A phase field system perturbed by noise
Consider the Kolmogorov operator \[ N_0=\tfrac 12 \text{Tr} [QD^2 \varphi]+ \langle x,A^*D\varphi\rangle +\langle\Gamma X+F(X), D\varphi\rangle \] for all \(\varphi\in\varepsilon_A(H)\), the linear span of all exponential functions \(e^{i\langle x,h\rangle}\) with \(h\in D(A^*)\). In this paper the authors prove the following main result: Assume that \(Q_1=Q_2= (-\Delta)^{-\beta}\) with \(\beta>1/2\). \(n\geq 3\) and that there exists \(M>0\) such that \(| e_k(\xi) |\leq M\), \(k\in \xi\in D\) where \(e_k\) are the eigenfunctions of the operator \(Q_1\). Then there exists a probability measure on \(H\) such that: \(\bullet\) The operator \(N_0\) defined by (1) is dissipative in \(L^2(H,\nu)\). \(\bullet\) The closure \(N\) of \(N_0\) is \(m\)-dissipative in \(L^2(H,\nu)\). \(\bullet\) \(\varepsilon_A(H)\) is a core for the semigroup \(d^{tN}\) generated by \(N\). \(\bullet\) \(\nu\) is an invariant measure for \(e^{tN}\). \(\bullet\) For any \(\varphi\in D(L_1)\cap C^1_b(H)\) we have \[ \int_H(N\varphi) \varphi d\nu=- \tfrac 12 \int_H | Q^{1/2}D \varphi|^2d\nu. \]
scientific article; zbMATH DE number 1836632