论文标题
使用显式驯服的指数Euler方案的一类Ergodic SPDE的不变分布的近似
Approximation of the invariant distribution for a class of ergodic SPDEs using an explicit tamed exponential Euler scheme
论文作者
论文摘要
我们考虑在单方面Lipschitz的连续性条件下，采用了由加性噪声驱动的一类抛物线半连续性部分微分方程的显式驯服指数欧拉方案的长期行为。该设置包含非线性以及多项式生长。首先，我们证明了数值方案的矩界限，最多与时间范围相对于多项式依赖性。其次，我们将此结果应用于弱意义上的误差估计，就时间步长和时间范围而言，将误差量化以相对于连续时间过程的不变分布的近似平均值。我们证明，使用显式驯服的指数欧拉方案近似不变分布的效率是合理的，因为计算成本不受当前界限的最多多项式生长的影响。据我们所知，这是文献中有关使用显式驯服方案使用非全球Lipschitz系数的SPDE的不变分布近似的结果。
We consider the long-time behavior of an explicit tamed exponential Euler scheme applied to a class of parabolic semilinear stochastic partial differential equations driven by additive noise, under a one-sided Lipschitz continuity condition. The setting encompasses nonlinearities with polynomial growth. First, we prove that moment bounds for the numerical scheme hold, with at most polynomial dependence with respect to the time horizon. Second, we apply this result to obtain error estimates, in the weak sense, in terms of the time-step size and of the time horizon, to quantify the error to approximate averages with respect to the invariant distribution of the continuous-time process. We justify the efficiency of using the explicit tamed exponential Euler scheme to approximate the invariant distribution, since the computational cost does not suffer from the at most polynomial growth of the moment bounds. To the best of our knowledge, this is the first result in the literature concerning the approximation of the invariant distribution for SPDEs with non-globally Lipschitz coefficients using an explicit tamed scheme.
