Ricci Curvature on Birth-Death Processes

Round 1

Reviewer 1 Report

The paper studies a series of outstanding problems in the program of discretization of Riemannian geometry. More precisely, the authors study curvature dimensions conditions on birth-death processes, that correspond to the so called linear graphs, that is weighted graphs supported on the line or the half line.

In particular they give a complete characterization of the Bakry-Emery CD(K,N) condition for linear graphs. They also prove a Bishop-Gromov type comparison theorem for normalized linear graphs. Furthermore, they obtain the volume doubling property as well as the Poincare inequality. A number of significant consequences of these results are also proven.

The paper represents a strong and important contribution to the field. Therefore I warmly recommend its publication.

Reviewer 2 Report

In the present  paper the authors study curvature dimension conditions on birth-death processes which correspond to linear graphs. It is shown that linear graphs with curvature decaying not faster than $−R ^2$  are stochastically complete. For normalized linear graphs with non-negative curvature, the volume doubling property and the Poincaré inequality are studied. The results obtained are illustrated by several examples.

The paper contains five sections: 1. Introduction; 2. Preliminaries (Curvature dimension conditions,  Ollivier curvature, Intrinsic metrics);  3. Physical linear graphs (Completeness and Stochastic completeness, Non-negative curvature on physical graphs);  4. Normalized linear graphs  (Bishop-Gromov volume comparison,  Poincaré inequality);   5. Applications  (From linear to weakly spherically symmetric graphs, Infinite graphs with positive curvature bounds).

The results are interesting and the manuscript is well-written and correct.

I consider that the paper is relevant on topic, original and important enough to be published as is, modulo correction of  some minor typos (see the pdf file).

Comments for author File: Comments.pdf