  In this paper, we study a phenomenological model for pattern formation in
electroconvection, and the effect of noise on the pattern. As such model we
consider an anisotropic Swift-Hohenberg equation adding an additive noise. We
prove the existence of a global solution of that equation on the two
dimensional torus. In addition, inserting a scaling parameter, we consider the
equation on a large domain near its change of stability. We observe numerically
that, under the appropriate scaling, its solutions can be approximated by a
periodic wave, which is modulated by the solutions to a stochastic
Ginzburg-Landau equation.
