Abstract
This chapter presents the basic relation between the linear potential theory and random walks. This fundamental connection relies on the observation that harmonic functions and martingales share a common cancellation property, expressed via mean value properties. We cover the following topics: the Laplace’s equation and harmonic functions, construction of the ball walk, values of the ball walk as harmonic functions, walk-regularity of boundary points, the exterior cone condition as sufficient for walk-regularity, relation to Perron solutions and relation to Brownian motion.
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Lewicka, M. (2020). The Linear Case: Random Walk and Harmonic Functions. In: A Course on Tug-of-War Games with Random Noise. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-030-46209-3_2
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DOI: https://doi.org/10.1007/978-3-030-46209-3_2
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