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

AbstractGiven a Dirichlet form E(·, ·) on the unit sphere S in Rn (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 Xt on the open unit ball B, it is shown that for any real function u ϵ L2(S) with E(u,u)<∞ the Xt-harmonic extension ũ has limit ǔ(θ) along a.a. paths Xt conditioned to exit from B at θ, for quasi-all θ ϵ S, where ǔ 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