Projection criterion for Brownian avoidance of affine subspaces

A Brownian motion starting outside a fixed affine subspace of codimension at least two almost surely never hits it. Choose a two-dimensional orthogonal projection annihilating the subspace's direction and with nonzero projected initial displacement. The projected process is planar Brownian motion starting away from zero, and a hit of the subspace would hit zero in that projection. The polar point for planar Brownian motion result excludes this. In three dimensions this shows avoidance of every fixed line when started outside it.