# Optimal Transport

## Table of Contents

## Discrete case

So the optimal transport problem becomes

subject to linear constraints

## villani09_optim_trans

### 7. Displacement interpolation

#### Notation

- denotes the action functional
- is a certain class of continuous curves
Cost function between an initial point and final point :

- Riemannian manifold
- Langrangian is defined on

#### Deterministic interpolation via action-minimizing curves

Action is classically given by the time-integral of a *Lagrangian$ along the path:

A continuous curve is

**absolutely continuous**if there exists a function s.t. for all intermediate times in ,- If is
*absolutely continuous*, then is differentiable a.e. and its derivative is integrable.

- If is

##### Examples

Let

- be a curve from to
- for some strictly convex

Then, by Jensen's inequality,

which is an equality only when , and thus the action minimizers are lines , i.e. *straight lines*:

Also, then we have

since the cost function is *defined* by the minimizers of .

#### Interpolation of random variables

- is a cost function associated with the Lagrangian action
- be two given laws
- Optimal coupling of
- Random action-minimizing path joining to
- => random variable is an interpolation of and ; or equivalently is an interpolation of and
- This is referred to as
**displacement interpolation** This should be constrasted with

*linear*interpolation

- This is referred to as
- A
**dynamical transference plan**is a probability measure on the space (i.e. space of curves). - A
**dynamical coupling**of two probability measures is a random curve s.t. and . A

**dynamical**is a prob. measure on on s.t.*optimal*transference plan- Equivalently, is the law of a random action-minimizing curve whose endpoints constitute an optimal coupling of .
- Such a random curve is called a
**dynamic optimal coupling**of .- By abuse of language, is sometimes referred to as the same.