On Uniform Positivity of Transition Densities of Small Noise Constrained Diffusions

TL;DR

This paper establishes uniform lower bounds on transition densities for small noise constrained diffusions in convex polyhedral domains, leading to an exponential leveling property for exit times under stability conditions.

Contribution

It provides new uniform lower bounds on transition densities for constrained diffusions with small noise, using heat kernel estimates and stability assumptions.

Findings

Uniform lower bounds on transition densities are derived.

An exponential leveling property for exit times is demonstrated.

Results depend on stability conditions and heat kernel estimates.

Abstract

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.