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Averaging principle for a class of stochastic reaction-diffusion equations. (English) Zbl 1176.60049
The authors consider the system of stochastic reaction-diffusion equations with a fast stochastic component
\begin{alignedat}{2} \frac{\partial u^\varepsilon}{\partial t}(t,\xi)&= {\mathcal A}u^\varepsilon(t,\xi) + f(\xi,u^\varepsilon(t,\xi),v^\varepsilon(t,\xi)), &&\quad t\geq 0,\;\xi\in [0,L], \\ \frac{\partial v^\varepsilon}{\partial t}(t,\xi)&= \frac{1}{\varepsilon}\left[{\mathcal A}v^\varepsilon(t,\xi) + g(\xi,u^\varepsilon(t,\xi),v^\varepsilon(t,\xi))\right] + \frac{1}{\sqrt{\varepsilon}}\frac{\partial w}{\partial t}(t,\xi), &&{}\\ u^\varepsilon(0,\xi)&= x(\xi), \quad v^\varepsilon(0,\xi) = y(\xi), &&\quad \xi\in [0,L], \\ {\mathcal N}_1 u^\varepsilon(t,\xi) &={\mathcal N}_2 v^\varepsilon(t,\xi) = 0, &&\quad t\geq 0,\;\xi\in [0,L] \end{alignedat} for each $$0< \varepsilon\ll 1$$. The linear operators $${\mathcal A}$$ and $${\mathcal B}$$, appearing respectively in the slow and in the fast equation, are second-order uniformly elliptic operators and $${\mathcal N}_1$$ and $${\mathcal N}_2$$ are operators acting on the boundary. The operator $${\mathcal B}$$, endowed with the boundary condition $${\mathcal N}_2$$, is self-adjoint and strictly dissipative. The reaction coefficients $$f$$ and $$g$$ are measurable mappings from $$[0,L]\times\mathbb R^2$$ into $$\mathbb R$$ and the noisy perturbation of the fast motion equation is given by a space-time white noise.
Under some assumptions providing existence of a unique invariant measure of the fast motion with the frozen slow component, they calculate the limiting slow motion. The study of solvability of Kolmogorov equations in Hilbert spaces and the analysis of regularity properties of solutions, allow them to generalize the classical approach to finite-dimensional problems of this type in the case of SPDE’s.

##### MSC:
 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 60F99 Limit theorems in probability theory 70K65 Averaging of perturbations for nonlinear problems in mechanics 70K70 Systems with slow and fast motions for nonlinear problems in mechanics 35K57 Reaction-diffusion equations 35R60 PDEs with randomness, stochastic partial differential equations 37H10 Generation, random and stochastic difference and differential equations
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