Weak Convergence Rates for an Explicit Full-Discretization of Stochastic Allen–Cahn Equation with Additive Noise
Weak Convergence Rates for an Explicit Full-Discretization of Stochastic Allen–Cahn Equation with Additive Noise
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DOI：
10.1007/s10915-020-01378-8
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发表时间：
2019-11
影响因子：
2.5
通讯作者：
Meng Cai;S. Gan;Xiaojie Wang
中科院分区：
文献类型：
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作者：
Meng Cai;S. Gan;Xiaojie Wang
我们离散的随机Allen-Cahn方程加性噪声的谱Galerkin方法在空间和驯服版本的指数欧拉方法在时间。由此产生的误差界进行了分析的时空全离散在强和弱的意义。与现有的工作不同，我们开发了一种新的和直接的方法进行弱误差分析，它不依赖于使用相关联的Kolmogorov方程或伊藤公式，因此是非马尔可夫的性质。因此，这种方法有可能被应用到非马尔可夫方程，如随机沃尔泰拉方程或其他类型的分数SPDE，其遭受缺乏柯尔莫哥洛夫方程。结果表明，所得到的弱收敛速度，在空间和时间方向上，基本上是两倍的强收敛速度。此外，它揭示了如何弱收敛速度依赖于噪声的规律性。最后通过数值实验验证了理论分析的正确性。
We discretize the stochastic Allen–Cahn equation with additive noise by means of a spectral Galerkin method in space and a tamed version of the exponential Euler method in time. The resulting error bounds are analyzed for the spatio-temporal full discretization in both strong and weak senses. Different from existing works, we develop a new and direct approach for the weak error analysis, which does not rely on the use of the associated Kolmogorov equation or Itô’s formula and is therefore non-Markovian in nature. Such an approach thus has a potential to be applied to non-Markovian equations such as stochastic Volterra equations or other types of fractional SPDEs, which suffer from the lack of Kolmogorov equations. It turns out that the obtained weak convergence rates are, in both spatial and temporal direction, essentially twice as high as the strong convergence rates. Also, it is revealed how the weak convergence rates depend on the regularity of the noise. Numerical experiments are finally reported to confirm the theoretical conclusion.