The Green Functions
Of Two Dimensional Random Walks Killed On A Line
And Their Higher Dimensional Analogues
Kohei Uchiyama, Tokyo Institute of Technology
Abstract
We obtain asymptotic estimates of the Green functions of random walks on the two-dimensional integer lattice that are killed on the horizontal axis. A basic asymptotic formula whose leading term is virtually the same as the explicit formula for the corresponding Green function of Brownian motion is established under the existence of second moments only. Some refinement of it is given under a slightly stronger moment condition. The extension of the results to random walks on the higher dimensional lattice is also given.
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