The theory of linear and nonlinear SPDEs driven by space-time white noise is well established in the one dimensional case. In addition, probabilistic convergence properties of space-time discretizations of such equations have been extensively studied in the literature. However, in the higher-dimensional case,the nature of white-noise driven PDEs changes drastically; it is known that even in the case of linear equations the solutions are no longer regular functions, but rather distribution-valued processes. For this reason, it is not obvious how to make sense of the nonlinear case and except for specfic cases where the differential operator is smooth enough (e.g. Cahn-Hilliard equation, Da Prato and Debussche, 1996), space-time white noise driven nonlinear PDEs in higher dimensions are generally recognized to be ill-posed in the mathematics com- munity.
In our work, we investigate this issue from the numerical perspective. In silico experiments of the 2D stochastic Allen-Cahn equation suggest that spatial discretizations of the equation lead to approximations that converge in probability to the zero-distribution, which in turn suggests that the original equation is ill-posed. We will present relevant numerical experiments, a numerical analysis of the employed schemes and we will discuss possible physical interpretations of the results as well as implications for applied scientists who wish to simulate higher-dimensional stochastic equations by means of standard numerical schemes.