Invariant measure for the stochastic Ginzburg Landau equation
This paper is about the stochastic Ginzburg-Landau equation with Dirichlet boundary conditions on a bounded domain \(D\) in \(\mathbb{R}^d\): \[ du- (1+ i\nu)\Delta_d udt- \alpha udt+ (1+ i\mu)|u|^{2\sigma} udt= \Phi\,dW,\tag{\(*\)} \] with \(\alpha\geq 0\), and \(\Phi\) an operator from \(L^2(D)\) to \(L^2(D)\) satisfying some assumptions. The author proves the following main result: Let be satisfied the following assumptions: (i) there exists \(v\in (0,1)\) such that \(A^{(-1+v)/2}\Phi\in L^0_2(H)\), where \(A= -(1+ i\nu)\Delta_d\), (ii) \(A\) and \(\Phi\) commute, (iii) if \((e_i)_{i\in \mathbb{N}}\) is the same \(H\) basis of eigenvectors diagonalizing \(\Phi\) and \(A\), then: \(|e_i(x)|\leq C\), \(\forall i\in \mathbb{N}\), \(\forall x\in D\). Then \((*)\) has a stationary solution in \[ X_\infty= L^2_{\text{loc}}((0,\infty); H)\cap C((0,\infty); D(A^{-\delta/2})), \] so there exists \(\xi\), a random variable square integrable in \(H\), and a martingale solution of \((*)\) \(\widetilde u^s\) in \(X_\infty\) such that for all \(t\geq 0\): \({\mathcal L}(\widetilde u^s(t))= {\mathcal L}(\xi)\) and such that \(\widetilde u^s\) is a stationary process.
scientific article; zbMATH DE number 2101923