Given a connected and locally finite network R=(V,E, c), let P be the stocastic matrix of the random walk on R. Take $ B \subseteq V$ such that $B^C$ is finite and connected. Take $h_B: B \rightarrow \mathbb{R}\, $ s.t. $\, h_B(x)=c \, \, \forall x \in \partial B$.
Prove that $h: V\rightarrow \mathbb{R} $, the harmonic extension of $h_B$, is s.t. $h(x)=c \, \forall x \in B$.
2026-03-30 05:25:28.1774848328
application of the discrete maximum principle for harmonic functions (markov chains)
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