Existence for semilinear parabolic stochastic equations
The author considers the stochastic differential equation \(dX-\Delta Xdt+\beta(X) dt\ni\sqrt{Q}dW_{t}\) in \((0,T)\times G=Q_{T}\), \(X(0)=x\) in \(G\), \(X=0\) on \((0,T)\times\partial G=\Sigma_{T}\). Here \(G\) is an open and bounded subset of \(\mathbb R^{d}\) with smooth boundary \(\partial G\), \(d\geq1\), and \(W_{t}\) is a cylindrical Wiener process in \(L^2(G)=H\). The operator \(Q\in L(H,H)\) is self-adjoint, positive and of finite trace, \(\beta:\;\mathbb R\to2^{\mathbb R}\) is a maximal monotone graph everywhere defined on \(\mathbb R\). The author proves that under suitable assumptions on \(Q\), considered stochastic differential equation has a unique strong solution.
scientific article; zbMATH DE number 5841653