The Hitting Times with Taboo for a Random Walk on an Integer Lattice

For a symmetric, homogeneous and irreducible random walk on d-dimensional integer lattice Z^d, having zero mean and a finite variance of jumps, we study the passage times (with possible infinite values) determined by the starting point x, the hitting state y and the taboo state z. We find the probability that these passages times are finite and analyze the tails of their cumulative distribution functions. In particular, it turns out that for the random walk on Z^d, except for a simple (nearest neighbor) random walk on Z, the order of the tail decrease is specified by dimension d only. In contrast, for a simple random walk on Z, the asymptotic properties of hitting times with taboo essentially depend on the mutual location of the points x, y and z. These problems originated in our recent study of branching random walk on Z^d with a single source of branching

Perturbation models

A Markov chain (with a discrete state space and a continuous parameter) is perturbed by forcing a chain to return to "permissible" states whenever it happens to enter "forbidden" states, with returns governed by a replacement distribution. The compensation method is employed to obtain the distribution for the modified chain, in terms of the original chain and the perturbation mechanism. Emphasis is placed on ergodic chains, and interpretation of results in terms of perturbation theory of semi-groups and the ergodic potential theory (based on the fundamental matrix of a chain) is mentioned.compensation method perturbation ergodic potential fundamental matrix Markov chains replacements resolvent equations

Ergodic potential

Potential Theory for ergodic Markov chains (with a discrete state spare and a continuous parameter) is developed in terms of the fundamental matrix of a chain. A notion of an ergodic potential for a chain is introduced and a form of Riesz decomposition theorem for measures is proved. Ergodic potentials of charges (with total charge zero) are shown to play the role of Green potentials for transient chains.ergodic potential fundamental matrix Markov chains potential theory resolvent Riesz decomposition semi-groups