Total Variation Error Bounds for the Approximation of the Invariant Distribution of Parabolic Semilinear SPDEs Using the Standard Euler Scheme

We study the long time behavior of the standard linear implicit Euler scheme for the discretization of a class of erdogic parabolic semilinear SPDEs driven by additive space-time white noise. When the nonlinearity is a gradient, the invariant distribution is of Gibbs form, but it cannot be approximated in the total variation sense by the standard Euler scheme. We prove that the numerical scheme gives an approximation in the total variation sense of a modified Gibbs distribution, which is the invariant distribution of a modified SPDE. The modified distribution and the modified equation depend on the time-step size. This original result goes beyond existing results in the literature where the weak error estimates for the approximation of the invariant distribution do not imply convergence in total variation when the time-step size vanishes. The proof of the main result requires regularity properties of associated infinite dimensional Kolmogorov equations.