2025/01/31 by Jairo K. Mengue, Mengue, Jairo K.
Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2501.19369
openalex publication_date 2025/01/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given two compact metric spaces X and Y, a Lipschitz continuous cost function c on X × Y and two probabilities μ\inP(X), ν\inP(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) = -DKL(π|μ× ν), where DKL is the Kullback-Leibler divergence, and then the pressure defined by the variational principle P(βA) = supπ∈ Π(μ,ν) [ \smallint βA dπ+ H(π)],where β>0 and A=-c. We will show that it admits a dual formulation and when β→+∞ 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= (1)/(T)) and β→+∞ is interpreted as a zero temperature limit.