Optimal transportation and pressure at zero temperature

Abstract

Given two compact metric spaces $X$ and $Y$, a Lipschitz continuous cost function $c$ on $X\times Y$ and two probabilities $\hat{I}¼\in \mathcal{P}(X),\hat{I}½\in \mathcal{P}(Y)$, we propose to study the Monge-Kantorovich problem and its duality from a zero temperature limit of a convex pressure function. We consider the entropy defined by $H(\ddot{I}€)=-D_{KL}(\ddot{I}€|\hat{I}¼\times \hat{I}½)$, where $D_{KL}$ is the Kullback-Leibler divergence, and then the pressure defined by the variational principle \[P(βA) = \sup_{π\in Î (μ,ν)} \left[ \smallint βA\,dπ+ H(π)\right],\]where ${\hat{I}}^2>0$ and $A=-c$. We will show that it admits a dual formulation and when ${\hat{I}}^2\to +\infty$ we recover the solution for the usual Monge-Kantorovich problem and its Kantorovich duality. Such approach is similar to one which is well known in Thermodynamic Formalism and Ergodic Optimization, where ${\hat{I}}^2$ is interpreted as the inverse of the temperature (${\hat{I}}^2=\frac{1}{T}$) and ${\hat{I}}^2\to +\infty$ is interpreted as a zero temperature limit.