This paper investigates longtime behaviors of the $\theta$-Euler-Maruyama method for the stochastic functional differential equation with superlinearly growing coefficients. We focus on the longtime convergence analysis in mean-square sense and weak sense of the $\theta$-Euler-Maruyama method, the convergence of the numerical invariant measure, the existence and convergence of the numerical density function, and the Freidlin-Wentzell large deviation principle of the method. The main contributions are outlined as follows. First, we obtain the longtime mean-square convergence of the $\theta$-Euler-Maruyama method and show that the mean-square convergence rate is $\frac12$. A key step in the proof is to establish the time-independent boundedness of high-order moments of the numerical functional solution. Second, based on the technique of the Malliavin calculus, we present the longtime weak convergence of the $\theta$-Euler-Maruyama method, which implies that the invariant measure of the $\theta$-Euler-Maruyama functional solution converges to the exact one with rate $1.$ Third, by the analysis of the test-functional-independent weak convergence and negative moment estimates of the determinant of the corresponding Malliavin covariance matrix, we derive the existence, convergence, and the logarithmic estimate of the density function of the $\theta$-Euler-Maruyama solution. At last, utilizing the weak convergence method, we obtain the Freidlin-Wentzell large deviation principle for the $\theta$-Euler-Maruyama solution on the infinite time horizon.

翻译：本文研究了$\theta$-Euler-Maruyama方法在系数超线性增长的随机泛函微分方程中的长时间行为。我们重点分析了$\theta$-Euler-Maruyama方法在均方意义和弱意义下的长时间收敛性、数值不变测度的收敛性、数值密度函数的存在性与收敛性，以及该方法满足的Freidlin-Wentzell大偏差原理。主要贡献如下：首先，我们获得了$\theta$-Euler-Maruyama方法的长时间均方收敛性，并证明均方收敛率为$\frac12$。证明的关键步骤是建立数值泛函解的高阶矩与时间无关的有界性。其次，基于Malliavin微积分技术，我们给出了$\theta$-Euler-Maruyama方法的长时间弱收敛性，这意味着$\theta$-Euler-Maruyama泛函解的不变测度以速率$1$收敛于精确不变测度。第三，通过分析与检验泛函无关的弱收敛性以及相应Malliavin协方差矩阵行列式的负矩估计，我们推导了$\theta$-Euler-Maruyama解密度函数的存在性、收敛性及其对数估计。最后，利用弱收敛方法，我们获得了在无穷时间区间上$\theta$-Euler-Maruyama解的Freidlin-Wentzell大偏差原理。