  In a companion paper, we investigated the question of the spatial origin of
the cosmic rays detected in the Solar neighborhood, in the case of standard
sources located in the Galactic disk. There are some reasons to believe that
there may also be a large number of sources located in the halo, for example if
the Galactic dark matter is made of supersymmetric particles or if Primordial
Black Holes are present. These exotic sources could enhance the $\bar{p}$,
$\bar{d}$ or positrons above the standard background, indicating the existence
of new physics. The spatial distribution of these hypothetical sources, though
an important ingredient to evaluate these exotic signals, is poorly known. The
aim of this paper is to point out that this discussion should not be
disconnected from that of the propagation properties in the Galaxy. More
precisely, we determine the regions of the halo from which a significant
fraction f of cosmic rays antiprotons and antideuterons detected in the Solar
neighborhood were emitted (we refer to these regions as f-volumes), for
different sets of propagation parameters consistent with B/C data, as derived
in Maurin et al (2002). It is found that some of them lead to rather small
f-volumes, indicating that the exotic cosmic rays could have a local origin (in
particular for a small halo or a large Galactic convective wind), coming from
the solar neighborhood or the Galactic center region. It is also found that the
dark matter density enhancement (spike) due to the accretion around the central
supermassive black hole gives a negligible contribution to the exotic charged
particle signal on Earth. The case of electrons and positrons is also
discussed.
