Abstract
This chapter introduces and investigates a dynamic random-walk model which is close to real-world applications. One-dimensional random walk can be used in various areas of science including physics, computer science, games theory, information theory, and mathematical finance. Physicists can use such a random walk model as a crude approximation to one-dimensional diffusion. A physical particle can be exposed to a greater number of molecular collisions which impart to it a random motion. The random walk consists of four reflecting barriers and one absorbing barrier. The hitting place and the hitting time of the absorbing boundary are the main parameters of interest in computer science. The random-walk model is more realistic. At the same time, they are space and time dependent. Random walks with probabilities can vary from place to place. The probabilities of transition of the walk for a long time can be perturbed without changing the asymptotic behavior of the probability of return to the origin.
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