Conditioned Two-Dimensional Simple Random Walk: Green’s Function and Harmonic Measure
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We study the Doob’s h-transform of the two-dimensional simple random walk with respect to its potential kernel, which can be thought of as the two-dimensional simple random walk conditioned on never hitting the origin. We derive an explicit formula for the Green’s function of this random walk and also prove a quantitative result on the speed of convergence of the (conditional) entrance measure to the harmonic measure (for the conditioned walk) on a finite set.
KeywordsTransience Doob’s h-transform Entrance measure
Mathematics Subject Classification (2010)Primary 60J10 Secondary 60G50 82C41
This work was partially supported by CNPq (301605/2015–7). The author is grateful to the anonymous referee for many comments and suggestions on the first version of this paper.
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