  Models for the steady state collisional evolution of low eccentricity
planetesimal belts identify debris disks with hot dust at 1AU, like eta Corvi
and HD69830, as anomalous since collisional processing should have removed most
of the planetesimal mass over their >1 Gyr lifetimes. This paper looks at the
effect of large planetesimal eccentricities (e>>0.3) on their collisional
lifetime and the amount of mass that can remain at late times M_{late}. For an
axisymmetric planetesimal disk with common pericentres and eccentricities e, we
find that M_{late} \propto e^{-5/3}(1+e)^{4/3}(1-e)^{-3}. For a scattered
disk-like population (i.e., common pericentres), in the absence of dynamical
evolution, the mass evolution at late times would be as if only planetesimals
with the largest eccentricity were present. Despite the increased remaining
mass, higher eccentricities do not increase the hot emission from the
collisional cascade until e>0.99, partly because most collisions occur near
pericentre thus increasing the dust blow-out diameter. However, at high
eccentricities (e>0.97) the blow-out population extending out from pericentre
may be detectable above the collisional cascade; higher eccentricities also
increase the probability of witnessing a recent collision. All of the imaging
and spectroscopic constraints for eta Corvi can be explained with a single
planetesimal population with pericentre at 0.75AU, apocentre at 150AU, and mass
5M_\oplus; however, the origin of such a high eccentricity population remains
challenging. The mid-IR excess to HD69830 can be explained by the ongoing
destruction of a debris belt produced in a recent collision in an eccentric
planetesimal belt, but the lack of far-IR emission requires small bound grains
to be absent from the parent planetesimal belt, possibly due to sublimation.
