Characterizations of random walks on random lattices and their ramifications

Abstract

We analyze the behavior of a particle moving along a d-dimensional lattice and representing a “doubly stochastic” random walk. Unlike the classical random walk, the lattice is randomly generated upon particle’s landing at a node. At any time, the particle is enclosed in a rectangular cylinder (i.e., with a bounded a-dimensional rectangle Ra at its base) it attempts to escape. Thus, the particle moves from one node to another at random epochs of time and, with some probability, leaves The particle may jump arbitrarily far beyond the boundary at its passage time. Furthermore, the particle’s location is not assumed to be known in real time, but only upon certain random epochs Of a key interest is the location of the particle at any time, where is the linear interpolation of particle’s positions at times τn’s. If denotes the first observed escape time (virtual first passage time), where we target the joint characteristic function of and itself, where or in tractable forms, thereby attempting to enhance as much as possible the probabilistic data lost due to the crudeness of the observations and to couple and with deterministic time intervals Among various applications, we discuss and treat antagonistic games of two active and several passive players as well as situations that occur in complex queueing systems.