weak optimal entropy transport problems and applications

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Ph.D. Dissertation

Weak Optimal Entropy TransportProblems and Applications

THANH SON TRINH

The Graduate SchoolSungkyunkwan UniversityDepartment of Mathematics

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Ph.D. Dissertation

Weak Optimal Entropy TransportProblems and Applications

THANH SON TRINH

The Graduate SchoolSungkyunkwan UniversityDepartment of Mathematics

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Weak Optimal Entropy TransportProblems and Applications

THANH SON TRINH

A Ph.D. Dissertation Submitted to the Department of Mathematicsand the Graduate School of Sungkyunkwan University

in partial fulﬁllment of the requirementsfor the degree of Ph.D. in Mathematics

October 2021

Supervised byNhan Phu Chung

Major Advisor

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This certiﬁes that the dissertationof THANH SON TRINH is approved.

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3Weak Optimal Entropy Transport Problems and Martingale Optimal

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

4Unbalanced Optimal Total Variation Transport Problems and

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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

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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

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Chapter 1

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 refersto 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

Monge-inf

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 γi:= πiγ = μi via adding

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penalizing divergences

Fi(γi|μi) :=

Fi(fi(xi))dμi(xi) + (Fi)

where γi= fiμ + γi⊥is the Lebesgue decomposition of γiwith respect to μi, and Fi :

[0, ∞) → [0, ∞] are given convex, lower semi-continuous entropy functions with theirrecession constants (Fi)

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

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

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

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

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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 (γx1)x1∈X1. Then the weak optimal transport problemis deﬁned as

V (μ1, μ2) := inf

C(x1, γx1)dμ1(x1) :γ ∈ Π(μ1, μ2)

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

transport problem is formulated as

EC(μ1, μ2) :=γ∈M(Xinf

1×X2)EC(γ|μ1, μ2), (1.3)

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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 andC(x1, ·) is convex for every x1 ∈ X1. Let Fi: [0, ∞) → [0, ∞], i = 1, 2 be admissible

entropy functions such that Fiis superlinear, i.e. (Fi)∞ = +∞ for i = 1, 2, and F2 hasproperty (BM) on X2. We deﬁne

Λ :=

(ϕ1, ϕ2)∈ Cb(X1, ˚D(F1◦))× Cb(X2, ˚D(F2◦)) : ϕ1(x1) + q(ϕ2)≤ C(x1, q),for every x1∈ X1, q ∈ P(X2)

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Then for every μi∈ M(Xi), 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 EC(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 functionand such that c is bounded from below. We consider the cost function C : X × P(X) →

(−∞, +∞] deﬁned by

C(x1, q) =

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 Beiglbăock,

Henry-Labord`ere and Penkner [8] and since then it has been studied intensively [9, 6, 7,21, 28].

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</div>

weak optimal entropy transport problems and applications

Ph.D. Dissertation

<b>Weak Optimal Entropy TransportProblems and Applications</b>

THANH SON TRINH

The Graduate SchoolSungkyunkwan UniversityDepartment of Mathematics

31dihh gia dat eu the uhue vu eohg tae dau gia puyeh xu duhg dat tai thi trah eco uhohg huyeh eco uhohg tihh hoc bihhThursday, June 13, 2024

Ph.D. Dissertation

<b>Weak Optimal Entropy TransportProblems and Applications</b>

THANH SON TRINH

The Graduate SchoolSungkyunkwan UniversityDepartment of Mathematics

31dihh gia dat eu the uhue vu eohg tae dau gia puyeh xu duhg dat tai thi trah eco uhohg huyeh eco uhohg tihh hoc bihhThursday, June 13, 2024

<b>Weak Optimal Entropy TransportProblems and Applications</b>

THANH SON TRINH

A Ph.D. Dissertation Submitted to the Department of Mathematicsand the Graduate School of Sungkyunkwan University

in partial fulﬁllment of the requirementsfor the degree of Ph.D. in Mathematics

October 2021

Supervised byNhan Phu Chung

Major Advisor

31dihh gia dat eu the uhue vu eohg tae dau gia puyeh xu duhg dat tai thi trah eco uhohg huyeh eco uhohg tihh hoc bihhThursday, June 13, 2024

31dihh gia dat eu the uhue vu eohg tae dau gia puyeh xu duhg dat tai thi trah eco uhohg huyeh eco uhohg tihh hoc bihhThursday, June 13, 2024

This certiﬁes that the dissertationof THANH SON TRINH is approved.

31dihh gia dat eu the uhue vu eohg tae dau gia puyeh xu duhg dat tai thi trah eco uhohg huyeh eco uhohg tihh hoc bihhThursday, June 13, 2024

31dihh gia dat eu the uhue vu eohg tae dau gia puyeh xu duhg dat tai thi trah eco uhohg huyeh eco uhohg tihh hoc bihhThursday, June 13, 2024

31dihh gia dat eu the uhue vu eohg tae dau gia puyeh xu duhg dat tai thi trah eco uhohg huyeh eco uhohg tihh hoc bihhThursday, June 13, 2024

31dihh gia dat eu the uhue vu eohg tae dau gia puyeh xu duhg dat tai thi trah eco uhohg huyeh eco uhohg tihh hoc bihhThursday, June 13, 2024

<b>Chapter 1</b>

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