  In this paper, we consider forward stochastic nonlinear parabolic equations,
with a control localized in the drift term. Under suitable assumptions, we
prove the small-time global null-controllability, with a truncated
nonlinearity. We also prove the statistical local null-controllability of the
true system. The proof relies on a precise estimation of the cost of
null-controllability of the stochastic heat equation and on an adaptation of
the source term method to the stochastic setting. The main difficulty comes
from the estimation of the nonlinearity in the fixed point argument due to the
lack of regularity (in probability) of the functional spaces where stochastic
parabolic equations are well-posed. This main issue is tackled through a
truncation procedure. As relevant examples that are covered by our results, let
us mention the stochastic Burgers equation in the one dimensional case and the
Allen-Cahn equation up to the three-dimensional setting.
