Oliver Curvature Bounds for the Brownian Continuum Random Tree

We compute bounds in the expected Ollivier curvature for the Brownian continuum random tree ${\mathcal{T}}_{\mathbb{e}}$. The results indicate that when the scale dependence of the Ollivier curvature is properly taken into account, the Ollivier-Ricci curvature of ${\mathcal{T}}_{\mathbb{e}}$ is bounded above by every element of $\mathbb{R}$ for almost all points of ${\mathcal{T}}_{\mathbb{e}}$. This parallels the well-known result that every continuum tree is a $CAT(K)$ space for all $K\in \mathbb{R}$.