This survey presents heat kernel regularization as a method for quantitative stability of Kantorovich potentials for the quadratic transport cost. We give a complete new heat kernel proof on Heisenberg groups equipped with the Carnot--Carath\'eodory distance. For source densities bounded above and away from zero on bou...
We study the stability of optimal transport for the squared Carnot--Carath\'eodory distance. We prove the sharp $W_1^{1/4}$ stability estimate for optimal transport maps on complete Riemannian and fat sub-Riemannian manifolds, assuming bounded source support and an upper density bound. On John domains with two-sided de...
Wasserstein barycenters provide a notion of weighted mean for probability measures on a metric space. We prove that the barycenter curvature-dimension condition ${\rm BCD}(K,N)$ is equivalent to ${\rm RCD}(K,N)$ for $K\in \mathbb R$ and $1<N<\infty$. This gives a new characterization of Riemannian curvature-dimension s...
Let $\mu_0$ and $\mu_1$ be $\kappa_0$- and $\kappa_1$-strongly log-concave probability measures on $\R^n$, and let $(\mu_t)_{t\in[0,1]}$ be their quadratic Wasserstein geodesic. We prove the sharp Poincar\'e constant estimate \[ \sqrt{C_P(\mu_t)} \leq \frac{1-t}{\sqrt{\kappa_0}} + \frac{t}{\sqrt{\kappa_1}}. \] The coef...
Bang-Xian Han, Zhuo-Nan Zhu· 1 citation
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