Convergence properties of harmonic measure distributions for planar domains
We establish sufficient conditions under which the harmonic measure distribution functionshn of a sequence of domains Dn converge pointwise to the distribution function h of the limiting domain D, at all points of continuity of h. In the case of a model example, we establish this convergence of the distribution functions. Here, the value of the function h(r) gives the harmonic measure of the part of the boundary of the domain that lies within distance r of a fixed basepoint in the domain, thus relating the geometry of the domain to the behaviour of Brownian motion in the domain.
Convergence properties of harmonic measure distributions for planar domains. Complex Var. Elliptic Equ., Vol. 53 (2008), 897--913. (coauthored with Lesley Ward)
Complex Variables of Elliptical Equations