Subordinate Average Structures on Random Walks

Abstract

In a random walk {N, p(x, y)} where N is an infinite graph and {p(x, y)} is a set of transition probabilities, if {p′(x, y)} is another set subordinate to the set {p(x, y)} such that p′(x, y) ≤ p(x, y) for all pairs (x, y) and p′(x, y) < p(x, y) for atleast one pair (x, y), then {N, p′(x, y)} can be identified as a Schr¨odinger network. In general, we consider the random walk {N, P ′} which is subordinate to {N, P} and discuss the relation between the classes of super-average functions defined by the transition probabilities sets {p(x, y)} and {p′(x, y)}. Moreover, we define {N, P ′} as parahyperbolic if 0 is the only bounded P ′-average function on N and study various potential-theoretic properties of parahyperbolic networks. We also give equivalent conditions for a random walk to be parahyperbolic. Finally, we discuss the relation between bounded P ′and P -average functions. average functions.