Existence and Uniqueness of Weak Solutions for the Stochastic Fractional Ginzburg–Landau Equation

In this study, we investigate the existence and uniqueness of weak solutions for a stochastic Ginzburg–Landau equation involving the fractional Laplacian. The primary focus is on establishing a proper mathematical framework to handle the coexistence of the nonlocal fractional Laplacian and stochastic perturbations. By employing the Galerkin method, we prove that the initial-boundary value problem admits a unique global weak solution for any F0-measurable L2(I)-valued random initial value with a finite second moment. We also utilize the properties of the fractional Laplacian and fractional Sobolev spaces to provide a proof of the existence of the uniqueness theorem. These results extend the analysis of the Ginzburg–Landau equations to models incorporating stochastic terms and the fractional Laplacian.