Log-concave measures

Definitions and mathematical statement

For a metric space $(X,d)$ and probability transition laws $(P_x)_{x\in X}$ with finite first moments, Ollivier's coarse Ricci curvature between distinct points is \[ \kappa(x,y)=1-\frac{W_1(P_x,P_y)}{d(x,y)},\qquad W_1(\mu,\nu)=\inf_{\pi\in\Pi(\mu,\nu)}\int_{X\times X}d(u,v)\,\mathrm{d}\pi(u,v), \] where $\Pi(\mu,\nu)$ is the set of probability couplings. For a diffusion one may use its time-$t$ transition laws $P_x^t$, obtaining $\kappa_t(x,y)$. Curvature depends on the transition process and the metric, not on a stationary measure alone.

A probability measure $\mu$ on ${\mathbb{R}}^N$ is log-concave if \[ \mu((1-s)A+sB)\geq\mu(A)^{1-s}\mu(B)^s\qquad(0<s<1) \] for compact sets $A,B$ (with the usual interpretation for zero measures). When $\mu$ has a Lebesgue density it has the form $e^{-V}$ with $V$ convex, allowing the value $+\infty$ off its support. In the smooth uniformly strictly log-concave setting the source refers to a positive uniform lower Hessian bound $\nabla^2V\geq\rho I$ for some $\rho>0$, together with its associated diffusion.

The problem asks what coarse-Ricci conclusions can be established for a general log-concave measure, beyond the supplied positive-curvature smooth uniformly strict case. In particular, it asks to analyze the curvature of Brownian motion in a convex body $K\subset{\mathbb{R}}^N$ conditioned not to leave $K$. No requested universal sign, numerical bound, dependence on $N$ or $K$, or transition-time regime is specified. The conditioning must also be fixed: survival to a finite horizon, the process conditioned to survive indefinitely via a limiting construction, and reflected Brownian motion are different processes. The record does not identify which transition laws define its curvature, so it states a research programme rather than a unique quantified bound.

Short English statement

Determine coarse Ricci curvature for processes associated with general log-concave measures, especially Brownian motion conditioned to stay in a convex body. A measure alone does not specify the transition laws, and the conditioning convention is missing.

Sources

• Local UnsolvedMath registry, permanent ID 4000001, exact source code catalogue entry, "Log-concave measures"; proposed by Yann Ollivier, 2008. Registry status partially solved, classification PARTIAL-PROGRESS.
• Yann Ollivier, Discrete Ricci curvature: Open problems (2008), Problem A, https://www.yann-ollivier.org/rech/publs/problems_curvmarkov.pdf. Supplied provenance: pdf-text extraction accessed July 29, 2026. The document presents the item as open; no later authoritative resolution check is recorded. No PDF was accessed here.
• The supplied research summary reports established positivity in the smooth strictly log-concave case and related functional inequalities, while describing the general log-concave and conditioned convex-body cases as not fully quantified. The original statement specifically says uniformly strictly log-concave; the summary's shorter wording is not used to drop uniformity. No precise later theorem or reference is supplied.
• Source provenance flags status and rights NEEDS_REVIEW: public access was verified, but redistribution permission was not established.