The Difference Between a Discrete and Continuous Harmonic Measure
AffiliationUniv Arizona, Dept Math
MetadataShow full item record
CitationJiang, J. & Kennedy, T. J Theor Probab (2017) 30: 1424. https://doi.org/10.1007/s10959-016-0695-3
Rights© Springer Science+Business Media New York 2016
Collection InformationThis item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at email@example.com.
AbstractWe consider a discrete-time, continuous-state random walk with steps uniformly distributed in a disk of radius h. For a simply connected domain D in the plane, let omega(h)(0, .; D) be the discrete harmonic measure at 0 is an element of D associated with this random walk, and omega(0, .; D) be the (continuous) harmonic measure at 0. For domains D with analytic boundary, we prove there is a bounded continuous function sigma(D)(z) on partial derivative D such that for functions g which are in C2+sigma (partial derivative D) for some alpha > 0 we have lim(h down arrow 0) integral(partial derivative D) g(xi)omega(h)( 0, vertical bar d xi vertical bar; D) - integral(partial derivative D) g(xi)omega(0, vertical bar d xi vertical bar; D)/h = integral(partial derivative D) g(z)sigma(D)(z)vertical bar dz vertical bar. We give an explicit formula for sigma(D) in terms of the conformal map from D to the unit disk. The proof relies on some fine approximations of the potential kernel and Green's function of the random walk by their continuous counterparts, which may be of independent interest.
Note12 month embargo; published online: 07 June 2016
VersionFinal accepted manuscript