An explicit finite-memory scheme for approximating and sampling invariant measures of stochastic functional differential equations with infinite delay

arXiv:2603.04724v2 Announce Type: replace Abstract: Efficient sampling and numerical approximation of invariant probability measures (IPMs) on infinite-dimensional function spaces are important problems in scientific computing. In this paper, we study the numerical approximation and sampling of IPMs associated with stochastic functional differential equations with infinite delay (SFDEswID). To this end, we develop a fully explicit ergodicity-preserving truncated Euler--Maruyama scheme for SFDEswID that requires only finite historical storage and accommodates superlinearly growing coefficients. We establish strong convergence of the numerical segment process and show that it admits a unique IPM and is exponentially ergodic in the Wasserstein distance. Building on these results, we prove the convergence of the numerical IPM to the exact one and derive an explicit convergence rate. As a consequence, we obtain a quantitative long-time sampling error estimate of order $O\left(e^{-\lambda_\varepsilon t_n}+\Delta^{\rho_\varepsilon}\right)$. The results provide a rigorous and computationally efficient framework for sampling IPMs and quantifying long-time sampling errors for stochastic systems with infinite delay.