Boundary Harnack principle for diffusions

The boundary Harnack principle (BHP) is a fundamental tool to understand the behaviour of positive harmonic functions near the boundary of a domain. For instance, the BHP implies a concrete description of the Martin boundary of a domain in geometric terms. Other applications of BHP include Carleson estimate, Fatou's theorem, and heat kernel estimates for diffusions killed upon exiting a domain. In this talk, I will discuss a recent extension of BHP that provides new examples of diffusions satisfying BHP even in \(\mathbb{R}^n\).

This is joint work with Martin Barlow.