Optimal transportation and pressure at zero temperature

Given two compact metric spaces $X$ and $Y$, a Lipschitz continuous cost function $c$ on $X \times Y$ and two probabilities $\mu \in\mathcal{P}(X),\,\nu\in\mathcal{P}(Y)$, 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 $H(\pi) = -D_{KL}(\pi|\mu\times \nu)$, where $D_{KL}$ is the Kullback-Leibler divergence, and then the pressure defined by the variational principle \[P(\beta A) = \sup_{\pi \in \Pi(\mu,\nu)} \left[ \smallint \beta A\,d\pi + H(\pi)\right],\]where $\beta>0$ and $A=-c$. We will show that it admits a dual formulation and when $\beta \to+\infty$ 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 $\beta$ is interpreted as the inverse of the temperature ($\beta = \frac{1}{T}$) and $\beta\to+\infty$ is interpreted as a zero temperature limit.