Central limit theorem for full discretization of parabolic SPDE

TL;DR

This paper proves a central limit theorem for the full discretization of parabolic SPDEs, showing that the normalized time-averaging estimator converges to a normal distribution with the same variance as the continuous case, and introduces a novel method involving a modified Poisson equation.

Contribution

It establishes a CLT for discretized parabolic SPDEs and introduces a new approach using a modified Poisson equation to handle regularity estimates.

Findings

Normalized estimator converges to a normal distribution.

Variance matches that of the continuous case.

Full discretization satisfies the weak law of large numbers.

Abstract

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.