无限延迟随机泛函微分方程不变测度的显式有限记忆逼近与采样方案
Approximation of invariant measures for stochastic functional differential equations with infinite delay via an explicit finite-memory scheme
AI总结：
针对无限延迟随机泛函微分方程，提出一种仅需有限历史存储、系数可超线性增长的显式保遍历截断欧拉-丸山方案，证明数值不变测度收敛并给出显式收敛速率。
摘要：
在无限维函数空间上高效采样和数值逼近不变概率测度是科学计算中的重要问题。本文研究了与无限延迟随机泛函微分方程相关的不变概率测度的数值逼近和采样。为此，我们为无限延迟随机泛函微分方程开发了一种完全显式的保遍历截断欧拉-丸山方案，该方案仅需有限历史存储且允许超线性增长系数。我们建立了数值段过程的强收敛性，并证明其具有唯一不变概率测度且在Wasserstein距离下指数遍历。基于这些结果，我们证明了数值不变测度收敛到精确不变测度，并推导出显式收敛速率。由此，我们获得了阶为$O\left(e^{-\lambda_\varepsilon t_n}+\Delta^{\rho_\varepsilon}\right)$的定量长时间采样误差估计。这些结果为无限延迟随机系统的不变测度采样和长时间采样误差量化提供了严格且计算高效的框架。
英文摘要：
Owing to the intrinsic computational limitation that infinite memory cannot be stored numerically, the approximation of invariant probability measures (IPMs) for stochastic functional differential equations (SFDEs) with infinite delay remains an open problem. This paper addresses this challenge by proposing an explicit finite-memory truncated Euler-Maruyama scheme that combines memory and state truncation for such SFDEs. We establish strong convergence of the numerical segment process over any finite time horizon and obtain a convergence order arbitrarily close to $1/2$. We further prove that this numerical segment process admits a unique IPM and that its IPM converges to the exact counterpart under the $L^2$-Wasserstein metric as the step size vanishes, with an explicit convergence rate. The framework applies to both linear and superlinear coefficients.
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