  This paper concerns the long-time asymptotics of diffusions with degenerate
coefficients at the domain's boundary. Degenerate diffusion operators with
mixed linear and quadratic degeneracies find applications in the analysis of
asymmetric transport at edges separating topological insulators. In one space
dimension, we characterize all possible invariant measures for such a class of
operators and in all cases show exponential convergence of the Green's kernel
to such invariant measures. We generalize the results to a class of
two-dimensional operators including those used in the analysis of topological
insulators. Several numerical simulations illustrate our theoretical findings.
