Harmonic measure on sets of co-dimension larger than 1

We'll describe an attempt, with J. Feneuil, S. Mayboroda, and M. Engelstein, to define and study a variant of the harmonic measure designed for open sets in $R^n$ with higher co-dimensional boundaries. For this the Laplacian is replaced with a degenerate elliptic operator (the coefficients tend to infinity at the boundary). We are interested in relations between the regularity of the boundary and the absolute continuity of the harmonic measure with respect to the Hausdorff measure.