Stability of the elliptic Harnack Inequality
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.
This is joint work with Mathav Murugan (UBC).
Professor Emeritus Martin Barlow, The University of British Columbia