Probabilistic Proof of Boundary Harnack Principle for #alpha#-Harmonic Functions

摘要

The boundary Harnack principle for fractional Laplacian is now known to hold in Lipschitz domains [5]. It states that if two nonnegative functions, harmonic with respect to a symmetric stable Levy process vanish continuously outside a Lipschitz domain, near a part of its boundary, then the ratio of the functions is bounded inside the domain, near this part of the boundary. We give a probabilistic proof of the assertion using elementary properties of the stable process.