论文标题
中央限制定理，用于完全离散的抛物线SPDE
Central limit theorem for full discretization of parabolic SPDE
论文作者
论文摘要
为了表征以量化方式完全离散化的千古限制与时间平均估计值之间的波动，我们为抛物线抛物线随机部分分化方程式完全离散地建立了一个中心限制定理。该定理表明，归一化的时间平均估计量会收敛到正态分布，方差与连续情况的方差相同，在这种情况下，用于归一化的量表对应于所考虑的完全离散化的时间强融合顺序。证据中的关键成分是从归一化的时间平均估计器中提取适当的Martingale差异序列总和，以便在其余部分的概率上融合到该总和的正态分布和零收敛到零。平衡融合方法的主要新颖性在于提出适当修改的泊松方程，从而具有无空间依赖性的规律性估计。作为副产品，已证明全部离散化可以实现大数量的弱法律，即，时间平时估计器在概率上会收敛到千古限制。
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.
