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 Thesis supervisor: Committee Chairman: Committee Member 1: Committee Member 2: Committee Member 3: The Graduate School Sungkyunkwan University December 2021 Contents Contents v Abstract vii Introduction Preliminaries 13 Weak Optimal Entropy Transport Problems and Martingale Optimal Entropy Transport Problems 19 3.1 Weak Optimal Entropy Transport Problems 19 3.2 Martingale Optimal Entropy Transport Problems 39 3.3 Examples 48 Unbalanced Optimal Total Variation Transport Problems and Barycenter Problems 55 4.1 Unbalanced Optimal Total Variation Transport Problems 55 4.2 Generalized Wasserstein Barycenters 68 Bibliography 81 ix Then for every μi ∈ M(Xi ), i = 1, we have that EC (μ1 , μ2 ) = = sup (ϕ1 ,ϕ2 )∈Λ i=1 Xi sup Xi (ϕ1 ,ϕ2 )∈ΛR i=1 = Fi◦ (ϕi )dμi sup ϕ∈Cb (X2 ) ϕi dμi F1◦ (RC ϕ)dμ1 + F2◦ (−ϕ)dμ2 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 > 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 EC (μ1 , μ2 ) = sup (ϕ1 ,ϕ2 )∈Λ i=1 Xi Fi◦ (ϕi )dμi 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) → (−∞, +∞] defined by C(x1 , q) = ⎧ ⎪ ⎪ ⎨ X c(x1 , x2 )dq(x2 ) if ⎪ ⎪ ⎩+∞ otherwise, X x2 dq(x2 ) = x1 , (1.5) for every x1 ∈ X, q ∈ 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 Beiglbă ock, Henry-Labord`ere and Penkner [8] and since then it has been studied intensively [9, 6, 7, 21, 28] Now we introduce our (MOET) problems Given μ, ν ∈ M(X), we denote by ΠM (μ, ν) the set of all measures γ ∈ M(X ) such that π1 γ = μ, π2 γ = ν and X ydπx (y) = x μalmost everywhere, where (πx )x∈X is the disintegration of γ with respect to μ We denote by MM (X ) the set of all γ ∈ M(X ) such that γ ∈ ΠM (π γ, π γ) Our (MOET) problem is defined as EM (μ1 , μ2 ) := inf γ∈M(X ) EC (γ|μ1 , μ2 ) = inf γ∈MM (X ) F(γi |μi ) + i=1 X×X c(x1 , x2 )dγ (1.6) We define ΛM ˚ 1◦ )) × Cb (X, D(F ˚ 2◦ )) : there exists h ∈ Cb (X) such that := (ϕ1 , ϕ2 ) ∈ Cb (X, D(F ϕ1 (x1 ) + ϕ2 (x2 ) + h(x1 )(x2 − x1 ) ≤ c(x1 , x2 ) for every x1 , x2 ∈ X Using the ideas of [34, Section 5], we investigate homogeneous formulations for our (MOET) problems We define the homogeneous marginal perspective cost H : (R × [0, +∞)) × (R × [0, +∞)) → (−∞, +∞] by ⎧ ⎪ ⎨ inf θ>0 {r1 F1 (θ/r1 ) + r2 F2 (θ/r2 ) + θc(x1 , x2 )} H(x1 , r1 ; x2 , r2 ) := ⎪ ⎩ F1 (0)r1 + F2 (0)r2 if c(x1 , x2 ) < ∞, otherwise with the convention that ri Fi (θ/ri ) = θ(Fi )∞ if ri = for i = 1, For μ1 , μ2 ∈ M(X) and γ ∈ M(X ) we define H(μ1 , μ2 |γ) := X×X H(x1 , 1 (x1 ); x2 , 2 (x2 ))dγ + i=1 where μi = i γi + μ⊥ i , i = 1, is the Lebesgue decomposition Fi (0)μ⊥ i (X), In our second main result in the first part of the thesis, we express our (MOET) problems in terms of duality, homogeneous marginal perspective functionals and homogeneous constraints Theorem 1.3 Let X be a compact subset of R and μ1 , μ2 ∈ M(X) Let Fi : [0, ∞) → [0, ∞], i = 1, be admissible entropy functions such that (F1 )∞ > 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 ∈ MM (X ) such that EC (γ|μ1 , μ2 ) < ∞ Then EM (μ1 , μ2 ) = sup(ϕ1 ,ϕ2 )∈ΛM 2 i=1 X Fi◦ (ϕi )dμi ⎧ ⎪ ⎪ ⎨0 If furthermore F2 (r) = I1 (r) := if r = 1, ⎪ ⎪ ⎩+∞ and c(x1 , x2 ) ≥ for every otherwise, x1 , x2 ∈ X then EM (μ1 , μ2 ) = = inf γ∈MM (X ) H(μ1 , μ2 |γ) inf p α∈HM,≤ (μ1 ,μ2 ) Y ×Y H(x1 , r1p ; x2 , r2p )dα + F1 (0)(μ1 − hp1 (α))(X) p The notions of homogeneous constraints HM,≤ (μ1 , μ2 ) and homogeneous marginals hp1 (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 Fi is not superlinear, i = 1, Given a, b > 0, we consider the total variation entropy function Fi (s) := a|s − 1|, i = 1, and the cost function b · C In this case, problem (1.3) will become Ea,b (μ1 , μ2 ) := inf γ∈M(X1 ×X2 ) Ea,b (γ|μ1 , μ2 ), (1.7) where Ea,b (γ|μ1 , μ2 ) := a |μ1 − γ1 | + a |μ2 − γ2 | + b X1 C(x1 , γx1 )dγ1 (x1 ) As Fi is not superlinear, to deal with problem (1.7) we need new techniques being different from the first part of the thesis We define ΦI := (ϕ1 , ϕ2 ) ∈ Cb (X1 ) × Cb (X2 ) : ϕ1 (x1 ), ϕ2 (x2 ) ≥ −a for every xi ∈ Xi , i = 1, (1.8) and ϕ1 (x1 ) + q(ϕ2 ) ≤ b · C(x1 , q) for every x1 ∈ X1 , q ∈ P(X2 ) Next, we define the functional J : R → (−∞, +∞] by ⎧ ⎪ ⎪ ⎪ +∞ if φ > a, ⎪ ⎪ φ − a|1 − s| ⎨ J(φ) = sup = φ if − a ≤ φ ≤ a, ⎪ s s>0 ⎪ ⎪ ⎪ ⎪ ⎩ −a otherwise (1.9) Then we define ΦJ := (ϕ1 , ϕ2 ) ∈ Cb (X1 ) × Cb (X2 ) : ϕ1 (x1 ), ϕ2 (x2 ) ≤ a for every xi ∈ Xi , i = 1, (1.10) and J(ϕ1 (x1 )) + q(J(ϕ2 )) ≤ b · C(x1 , q) for every x1 ∈ X1 , q ∈ P(X2 ) Our main result for the second part is a Kantorovich duality of problem (1.7) Theorem 1.4 Let X1 , X2 be 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 μi ∈ M(Xi ), i = 1, we have E a,b (μ1 , μ2 ) = = sup (ϕ1 ,ϕ2 )∈ΦI i=1 Xi sup (ϕ1 ,ϕ2 )∈ΦJ i=1 Xi I(ϕi (xi ))dμi (xi ) ϕi (xi )dμi (xi ),