We construct a compact metric space with an atomless probability measure and a bounded Lipschitz cost for which the entropic optimal-transport minimisers have no zero-temperature weak limit. More precisely, $P_\varepsilon$ does not converge as $\varepsilon\downarrow0$. In the example, every unregularised minimiser is singular with respect to $\mu\otimes\mu$, so that the entropy on the optimal face is identically $+\infty$. We describe the cluster set by \[ \operatorname{Clust}(P_\varepsilon)=\{P_w:w\in\mathcal W\}, \] where $P_w$ is the mixture of the two zero-cost graph couplings with weight $w$, and where $\mathcal W\subset[0,1]$ is a non-degenerate compact interval. We then compute two explicit points $w^-<w^+$ in this interval. This shows that compactness, atomlessness, and Lipschitz regularity of the cost do not imply zero-temperature convergence. We also present a compactness theorem for the general problem. If $C\in L^1(\mu\otimes\nu)$ is continuous and bounded from below on Polish spaces, then the zero-temperature cluster set is a nonempty weakly compact connected subset of the optimal face. In the proof, we apply the cluster-point theorem of Bernton, Ghosal, and Nutz and the continuity of $\varepsilon\mapsto\pi_\varepsilon$. Finally, we give local and exterior first-order criteria for full convergence and cluster membership. We show that nonconvergence is possible, but only through a connected continuum of optimal plans.
We fix an integer $d\ge1$ and a symmetric positive-definite matrix $Q\in\mathbb{R}^{d\times d}$. Let $V:\mathbb{R}^d\to\mathbb{R}$ be finite, set \[ Z_\mu:=\int_{\mathbb{R}^d}e^{-V(x)}\,dx\in(0,\infty), \qquad d\mu(x):=Z_\mu^{-1}e^{-V(x)}\,dx, \] and assume that $\mu$ has finite second moment and that \[ x\longmapsto \frac12\langle Qx,x\rangle-V(x) \] is convex. Let $\nu$ be a compactly supported log-concave probability measure with support $K$, and let $\nabla\Phi$ be the Brenier map from $\mu$ to $\nu$. For $v\in\mathbb{R}^d$, define \[ w_K(v):= \sup_{y\in K}\langle y,v\rangle - \inf_{y\in K}\langle y,v\rangle. \] We prove that \[ \partial_{vv}\Phi \le 0.587 \sqrt{\langle Qv,v\rangle}\,w_K(v) \qquad(v\in\mathbb{R}^d) \] in the sense of distributions. We show that $\nabla\Phi$ has an everywhere-defined globally Lipschitz representative such that \[ \operatorname{Lip}(\nabla\Phi) \le 0.587 \sqrt{\|Q\|_{\mathrm{op}}}\,\operatorname{diam}(K). \] The directional Hessian estimate is affinely covariant, whereas the global Lipschitz estimate is dimension-free. The result also applies to singular or lower-dimensional targets. In particular, it removes the $\sqrt d$ loss in Kolesnikov's estimate for the Brenier map from Gaussian measure to normalised Lebesgue measure on a convex body. We also prove new bounds that depend only on the support for compactly supported semi-log-concave targets, which includes targets with bounded negative curvature.
Maja Gwóźdź· 1 citation· ⚡1
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