Epitaxial growth processes on a GaAs(111)A surface were investigated by using first-principles pseudopotential calculations. Stable adsorption sites of adatoms, stable microstructures formed by adatoms, and interactions among the adatoms were investigated. The formation energies of various microstructures were estimated in order to investigate relative stability of the microstructures. We found that formation of a certain stable microstructure is essential to initiate the epitaxial growth. Based on the calculation results, we propose an epitaxial growth mechanism on the (111)A surface, which can qualitatively explain the characteristic properties of the surface.