Long time asymptotics of mixed-type Kimura diffusions

TL;DR

This paper analyzes the long-term behavior of degenerate diffusions with mixed degeneracies, characterizing invariant measures and exponential convergence, with applications to topological insulators.

Contribution

It provides a comprehensive characterization of invariant measures and convergence rates for mixed-type Kimura diffusions, extending to two-dimensional cases relevant to physics.

Findings

Invariant measures are fully characterized for one-dimensional cases.

Exponential convergence of Green's kernel to invariant measures is established.

Numerical simulations support theoretical results.

Abstract

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.