Does Anisotropic "Inflation" Produce a Small Statistical Anisotropy?

Anisotropic inflation is an interesting model with an U(1) gauge field and it predicts the statistical anisotropy of the curvature perturbation characterized by a parameter $g_{\ast }$. However, we find that the background gauge field does not follow the classical attractor solution due to the stochastic effect. We develop the stochastic formalism of a vector field and solve Langevin and Fokker-Planck equations. It is shown that this model is excluded by the CMB constraint $g_{\ast }\le {10}^{-2}$ with a high probability about $99.999\mathrm{\%}$.