# Optimal Transport

## 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.
##### 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

• 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 optimal transference plan is a prob. measure on on s.t.

• 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.