Numerical Central Limit Theorem

In Sect. 1.3.4 , we have investigated the strong law of large numbers (SLLN) and the central limit theorem (CLT) for the functional solution of SFDEs, thereby clarifying the connection between these probabilistic limit theorems and the corresponding invariant measure. It is well-known that the functional solution of an autonomous SFDE, viewed in its natural infinite-dimensional state space, constitutes a time-homogeneous Markov process. However, due to its inherent dependence on the path history, the ergodicity and longtime behavior of such systems are routinely analyzed within this infinite-dimensional framework. For equations of this type, often characterized by highly degenerate coefficients, a natural question arises: can numerical discretizations preserve the probabilistic limit theorems of the original equation, particularly the SLLN and CLT?