On Uniform Positivity of Transition Densities of Small Noise Constrained Diffusions

Constrained diffusions in convex polyhedral domains with a general oblique reflection field, and with a diffusion coefficient scaled by a small parameter, are considered. Using an interior Dirichlet heat kernel lower bound estimate for second order elliptic operators in bounded domains from [13], certain uniform in the scaling parameter lower bounds on transition densities of such constrained diffusions are established. These lower bounds together with results from [1] give, under additional stability conditions, an exponential leveling property, as the scaling parameter approaches zero, for exit times from suitable bounded domains.