For each $\delta > 0$ there is a parabolic operator in the half-plane $R_x \times R^+_t$ whose parabolic measure is supported by a boundary set of dimension $< \delta$.
@article{1176991599,
author = {Kaufman, Robert and Wu, Jang-Mei},
title = {An Example on Highly Singular Parabolic Measure},
journal = {Ann. Probab.},
volume = {16},
number = {4},
year = {1988},
pages = { 1821-1831},
language = {en},
url = {http://dml.mathdoc.fr/item/1176991599}
}
Kaufman, Robert; Wu, Jang-Mei. An Example on Highly Singular Parabolic Measure. Ann. Probab., Tome 16 (1988) no. 4, pp. 1821-1831. http://gdmltest.u-ga.fr/item/1176991599/