Hitting Point Distribution of Two-Dimensional Random Walk

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  • Hitting Point Distribution of Two Dimen

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It is known that each path of the 2-dimensional (standard) random walk starting at arbitrary point on the plane passes the x-axis in the long run with probability 1. The purpose of this article is to calculate the probability distribution of the point where a particle of random walk for the first time hits the x-axis. The distribution is formulat- ed in terms of the starting point of the particle. It is also applied to the solution of the discrete Dirichlet problem for the half plane. Since the random walk can be considered as a discretization of the Brownian motion, the above distribution should be closely related to the Cauchy distribution, which is well known as the hitting point distribution of the 2-dimensional Brownian motion to the x-axis. This relation is also mentioned later. In the last part, the 2-dimensional biased random walk which moves at each step in four directions with different probabilities is considered.

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