A stochastic approach to quasi-everywhere boundary convergence of harmonic functions

Abstract

Given a Dirichlet form E (·, ·) on the unit sphere S in R n ( n ⩾ 2) associated to a continuous, symmetric convolution semigroup of measures on a group G of isometries on S and given a ( G-invariant) Markov process X t on the open unit ball B, it is shown that for any real function u ϵ L 2( S) with E ( u, u)<∞ the X t -harmonic extension u ̃ has limit u ̌ (θ) along a.a. paths X t conditioned to exit from B at θ, for quasi-all θ ϵ S, where u ̌ is a quasi-continuous version of u. This extends in several ways classical results due to Beurling and Broman about the existence of radial limits quasi-everywhere for a harmonic function in the open unit disc in the plane with a finite Dirichlet integral.