Unavoidable sets and harmonic measures living on small sets

Abstract

Unavoidable sets for Brownian motion are investigated.and a new criterion for unavoidability is established. This, in combination with the existence of small non-polar set of Cantor type, leads to domains such that harmonic measures are living on very small sets (smallness in the sense of Hausdorff measures).

The results are obtained in very general potential theoretic settings.

Keywords

• Brownian motion
• harmonic measure
• unavoidable sets
• Hausdorff measure