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 \gg 0.3 ) on their collisional lifetime and the amount of mass that can remain at late times M _ { late } . Assuming an axisymmetric planetesimal disk with common pericentre distances 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. , with common pericentre distances but range of eccentricities ) , in the absence of dynamical evolution , the mass evolution at late times would be as if only planetesimals with the largest eccentricity were present in the disk . Despite the increased remaining mass , higher eccentricities do not increase the amount of 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 outwards 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 5 M _ { \oplus } ; however , the origin of such a high eccentricity population remains challenging . The mid-infrared 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-infrared emission would require small bound grains to be absent from the parent planetesimal belt , possibly due to sublimation . The model presented here is applicable wherever non-negligible planetesimal eccentricities are implicated and can be readily incorporated into N-body simulations .