Harnack inequalities are an important tool in probability theory, analysis, and partial differential equations. I will talk about a stability theorem for the elliptic Harnack inequality: if two weighted graphs are equivalent, then the elliptic Harnack inequality holds for harmonic functions with respect to one of the graphs if and only if it holds for harmonic functions with respect to the other graph. This is easily extended to metric measure spaces. I'll also give a characterization of the elliptic Harnack inequality.
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