The present work studies the effect of additive noise on two high-dimensional systems. The first system under study is two-dimensional, evolves close to the deterministic stability threshold and exhibits an additive noise-induced shift of the control parameter when driving one variable by uncorrelated Gaussian noise. After a detailed analytical and numerical study of this effect, the work further focusses on the extended Swift-Hohenberg equation subjected to global noise, i.e. noise constant in space and uncorrelated in time. This spatial system generalizes the two-dimensional system and thus reveals phase transitions induced by additive global noise. Numerical studies confirm this effect. Further closer investigations reveal that the occurence of the noise-induced shift is subjected to the model nonlinearity and the shifts sign depends on the sign of the nonlinearity prefactors.