On a Stability Property of Harmonic Measures
Peter March · Birkhäuser Boston eBooks · 1989
The purpose of this note is to prove that the exit distributions of a diffusion from a somewhat smooth domain are stable under a large class of perturbations. That this need not be so in general had been observed already by M.V. Keldyš [5], who constructed a Jordan domain D containing the origin, with the property that if D n ,n ≥ 1, is any sequence of smooth domains such that $$ \overline D \subset {D_n} \subset {D_{{n - 1}}}\;{\text{and}}\;\overline D = \mathop{ \cap }\limits_n {D_n} $$ (1) then the classical harmonic measures h Dn (0,dy),n ≥ 1, do not converge weakly to the harmonic measure h D (0,dy). Of course, these measures are the exit distributions of Brownian motion starting from the origin.