Probability of Jump Across the Border for Random Walk in a Half-Plane and a Branching Process with Interaction

Authors: Kalinkin A.V. Published: 15.04.2015
Published in issue: #2(59)/2015  
DOI: 10.18698/1812-3368-2015-2-38-52

Category: Mathematics and Mechanics | Chapter: Probability Theory and Mathematical Statistics  
Keywords: absorption probability of a random walk, Markov process with discrete states, exponential generating function, ordinary linear differential equation of infinite order with the parameter, exact solution

Random walk on the integer-valued lattice of a half-plane is considered. Probability of a random walk stop at the border as well as probability of jump across the border, are studied. In order to solve the problem analytically, an auxiliary Markov process with continuous time is defined on the integer-valued lattice of a quadrant. "Embedded Markov chain" for this process coincides with the random walk. The method of exponential generating function is proposed by the author to solve a stationary first (backward) system of Kolmogorov differential equations for the Markov branching process with interaction. Explicit representation for the probability of stopping at the border-strip is obtained under the assumption that the random walk jumps are directed to the half-plane. This representation is a generalization of the case when boundary is a line, and there is no jump across the border.


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