Optimal transportation and pressure at zero temperature
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arXiv
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| Format: | Preprint |
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2025
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| author | Mengue, Jairo K. |
| author_facet | Mengue, Jairo K. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_19369 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal transportation and pressure at zero temperature Mengue, Jairo K. Dynamical Systems Functional Analysis Probability 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. |
| title | Optimal transportation and pressure at zero temperature |
| topic | Dynamical Systems Functional Analysis Probability |
| url | https://arxiv.org/abs/2501.19369 |