In this thesis, we develop a scattering theory for the asymmetric transport observed at interfaces separating two-dimensional topological insulators. Starting from the spectral decomposition of an unperturbed confining Hamiltonian, we present a limiting absorption principle and construct a generalized eigenfunction expansion for perturbed systems. We then relate the interface conductivity, a current observable quantifying the transport asymmetry, to the scattering matrix associated to the generalized eigenfunctions. In particular, we show that the interface conductivity is concretely expressed as a difference of transmission coefficients and is stable against perturbations. We apply the theory to systems of perturbed Dirac equations with asymptotically linear domain wall. In the presence of random perturbations in the Hamiltonians, the limiting behavior of the scattering matrix entries as the thickness L of the random medium increases gives rise to a second order diffusion operator by the diffusion approximation theory. We call such diffusion operator a mixed type generalized Kimura diffusion operator. We model the operator andprovide the degenerate Hölder space-type estimates for model operators. With the analysis of perturbation term we establish the existence of solutions. We also give proofs of the existence and regularity of the global heat kernel. We also concern the long-time asymptotics of this degenerate diffusion operators with mixed linear and quadratic degeneracies. 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.