  We study the localization and oscillation properties of the Majorana fermions
that arise in a two-dimensional electron gas (2DEG) with spin-orbit coupling
(SOC) and a Zeeman field coupled with a d-wave superconductor. Despite the
angular dependence of the d-wave pairing, localization and oscillation
properties are found to be similar to the ones seen in conventional s-wave
superconductors. In addition, we study a microscopic lattice version of the
previous system that can be characterized by a topological invariant. We derive
its real space representation that involves nearest and
next-to-nearest-neighbors pairing. Finally, we show that the emerging chiral
Majorana fermions are indeed robust against static disorder. This analysis has
potential applications to quantum simulations and experiments in high-$T_c$
superconductors.
