Central limit theorem for full discretization of parabolic SPDE

In order to characterize the fluctuation between the ergodic limit and the time-averaging estimator of a full discretization in a quantitative way, we establish a central limit theorem for the full discretization of the parabolic stochastic partial differential equation. The theorem shows that the normalized time-averaging estimator converges to a normal distribution with the variance being the same as that of the continuous case, where the scale used for the normalization corresponds to the temporal strong convergence order of the considered full discretization. A key ingredient in the proof is to extract an appropriate martingale difference series sum from the normalized time-averaging estimator so that the convergence to the normal distribution of such a sum and the convergence to zero in probability of the remainder are well balanced. The main novelty of our method to balance the convergence lies in proposing an appropriately modified Poisson equation so as to possess the space-independent regularity estimates. As a byproduct, the full discretization is shown to fulfill the weak law of large numbers, namely, the time-averaging estimator converges to the ergodic limit in probability.