###### tags: `one-offs` `stochastic processes` `transport`

# Forwards, Backwards, and Stochastic Formulations of Optimal Transport

**Overview**: In this note, I discuss some appealing aspects of i) viewing deterministic systems as special cases of stochastic systems, and ii) considering the time-reversal of stochastic systems.

## Thinking Forward and Backward

Coming from the world of MCMC algorithms, in which most algorithms are constructed to be invariant under time-reversal (or at least to admit some symmetry under time-reversal), I tend to like the idea that in order to understand (stochastic) dynamical systems, it is often very helpful to think about how they behave when run backwards in time. Perhaps this seems natural after the fact, but I think that it's not so obvious on the way in.

The brief idea is that
1. if the system looks the same forwards and backwards, then there are various aspects of the system which become easier to study (for an analogy, consider how useful it is to know that a matrix or operator is self-adjoint), and
2. if it does **not** look the same in both directions, then the differences between them can be informative about e.g. the flux of probability mass around the state space. Additionally, it can be fruitful to combine the forward and backward modes of the system in interesting ways.

## Thinking Deterministic and Stochastic

A related observation is that I have come to find that most of my current interest in topics surrounding Optimal Transport has seeped towards the { Schrödinger Bridge, Stochastic Control, ... } side of things. The field of OT is generally very cool, but from the perspective of somebody who likes Markov processes a great deal, these particular aspects have a lot to offer. One potential cause is that standard OT theory has many links to geometry and metric structures, which I have historically not understood so well, whereas the stochastic formulations are easier to generalise to arbitrary spaces (e.g. discrete spaces), and this has long held great appeal to me.

Another byproduct is the recurring perspective that deterministic processes are 'special cases' of random processes. Sometimes this distinction is actually quite useful to remember, but it's easy to see how it could become an annoying reflex. One certainly needs to be a bit careful about such habits.