Ricci Curvature on Birth-Death Processes

Abstract

In this paper, we study curvature dimension conditions on birth-death processes which correspond to linear graphs, i.e., weighted graphs supported on the infinite line or the half line. We give a combinatorial characterization of Bakry and Émery’s CD(K,n) condition for linear graphs and prove the triviality of edge weights for every linear graph supported on the infinite line Z with non-negative curvature. Moreover, we show that linear graphs with curvature decaying not faster than −R2 are stochastically complete. We deduce a type of Bishop-Gromov comparison theorem for normalized linear graphs. For normalized linear graphs with non-negative curvature, we obtain the volume doubling property and the Poincaré inequality, which yield Gaussian heat kernel estimates and parabolic Harnack inequality by Delmotte’s result. As applications, we generalize the volume growth and stochastic completeness properties to weakly spherically symmetric graphs. Furthermore, we give examples of infinite graphs with a positive lower curvature bound.

Keywords

• Discrete Ricci curvature
• Bakry-Émery curvature
• birth-death processes