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**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.

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.

Reviewer: Martin Ondreját (Praha)

### MSC:

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

60G52 | Stable stochastic processes |

60J27 | Continuous-time Markov processes on discrete state spaces |