RANDOM-WALKS ON FINITELY STRUCTURED TRANSFINITE NETWORKS

A general theory for random walks on transfinite networks whose ranks are arbitrary natural numbers is established herein. In such networks, nodes of higher ranks connect together transfinite networks of lower ranks. The probabilities for transitions through such nodes are obtain ed as extensions of the Nash-Williams rule for random walks on ordinar y infinite networks. The analysis is based on the theory of transfinit e electrical networks, but it requires that the transfinite network ha ve a structure that generalizes local-finiteness for ordinary infinite networks. The shorting together of nodes of different ranks are allow ed; this complicates transitions through such nodes but provides a con siderably more general theory. It is shown that, with respect to any f inite set of nodes of any ranks, a transfinite random walk can be repr esented by an irreducible reversible Makov chain, whose state space is that set of nodes.