论文标题
随机allen-cahn方程的弱误差分析
Weak error analysis for the stochastic Allen-Cahn equation
论文作者
论文摘要
我们证明了强大的率。弱点$ {\ MATHCAL O}（τ）$用于与随机allen-cahn方程的结构保持时间离散化（带有$τ$的$τ$）。 $ d = 1,2,3 $尺寸中的乘法有色噪声。直接的变分参数利用了第一个环境中立方非线性的单方面Lipschitz特性，以解决一阶强率。它是相同的属性，可以在相关的kolmogorov方程的解决方案的衍生物中达到统一的边界，然后在存在乘法噪声的情况下导致弱速率$ {\ Mathcal O}（τ）$。因此，我们获得了在存在乘法噪声的情况下以强误差而闻名的收敛速率的两倍。
We prove strong rate resp. weak rate ${\mathcal O}(τ)$ for a structure preserving temporal discretization (with $τ$ the step size) of the stochastic Allen-Cahn equation with additive resp. multiplicative colored noise in $d=1,2,3$ dimensions. Direct variational arguments exploit the one-sided Lipschitz property of the cubic nonlinearity in the first setting to settle first order strong rate. It is the same property which allows for uniform bounds for the derivatives of the solution of the related Kolmogorov equation, and then leads to weak rate ${\mathcal O}(τ)$ in the presence of multiplicative noise. Hence, we obtain twice the rate of convergence known for the strong error in the presence of multiplicative noise.
