Introduction to multi-marginal optimal transport on sub-riemannian manifold :Hội nghị khoa học trẻ lần 4

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INTRODUCTION TO MULTI-MARGINAL OPTIMAL TRANSPORT ON SUB￾RIEMANNIAN MANIFOLD

THANH SON TRINH

Faculty of Information Technology, Industrial University of Ho Chi Minh City

[email protected]

Abstract. In this paper, we introduce the multi-marginal optimal transport on sub-Riemannian manifold

to minimize the total cost, which is defined to be the sum of the squared sub-Riemannian distances. As the

works of Kim and Pass on Riemannian manifold [13]; and Pass, Pinamonti and Vedovato on Heisenberg

group [16], we give a Kantorovich dual formula and prove the existence of a solution to our problem under

structure of sub-Riemannian manifold.

Keywords. Multi-marginal, Kantorovich duality, Sub-Rieamannian manifold.

1. INTRODUCTION

In 2010s, the theory of classical optimal transport problems was developed by many authors

[4,5,6,7,9,19,20,21]. It has many applications in economics, image processing, PDEs, probability and

statistics, and logarithmic Sobolev inequalities. For more details, readers can see [3,7,8,15,17,18,19]. The

primal optimal transport problem was introduced by Kantorovich in 1940s [9,10]. This problem is presented

in the form

 

 

1 2 1 2

1 2 1 2

,

inf ( , ) , ,

X X

c x x d x x

  



 



where

1

and

2 

are Borel probability measures on Polish (separable and complete) metric spaces

X1

and

X2

, respectively;

1 2 c X X : ( , ]     is a cost function, and we denote by

 1 2 , 

the set of

all Borel probability measures

 on

X X 1 2 

with the first and the second marginals are

1

and

2  ,

respectively, this means that

     A X A X A A 1 2 1 1 1 2 2 2      ( ), ( )  

,

for every Borel subsets

Ai

of

Xi

, i 1,2 .

In 2015, Kim and Pass have generalized the primal optimal transport problem by introducing and

investigating the multi-marginal optimal transport problem on Riemannian manifolds in paper [13]. Let

1

, ,  k

be

k

Borel probability measures on a manifold

M

, this problem is written in the following

form

 

   

1

1 1

,...,

inf ,..., ,..., k

k

M k k c x x d x x

  



 

,

where

  1

,..., k 

is the set of all Borel probability measures



on

k M M M   

with marginals

1

,...,  k

, this means that