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.

在本文中，我们考虑控制位于漂移项中的正向随机非线性抛物方程。在适当的假设下，我们证明了具有截断的非线性的小时间全局零控制性。我们还证明了真实系统的“统计”局部空可控性。该证明依赖于对随机热方程的零可控制性成本的精确估计，并且取决于源项方法对随机环境的适应性。主要困难来自对定点参数的非线性的估计，这是由于随机抛物线方程处于适当位置的函数空间缺乏规则性（概率）。这个主要问题是通过截断程序解决的。作为我们的结果涵盖的相关示例，