On Some Random Walk Games with Diffusion Control

Abstract

Random walks with discrete time steps and discrete state spaces have widely been studied for several decades. We investigate such walks as games with “Diffusion Control”: a player (=controller) with certain intentions influences the random movements of the particle. In our models the controller decides only about the step size for a single particle. It turns out that this small amount of control is sufficient to cause the particle to stay in “premium regions” of the state space with surprisingly high probabilities.

Appendix: Data for (m, n)-Casinos

Below, only the data for those pairs (m, n) are presented where m and n have greatest common divisor 1. In Table 1, for each tuple the probabilities for $t = 64, 256, {\text {and}} \, 1024$ are listed, as well as the suspected limit value $\frac{n}{n+m}$. In Table 2, for each tuple (m, n) the values of the stationary distributions for $p=125$, 250 and 500 are listed as well as the suspected limit value $\frac{n^2}{n^2+m^2}$.

Table 1. (m, n)-casino on $\mathbb Z$

Table 2. (m, n)-casino on a circle with 2p cells