Volume doubling, Poincaré inequality and Gaussian heat kernel estimate for non-negatively curved graphs

Abstract

Abstract Studying the heat semigroup, we prove Li–Yau-type estimates for bounded and positive solutions of the heat equation on graphs. These are proved under the assumption of the curvature-dimension inequality <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>CDE</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:mn>0</m:mn> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathrm{CDE}^{\prime}(n,0)} , which can be considered as a notion of curvature for graphs. We further show that non-negatively curved graphs (that is, graphs satisfying <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>CDE</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:mn>0</m:mn> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathrm{CDE}^{\prime}(n,0)} ) also satisfy the volume doubling property. From this we prove a Gaussian estimate for the heat kernel, along with Poincaré and Harnack inequalities. As a consequence, we obtain that the dimension of the space of harmonic functions on graphs with polynomial growth is finite. In the Riemannian setting, this was originally a conjecture of Yau, which was proved in that context by Colding and Minicozzi. Under the assumption that a graph has positive curvature, we derive a Bonnet–Myers-type theorem. That is, we show the diameter of positively curved graphs is finite and bounded above in terms of the positive curvature. This is accomplished by proving some logarithmic Sobolev inequalities.