On uniform positivity of transition densities of small noise constrained diffusions

Constrained diffusions in convex polyhedral cones with a general oblique reflection field, and with a diffusion coefficient scaled by a small parameter ∈ > 0, are considered. Using an interior Dirichlet heat kernel lower bound estimate for second order elliptic operators in bounded domains from [14], certain uniform in ∈ lower bounds on transition densities of such constrained diffusions are established. These lower bounds together with results from [2] give, under additional stability conditions, an exponential leveling property as ∈ → 0 for exit times from suitable bounded domains.