  Weakly stationary random processes of $k$-dimensional affine subspaces
(flats) in $\mathbb{R}^n$ are considered. If $2k\geq n$, then intersection
processes are investigated, while in the complementary case $2k<n$ a proximity
process is introduced. The intensity measures of these processes are described
in terms of parameters of the underlying $k$-flat process. By a translation
into geometric parameters of associated zonoids and by means of integral
transformations, several new uniqueness and stability results for these
processes of flats are derived. They rely on a combination of known and novel
estimates for area measures of zonoids, which are also developed in the paper.
Finally, an asymptotic second-order analysis as well as central and non-central
limit theorems for length-power direction functionals of proximity processes
derived from stationary Poisson $k$-flat process complement earlier works for
intersection processes.
