Introduction to Optimal Mass Transport

Optimal Mass Transport has emerged as an extremely fruitful mathematical framework applicable across a spectrum of domains, from robotics and machine learning to statistical mechanics. It has recently garnered considerable attention in the field of Stochastic Thermodynamics, where Optimal Mass Transport results may be used to characterize minimal entropy production in finite time. In this chapter, we introduce Optimal Mass Transport to the non-expert reader with an eye towards its applications in Stochastic Thermodynamics. Beginning with its inception by Gaspard Monge in 1781, we trace its history through Kantorovich, to Benamou and Brenier. The presented theory of Optimal Mass Transport culminates with a pseudo-Riemannian geometry on the space of probability distributions that enables the measurement of path-lengths and distances. To conclude, we provide an explicitly solvable Gaussian example that serves to ground the theory.