Optimal transportation and pressure at zero temperature
Jairo K. Mengue
TL;DR
The paper formulates a zero-temperature, entropy-regularized view of the Monge-Kantorovich problem by introducing a pressure functional $P(A)=\sup_{\pi\in\Pi(\mu,\nu)}\left[\int A\,d\pi+H(\pi)\right]$ with $A=-c$ and entropy defined via Kullback-Leibler divergence. It proves the existence of a dual pair $(\varphi_\beta,\psi_\beta)$ at finite inverse temperature $\beta$, constructs Gibbs-type plans $\pi_\beta$, and shows that as $\beta\uparrow\infty$ these converge to solutions of the classical MK problem and its Kantorovich dual, providing a thermodynamic formalism perspective for transport. The work also establishes regularity and convergence properties, a Kantorovich-Rubinstein corollary in the metric case, a large deviation principle for the limiting plans with rate function $I(x,y)=c(x,y)-\varphi(x)-\psi(y)$, and an entropy-maximizing characterization of accumulation points of $\pi_\beta$ among MK optimizers. Overall, the approach links transport theory to thermodynamic formalism and ergodic optimization, offering a zero-temperature interpretation and quantitative convergence/LD results with potential for broader applications in statistical transport models.
Abstract
Given two compact metric spaces $X$ and $Y$, a Lipschitz continuous cost function $c$ on $X \times Y$ and two probabilities $μ\in\mathcal{P}(X),\,ν\in\mathcal{P}(Y)$, 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 $H(π) = -D_{KL}(π|μ\times ν)$, where $D_{KL}$ is the Kullback-Leibler divergence, and then the pressure defined by the variational principle \[P(βA) = \sup_{π\in Π(μ,ν)} \left[ \smallint βA\,dπ+ H(π)\right],\]where $β>0$ and $A=-c$. We will show that it admits a dual formulation and when $β\to+\infty$ 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 $β$ is interpreted as the inverse of the temperature ($β= \frac{1}{T}$) and $β\to+\infty$ is interpreted as a zero temperature limit.