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 $V \subset {{\mathbf {R}}^d}$ with finite Dirichlet integral wrt $Gm$, i.e. $(\ast )$ \[ \int _V|\nabla h(y)|^2G(x,y) dm(y) < \infty \quad {\text {for}}\;x \in V,\] 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 ${h^\ast }$ 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 $( \ast )$, one of these being that $h$ is the harmonic extension to $V$ of a random "boundary" function ${h^\ast }$ (of a certain type), i.e. $h(x) = {E^x}[{h^\ast }]$ for all $x \in V$. Another application is that the polar sets are removable singularity sets for finely harmonic functions satisfying $( \ast )$. This is in contrast with the situation for finely harmonic functions with finite Dirichlet integral wrt $m$.