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. (Based on joint work with Mathav Murugan.)

• Speaker: Martin Barlow (UBC)
• Monday 05 September 2022, 16:30–17:30
• Venue: MR9, Centre for Mathematical Sciences.
• Series: Probability; organiser: Jason Miller.