Ergodicity for Ginzburg-Landau equation with complex-valued space-time white noise on two-dimensional torus

We investigate the global well-posedness and ergodicity of the complex Ginzburg-Landau equation with a general nonlinear term on the two-dimensional torus, driven by complex-valued space-time white noise. Due to the roughness of noise, the solution to this singular equation is a distribution-valued stochastic process. As a result, the nonlinear term is ill-defined and requires renormalization. We establish global well-posedness by combining the fixed point theorem with an estimate that decays over time. Moreover, we prove ergodicity by applying the Krylov-Bogoliubov theorem along with an asymptotic coupling argument. A crucial tool in our proof is the theory of complex multiple Wiener-Ito integrals, which enables direct estimates for random distributions themselves and provides a systematic framework for estimating complex Wick products.