i)

space of all nonnegative and finite Borel measures on X1× X2

In [34], the authors showed that under certain mild conditions of entropy functions

F i, the problem (1.1) always has minimizing solutions and they established the following

Here ˚D(F ) is the interior of D(F ) := {r ≥ 0 : F (r) < ∞}, f1 ⊕ f2 ≤ c means that

f (x1) + f2(x2)≤ c(x1, x2) for every x1 ∈ X1, x2 ∈ X2, and the definitions of F i ◦ and R ∗ i

will be presented in (2.1), (2.3) and (2.5)

2

On the other hand, in 2014, Gozlan, Roberto, Samson and Tetali [25] introducedweak optimal transport problems encompassing the classical Monge-Kantorovich optimaltransport and weak transport costs introduced by Talagrand and Marton in the 90’s.After that, theory of weak optimal transport problems and its applications have beeninvestigated further by a numerous authors [3, 2, 7, 5, 6, 22, 23, 24, 44] In [25], the authorsalso established a Kantorovich type duality for their weak optimal transport problem asfollows.

LetP(X2) be the space of all Borel probability measures on X2and C : X1×P(X2)→

[0, ∞] be a lower semi-continuous function such that C(x, ·) is convex for every x ∈ X1.

Given μ1 ∈ P(X1), μ2 ∈ P(X2) and γ ∈ Π(μ1, μ2), we denote its disintegration with

respect to the first marginal γ1by (γ x1)x1∈X1 Then the weak optimal transport problem

port problems [25] Given μ1 ∈ M(X1), μ2 ∈ M(X2), our primal weak optimal entropy

transport problem is formulated as

E C (μ1, μ2) :=γ∈M(Xinf

1×X2 )E C(γ|μ1, μ2), (1.3)3

whereE C(γ|μ1, μ2) :=2

i=1 F i (γ i |μ i) +

X1C(x1, γ x1)dγ1(x1).

We say that an admissible entropy function F : [0, ∞) → [0, ∞] has property (BM) on

a metric space X if for every ψ ∈ C b (X) satisfying that sup x∈X ψ(x) < F (0), there exists

a Borel bounded function s : X → (0, ∞) such that

R(s(x)) + R ∗ (ψ(x)) = s(x)ψ(x), for every x ∈ X, (1.4)

where R and R ∗ will be defined in (2.5) and (2.3)

Our first main result is a Kantorovich duality for our weak optimal entropy transportproblem

Theorem 1.1. Let X1, X2be locally compact, Polish metric spaces Let C : X1×P(X2)→

(−∞, +∞] be a lower semi-continuous function such that C is bounded from below and C(x1, ·) is convex for every x1 ∈ X1 Let F i : [0, ∞) → [0, ∞], i = 1, 2 be admissible

entropy functions such that F i is superlinear, i.e (F i) ∞ = +∞ for i = 1, 2, and F2 has property (BM) on X2 We define

Then for every μ i ∈ M(X i ), i = 1, 2 we have that

Furthermore, if X1 and X2 are compact then we can relax condition (BM) of F2 for

our duality formula

Theorem 1.2. Assume that X1, X2 are compact and (F1) ∞ + (F2) ∞ + inf C > 0 Let

μ1∈ M(X1), μ2∈ M(X2) If problem (1.3) is feasible, i.e there exists γ ∈ M(X1× X2)

such that E C(γ|μ1, μ2) < ∞ then we have

Next we will present martingale optimal entropy transport (MOET) problems Let

X1 = X2 = X ⊂ R and c : X × X → (−∞, +∞] be a lower semi-continuous function and such that c is bounded from below We consider the cost function C : X × P(X) →

In this case, for every μ1, μ2 ∈ P(X) the problem (1.2) will become the martingale

optimal transport problem It was introduced first for the case X = R by Beiglb¨ock,

Henry-Labord`ere and Penkner [8] and since then it has been studied intensively [9, 6, 7,

21, 28]

5

Now we introduce our (MOET) problems Given μ, ν ∈ M(X), we denote by Π M (μ, ν)

the set of all measures γ ∈ M(X2) such that π1 γ = μ, π2

 γ = ν andX ydπ x (y) = x almost everywhere, where (π x)x∈X is the disintegration ofγ with respect to μ We denote

μ-byM M (X2) the set of allγ ∈ M(X2) such thatγ ∈ Π M (π1γ, π2γ) Our (MOET) problem

with the convention that r i F i (θ/r i ) = θ(F i) ∞ if r i = 0 for i = 1, 2.

For μ1, μ2∈ M(X) and γ ∈ M(X2) we define

6

In our second main result in the first part of the thesis, we express our (MOET) lems in terms of duality, homogeneous marginal perspective functionals and homogeneousconstraints.

prob-Theorem 1.3. Let X be a compact subset of R and μ1, μ2 ∈ M(X) Let F i : [0, ∞) → [0, ∞], i = 1, 2 be admissible entropy functions such that (F1)

∞ > 0 and F2 is superlinear Let c : X ×X → (−∞, +∞] be a lower semi-continuous function and such that c is bounded from below, and define C as (1.5) Assume that problem (1.6) is feasible, i.e there exists

γ ∈ M M (X2) such that E C(γ|μ1, μ2) < ∞ Then

The notions of homogeneous constraints H M,≤ p (μ1, μ2) and homogeneous marginals

h p1(α) will be defined in (3.8) and (3.7), respectively.

In the second part of the thesis, we investigate problem (1.3) for a special case that

F i is not superlinear, i = 1, 2 Given a, b > 0, we consider the total variation entropy function F i (s) := a|s − 1|, i = 1, 2 and the cost function b · C In this case, problem (1.3)

where Ea,b(γ|μ1, μ2) := a |μ1− γ1| + a |μ2− γ2| + bX1C(x1, γ x1)dγ1(x1).

As F i is not superlinear, to deal with problem (1.7) we need new techniques being

different from the first part of the thesis We define

Our main result for the second part is a Kantorovich duality of problem (1.7)

Theorem 1.4. Let X1, X2be locally compact, Polish metric spaces Let C : X1×P(X2)→

[0, ∞] be a lower semi-continuous function such that C(x1, ·) is convex for every x1∈ X1 Then for every μ i ∈ M(X i ), i = 1, 2 we have

for locally compact spaces.

We now present consequences of Theorem 1.4 The first one is that we can get a version

of [12, Corollary 2.6] Let X1= X2= X be a Polish space, μ1, μ2∈ M(X), a, b > 0 and

c1: X × X → [0, +∞] be a lower semi-continuous function We define ˆ X := X ∪ { ˆ ∞} by

attaching an isolated point ˆ∞ to X We endow ˆ X with the topology induced from the

topology of X and the isolated point ˆ ∞ We extend the cost function

Corollary 1.5. Given a locally compact, Polish metric space X, μ1, μ2∈ M(X), a, b > 0, and a lower semi-continuous function c1: X × X → [0, +∞] Then

Let (X, d) be a metric space For a function f : X → R, we denote

Note that Corollary 1.6 (1) is proved for the case p = 1 in [14], and Corollary 1.6 (2)

is a main result of [39] proved by a different method there

Next, we apply Corollary 1.6 to study barycenters of generalized Wasserstein distances

In 2002, Sturm investigated barycenters in nonpositive curvature spaces as he showed the

10

existence, uniqueness and contraction of barycenters in such spaces [Sturm] Because

Wassertein spaces are not in the framework of nonpositive curvature spaces, to studythe existence, uniqueness and properties of Wasserstein barycenters overRn, Agueh and

Carlier introduced dual problems of the primal barycenter problem and used convex ysis to handle them [1] Recently, barycenters in Hellinger-Kantorovich spaces, siblings ofWasserstein spaces, have been investigated in [13, 19]

anal-On the other hand, in 2014, Piccoli and Rossi introduced generalized Wasserstein tances [38] and established a duality Kantorovich-Rubinstein formula and a generalizedBenamou-Breiner formula for them [39] Combining Corollary 1.6 with the streamline ofAgueh and Carlier’s work [1], we study the existence and consistency of generalized Wasser-stein barycenters More precisely, first we show the existence of generalized Wassersteinbarycenters whenever starting measures have compact supports Secondly, we introduceand investigate a dual problem of the barycenter problem Although our barycenters arenot unique, we still can establish their consistency as Boissard, Le Gouic and Loubes did

dis-in the Wasserstedis-in case [11]

The main results in this thesis can be found in [15] and [16]

My thesis is organized as follows In chapter 2, we review notations, generalized stein distance and properties of entropy functionals In sections 3.1 and 3.2 we proveTheorems 1.1, 1.2 and 1.3 We will illustrate examples of our results including the onesthat cover both problems (1.1) and (1.2) in section 3.3 In section 4.1, we prove Theorem1.4, Corollaries 1.5 and 1.6 Finally, we study our primal barycenter problem and its dualproblems in section 4.2

Wasser-11

Chapter 2

Preliminaries

Let (X, d) be a metric space We denote by M(X) (reps P(X)) the set of all positiveBorel measures (reps probability Borel measures) with finite mass We denote by Cb(X)the space of all real valued continuous bounded functions on X

For any µ ∈ M(X), setting |µ| := µ(X) Let M be a subset of M(X) We say that

M is bounded if there exists C > 0 such that |µ| ≤ C for every µ ∈ M , and M is equallytight if for every ε > 0, there exists a compact subset Kε of X such that µ (X\Kε) ≤ εfor every µ ∈ M

A metric space X is Polish if it is complete and separable We recall the Prokhorov’sTheorem

Theorem 2.1 (Prokhorov’s Theorem) Let (X, d) be a Polish metric space Then a subset

M ⊂ M(X) is bounded and equally tight if and only if M is relatively compact under theweak*- topology

Let µ1, µ2 ∈ M(X) If µ1(A) = 0 yields µ2(A) = 0 for any Borel subset A of X then

we say that µ2 is absolutely continuous with respect to µ1 and write µ2 ≪ µ1 We call

13

that µ1⊥ µ2 if there exists a Borel subset A of X such that µ1(A) = µ2(X\A) = 0.Let µ, γ ∈ M(X) then there are a unique measure γ⊥ ∈ M(X) and a unique σ ∈

L1+(X, µ) such that γ = σµ + γ⊥, and γ⊥⊥ µ It is called the Lebesgue decomposition of

Let f : X1 → X2 be a Borel map and µ ∈ M(X1) We denote by f♯µ ∈ M(X2) thepush-forward measure defined by

f♯µ(B) := µ(f−1(B)),

for every Borel subset B of X2

Let (X, d) be a metric space and p ≥ 1 We call Mp(X) (reps Pp(X)) the set ofmeasures µ ∈ M(X) (reps P(X)) satisfying that there exists x0∈ X such that

disinte-γ1(B1) = γ(B1× X2) and γ2(B2) = γ(X1× B2) for Borel sets Bi⊂ Xi

14

Given µ1 ∈ Mp(X), µ2 ∈ Mp(X) with the same mass, the p-Wasserstein distancebetween µ1 and µ2 is defined by

Definition 2.2 Let X be a Polish metric space and let a, b > 0, p ≥ 1 For every µ1, µ2 ∈M(X), the generalized Wasserstein distance fWpa,b between µ1 and µ2 is defined by

where D(F ) := {s ∈ [0, ∞)|F (s) < ∞} We also denote by ˚D(F ) the interior of D(F ).Let F ∈ Adm(R+), we define function F◦: R → [−∞, ∞] by

F◦(φ) := inf

s≥0 φs + F (s)
for every φ ∈ R (2.1)Given F ∈ Adm(R+) we define the recession constant F∞′ by

The Legendre conjugate function F∗: R → (−∞, +∞] is defined by

F∗(φ) := sup

s≥0

(sφ − F (s)) (2.3)Then it is clear that F◦(φ) = −F∗(−φ), for every φ ∈ R Note that ˚D(F∗) =(−∞, F∞′ ) and F∗ is continuous and non-decreasing on (−∞, F∞′ ) [34, page 989] andhence we get that

˚

D(F◦) = (−F∞′ , +∞) and F◦ is non-decreasing on (−F∞′ , +∞) (2.4)Next, we define the reverse density function R : [0, ∞) → [0, ∞] of a given F ∈Adm(R+) by

16

We also define the functional R : M(X) × M(X) → [0, ∞] by

R(µ|γ) :=

Z

X

R(ϱ)dγ + R′∞µ⊥(X),

where µ = ϱγ + µ⊥ is the Lebesgue decomposition of µ with respect to γ

Then by [34, Lemma 2.11] for every µ, γ ∈ M(X) we have that

ρ(x) ∈ D(R), ψ(x) ∈ D(R∗), R(ρ(x)) + R∗(ψ(x)) = ρ(x)ψ(x),

for µ-a.e in X

17

Lemma 2.5 ([34, Theorem 2.7 and Remark 2.8]) Let X be a Polish space, γ, µ ∈ M(X)and F ∈ Adm(R+) Then

X

R∗(ψ)dγ : ψ ∈ Cb(X, ˚D(R∗))



18

Chapter 3

Weak Optimal Entropy Transport Problems and Martingale Optimal Entropy Transport Problems

We consider a function C : X1 × P(X2) → (−∞, +∞] which is lower semi-continuous,bounded from below and satisfies that for every x ∈ X1, C(x, ·) is convex, i.e