Which values of the volume growth and escape time exponent are possible for a graph?
Many mathematical questions pertaining to heat kernels on manifolds can be posed with respect to locally finite weighted graphs on measurable spaces, where the level of technical difficulties is lower, but the same basic principles apply. Typically, one relates geometric properties, like e.g. the graph distance, Ahlfors regularity or volume doubling (those are encoded in the one-parameter family of volume growth functions \(V_{\alpha }\)), and analytic ones (e.g. Sobolev and Harnack inequalities) with global properties of the random walk and its transition density. In the present paper, conditions are established for the numerical range of two exponents: \(\alpha \) in the Ahlfors regularity condition and \(\beta \) in the escape time exponent for random walks (and continuous time random walks) on infinite weighted graphs.
scientific article; zbMATH DE number 2104136
scientific article