(untitled) (European Journal of Pure and Applied Mathematics, 2025)
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.