Finely harmonic functions with finite Dirichlet integral with respect to the Green measure

Abstract

We consider finely harmonic functions h on a fine, Greenian domain V c Rd with finite Dirichlet integral wrt Gm, i.e. ( * ) f |IVh (Y)2 G(x, y) dm (y) < oo for x c V, where m denotes the Lebesgue measure, G(x, y) the Green function. We use Brownian motion and stochastic calculus to prove that such functions h always have boundary values h* along a.a. Brownian paths. This partially extends results by Doob, Brelot and Godefroid, who considered ordinary harmonic functions with finite Dirichlet integral wrt m and Green lines instead of Brownian paths. As a consequence of Theorem 1 we obtain several properties equivalent to (*), one of these being that h is the harmonic extension to V of a random "boundary" function h* (of a certain type), i.e. h(x) = EX[h*] for all x c V. Another application is that the polar sets are removable singularity sets for finely harmonic functions satisfying (*). This is in contrast with the situation for finely harmonic functions with finite Dirichlet integral wrt m.