= 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.
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