A Khasminskii Type Averaging Principle for Stochastic Reaction–diffusion Equations by Sandra Cerrai

for some parameter 0 < ε 1 and some mappings b :R × R → R and g :R ×Rk → R . Under reasonable conditions on b and g, it is clear that as the parameter ε goes to zero, the first component X̂ε(t) of the perturbed system (1.1) converges to the constant first component x of the unperturbed system, uniformly with respect to t in any bounded interval [0, T ], with T > 0. But in applications that is more interesting is the behavior of X̂ε(t) for t in intervals of order ε−1 or even larger. Actually, it is indeed on those time scales that the most significant changes happen, such as exit from the neighborhood of an equilibrium point or of a periodic trajectory. With the natural time scaling t → t/ε, if we set Xε(t) := X̂ε(t/ε) and Yε(t) := Ŷε(t/ε), (1.1) can be rewritten as ⎧⎪⎪⎨ ⎪⎪⎩ dXε