This article presents a self-diffusion theory, including analytical expressions for activation energy, pre-exponential factor, and self-diffusion coefficient, dependent on temperature, pressure, the concentration of substitutional atoms, the concentration of interstitial atoms, and strain for substitutional and interstitial ternary alloys with the FCC structure, based on the statistical moment method (SMM). The theoretical results are applied to numerical calculations for the AuCuSi alloy. The SMM numerical results for AuCuSi are compared with those for AuSi, AuCu, and Au. The variation of self-diffusion quantities with temperature and strain in AuCuSi follows the same patterns as in AuCu, AuSi, and Au. The variation with the interstitial atom concentration in AuCuSi follows the same pattern as in AuSi, while the variation with the substitutional atom concentration follows the same pattern as in AuCu. The SMM numerical results for Au's self-diffusion quantities agree well with experimental data and other calculation results. Other SMM numerical results for self-diffusion quantities are novel and predict future experimental outcomes.