  We study the stochastic complex Ginzburg-Landau equation (SCGL) with an
additive space-time white noise forcing on the two-dimensional torus. This
equation is singular and thus we need to renormalize the nonlinearity in order
to give proper meaning to the equation. Unlike the real-valued stochastic
quantization equation, SCGL is complex valued and hence we are forced to work
with the generalized Laguerre polynomials for the sake of renormalization. In
handling nonlinearities of arbitrary degree, we derive a useful algebraic
identity on the renormalization in the complex-valued setting and prove that
the renormalized SCGL is locally well-posed. We prove global well-posedness
using an energy estimate and almost sure global well-posedness under different
conditions using an invariant measure argument.
