Some inequalities for superharmonic functions on graphs
In a discrete context the author proves an analogue of an inequality for superharmonic functions on Riemannian manifolds of at most quadratic volume growth that is due to \textit{S. Y. Cheng} and \textit{S.-T. Yau} [Commun. Pure Appl. Math. 28, 333-354 (1975; Zbl 0312.53031)]. For that purpose he considers functions on a weighted graph or network, that is a graph \((X,E)\) with vertices \(X\) and a symmetric set of edges \(E\subset X \times X\), a reference measure \(\mu\) on \(X\) and a Markovian kernel \(P(x,y)\) on \(X\) such that \(P(x,y)>0\) iff \((x,y)\in E\) and which is reversible with respect to \(\mu\). Then \(P\) defines an associated Laplace operator by \[ \Delta u(x)= (I-P)u(x) =\sum_{(x,y) \in E} \bigl(u(x)- u(y)\bigr) P(x,y) \] and an operator \[ \nabla u(x)= \left(\sum_{(x,y) \in E} \bigl|u(x)-u(y) \bigr|^2 P(x,y) \right)^{1/ 2}. \] A function \(u\) is called (super-)harmonic if \(\Delta u=0\) \((\Delta u\geq 0)\). If one defines in a proper way a volume \(W_x(r)\) of balls \(B_x(r)\) in terms of the kernel \(P\) and \(\mu\) and if \(r_0<r_1 <\cdots <r_l=R\) are integers such that \(W_x (r_i)\leq cr^2_i\), then one version of the analogue of the Cheng-Yau inequality for positive superharmonic functions \(u\) on \(B_x (R +1)\) reads as follows \[ \sum_{y\in B_x (r_0)} \bigl|\nabla \log u(y) \bigr |^2 \mu(y) \leq 2 \theta \left(\sum_{y\in B_x(r_0)} {-\nabla u(y) \over u(y)} \mu(y)+ {c\over \sum^{l-1}_0 (1-{r_i \over r_{i+1}})^2} \right) \] where \(\theta= \max_{(y,z)\in E\atop x\in B_x (R)} u(z)/u(y)\). One consequence of the inequality are sufficient conditions in terms of the volume growth for the recurrency of the associated Markov chain. Another implication are Harnack inequalities for positive harmonic functions in the case of graphs with at most quadratic volume growth and a certain connectedness property.
scientific article; zbMATH DE number 1019559