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Martingales, Harmonic Functions and Poisson–Dirichlet Problems

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Markov Chains

Abstract

In this chapter, we introduce several notions of potential theory for Markov chains. Harmonic and superharmonic functions on a set A are defined in Section 4.1, and Theorem 4.1.3 establishes links between these functions and the return (or hitting) times to the set A. In Section 4.2, we introduce the potential kernel and prove the maximum principle, Theorem 4.2.2, which will be very important in the study of recurrence and transience throughout Part II.

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Correspondence to Randal Douc .

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Douc, R., Moulines, E., Priouret, P., Soulier, P. (2018). Martingales, Harmonic Functions and Poisson–Dirichlet Problems. In: Markov Chains. Springer Series in Operations Research and Financial Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-97704-1_4

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