Weak Optimal Entropy Transport Problems

Abstract

In this paper, we introduce weak optimal entropy transport problems that cover both optimal entropy transport problems and weak optimal transport problems introduced by Liero, Mielke, and Savaré [27]; and Gozlan, Roberto, Samson and Tetali [20], respectively. Under some mild assumptions of entropy functionals, we establish a Kantorovich type duality for our weak optimal entropy transport problem. We also introduce martingale optimal entropy transport problems, and express them in terms of duality, homogeneous marginal perspective functionals and homogeneous constraints.

Unbalanced optimal total variation transport problems and generalized Wasserstein barycenters

“…In the 2010s, various generalizations of classical optimal transport problems and Wasserstein distances have been introduced and investigated by numerous authors [2,4,6,7,13,17,18,20,21]. Recently, in 2021 we introduced unbalanced optimal entropy problems [10] which cover both optimal entropy transport problems in [18] and weak optimal transport problems in [13]. In [10], under certain conditions of entropy functionals we establish a Kantorovich duality for our unbalanced optimal transport problem.…”

“…Recently, in 2021 we introduced unbalanced optimal entropy problems [10] which cover both optimal entropy transport problems in [18] and weak optimal transport problems in [13]. In [10], under certain conditions of entropy functionals we establish a Kantorovich duality for our unbalanced optimal transport problem. Before stating our first main result, let us review our unbalanced optimal entropy problems.…”

“…In the first part of the paper, we investigate problem (1.1) for a special case that F i is not superlinear, i = 1, 2. Given a, b > 0, we consider the total variation entropy function As F i is not superlinear, to deal with problem (1.2) we need new techniques being different from [10,18]. We define…”