We prove existence and uniqueness of a solution for the stochastic Allen–Cahn equation with logarithmic potential and multiplicative Wiener noise, under homogeneous Neumann boundary condition. The existence of a solution is obtained in the variational sense by means of an approximated equation involving a Yosida regularization of the nonlinearity. The noise is assumed to vanish at the extremal points of the physical relevant domain and to satisfy a suitable Lipschitz-continuity property, allowing to prove uniform estimates of the approximated solution. The passage to the limit is then carried out and continuous dependence on the initial datum of the solution is verified. Under an additional assumption, the existence of an analytically strong solution is proved. Finally, estimates for the derivatives of the logarithmic potential are derived, in view of the study of an optimal control problem associated to the stochastic Allen–Cahn equation.

我们证明了在齐次Neumann边界条件下具有对数势和可乘Wiener噪声的随机Allen-Cahn方程解的存在性和唯一性。通过涉及非线性的Yosida正则化的近似方程，可以从变分的意义上获得解的存在。假定噪声在物理相关域的极点处消失并满足适当的Lipschitz连续性，从而可以证明近似解的统一估计。然后进行传递到极限，并验证对溶液初始数据的连续依赖性。在另外的假设下，证明存在解析力强的解决方案。最后，得出对数势导数的估计值，