Large deviations for the stochastic derivative Ginzburg–Landau equation with multiplicative noise

This paper first proves the existence of a unique mild solution to the stochastic derivative Ginzburg–Landau equation. The fixed point theorem for the corresponding truncated equation is used as the main tool. Since we restrict our study to the one-dimensional case, it is not necessary to introduce another Banach space and thus the estimates of the stochastic convolutions in the Banach space are avoided. Secondly, we also consider large deviations for the stochastic derivative Ginzburg–Landau equation perturbed by a small noise. Since the underlying space considered is Polish, using the weak convergence approach, we establish a large deviations principle by proving a Laplace principle.