Isoperimetric inequalities for random walks (Q1398029)

On a countably infinite, locally finite and connected graph a reversible Markov chain is considered whose one step transition probabilities are uniformly bounded below. The main theorem states a set of equivalent conditions for diagonal upper bounds on the \(n\)-step transition function. It is based on the corresponding work of \textit{P. Mathieu} about Sobolev inequalities and hitting times [Potential Anal. 9, 293--300 (1998; Zbl 0922.60066)]. Later it is generalized along the lines of the work of \textit{G. Carron} on isoperimetric inequalities for manifolds [in: Actes de la table ronde de géométrie différentielle en l'honneur de Marcel Berger. Sémin. Congr. 1, 205--232 (1996; Zbl 0884.58088)]. Theorem~4.1 states a weak isoperimetric inequality: Among all sets of fixed volume the upper level sets of the Green's function maximize the mean exit time.