Before revision:  Using an analytical method of small noise expansion, I prove that when weak and rapid noise enters the system through a reaction of Michaelis-Menten type (reactionrate function monotonically increasing and concave) , then the steady states of the system always shift to the right as noise intensity increases.
After revision:  Using an analytical method of small noise expansion, I prove that when weak and rapid noise enters the system through a reaction of Michaelis-Menten reaction , then the steady states of the system always shift to the right as noise intensity increases.