Abstract
This paper considers a class of stochastic fractional-space diffusion equations with polynomials. We establish a limiting equation that specifies the critical dynamics in a rigorous way. After this, we use the limiting equation, which is an ordinary differential equation, to approximate the solution of the stochastic fractional-space diffusion equation. This equation has never been studied before using a combination of additive noise and fractional-space, therefore we generalize some previously obtained results as special cases. Furthermore, we use Fisher's and Ginzburg-Landau equations to illustrate our results. Finally, we look at how additive noise affects the stabilization of the solutions.