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Stability of the elliptic Harnack Inequality

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A manifold has the Liouville property if every bounded harmonic function is constant. A theorem of T.\ Lyons is that the Liouville property is not preserved under mild perturbations of the space. Stronger conditions on a space, which imply the Liouville property,are the parabolic and elliptic Harnack inequalities (PHI and EHI ). In the early 1990s Grigor’yan and Saloff-Coste gave a characterisation of the parabolic Harnack inequality (PHI), which immediately gives its stability under mild perturbations. In this talk we prove the stability of the EHI . The proof uses the concept of a quasi symmetric transformation of a metric space, and the introduction of these ideas to Markov processes suggests a number of new problems. (Based on joint work with Mathav Murugan.)

This talk is part of the Probability series.

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