Two-time-scale stochastic partial differential equations driven by $$\alpha$$-stable noises: averaging principles.(English)Zbl 1360.60118

The authors consider three equations in Hilbert spaces driven by additive infinite-dimensional $$\alpha$$-stable noises for which they prove an averaging principle. Hereafter, $$H$$ is a Hilbert space, $$A$$ is a self-adjoint operator on $$H$$ with a strictly negative discrete spectrum and $$L$$ is a cylindrical $$\alpha$$-stable process on $$H$$.
The first problem has the form $dX^\varepsilon=\{AX^\varepsilon+b(X^\varepsilon,r^\varepsilon)\}\,dt+dL,\qquad X^\varepsilon(0)=x_0,\quad r^\varepsilon(0)=r_0,\tag{1}$ where $$b$$ is a Lipschitz-continuous non-linearity, $$r^\varepsilon$$ is a continuous-time finite-state-space Markov chain whose generator is a sum of $$\varepsilon^{-1}\widetilde{\mathbb Q}$$ and $$\widehat{\mathbb Q}$$, and $$\widetilde{\mathbb Q}$$ is weakly irreducible. Under an additional assumption on the eigenvalues of $$A$$ and on the covariance of $$L$$, it is proved that the mild solutions of (1) converge in $$L^\infty(0,T;L^p(\Omega;H))$$ to the mild solution of $d\overline X=\{A\overline X+\overline b(\overline X)\}\,dt+dL,\qquad \overline X=x_0,$ with a polynomial rate, the non-linearity $$\overline b$$ being a suitable average of $$b$$ in the second variable.
In the second problem, the matrix $$\widetilde{\mathbb Q}$$ is assumed to have a diagonal structure with irreducible blocks. In that case, it is proved that the mild solutions of (1) converge in $$L^p(\Omega;H)$$ to the mild solution of $d\overline X=\{A\overline X+\overline b(\overline X,\overline r)\}\,dt+dL,\qquad \overline X=x_0,\quad\overline r(0)=\overline r_0,$ for every fixed time $$t\geq 0$$. Here the non-linearity $$\overline b$$ is again a suitable average of $$b$$ in the second variable and $$\overline r$$ is a finite-state-space Markov chain whose generator is described exactly in the paper.
Finally, two equations \begin{aligned} dX^\varepsilon & =\{AX^\varepsilon+b(X^\varepsilon,Y^\varepsilon)\}\,dt+dL, \;\;X^\varepsilon(0)=x_0, \\ dY^\varepsilon&=\varepsilon^{-1}\{BY^\varepsilon+f(X^\varepsilon,Y^\varepsilon)\}\,dt+\varepsilon^{-1/\beta}\,dZ,\;\;Y^\varepsilon(0)=y_0, \end{aligned}\tag{2} are considered, where $$B$$ has the same properties as $$A$$, $$b$$ is uniformly bounded and Lipschitz-continuous, $$f$$ has a bounded differential in every variable and $$Z$$ is a cylindrical $$\beta$$-stable process on $$H$$. Under an additional assumption on the eigenvalues of $$A$$ and $$B$$ and on the covariances of $$L$$ and $$Z$$, and provided that the norm of $$\nabla_2f$$ is small, it is proved that the mild solutions of (2) converge in $$L^p(\Omega;H))$$ to the mild solution of $d\overline X=\{A\overline X+\overline b(\overline X)\}\,dt+dL,\qquad \overline X=x_0,$ for every fixed time $$t\geq 0$$, where the non-linearity $$\overline b$$ is a suitable average of $$b$$ in the second variable.

MSC:

 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 60G52 Stable stochastic processes 60J27 Continuous-time Markov processes on discrete state spaces
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