Abstract Constrained diffusions, with diffusion matrix scaled by small ϵ > 0 , in a convex polyhedral cone G ⊂ R k , are considered. Under suitable stability assumptions small noise asymptotic properties of invariant measures and exit times from domains are studied. Let B ⊂ G be a bounded domain. Under conditions, an “exponential leveling” property that says that, as ϵ → 0 , the moments of functionals of exit location from B , corresponding to distinct initial conditions, coalesce asymptotically at an exponential rate, is established. It is shown that, with appropriate conditions, difference of moments of a typical exit time functional with a sub-logarithmic growth, for distinct initial conditions in suitable compact subsets of B , is asymptotically bounded. Furthermore, as initial conditions approach 0 at a rate ϵ 2 these moments are shown to asymptotically coalesce at an exponential rate.