AI 中文总结
本文提出一种蒙特卡洛方法，通过欧拉方案近似被杀死布朗运动，求解有界域上时间非局部扩散问题，并给出误差界与数值验证。
AI 中文摘要
我们开发并分析了一种蒙特卡洛方法，用于采样被杀死的时间异常扩散过程，该过程通过将带漂移的布朗运动按一个子排序器的逆进行时间变换而获得。该方法针对有界域上的时间非局部（包括时间分数阶）柯西-狄利克雷问题的概率表示。由于逆子排序器在广泛的类别中可以精确采样，而布朗退出时间在任意域中通常不可用，我们通过带有离散边界检测的欧拉方案来逼近被杀死的布朗分量。我们证明了平方根弱误差界，该误差界显式依赖于子排序器的拉普拉斯指数，并推导了所得蒙特卡洛估计量的均方和中心极限结果。一个在圆盘中的数值示例和一个在高维各向异性壳中的数值示例说明了理论速率、计算时间以及该方法的无网格特性。
英文摘要
We develop and analyze a Monte Carlo method for sampling killed anomalous diffusions obtained by time-changing Brownian motion with drift by the inverse of a subordinator. The method targets probabilistic representations of time-nonlocal, including time-fractional, Cauchy--Dirichlet problems on bounded domains. Since inverse subordinators can be sampled exactly in broad classes, while Brownian exit times are generally unavailable in arbitrary domains, we approximate the killed Brownian component by an Euler scheme with discrete boundary detection. We prove a square-root weak error bound with explicit dependence on the Laplace exponent of the subordinator, and derive mean-square and central limit results for the resulting Monte Carlo estimator. A numerical example in the disk and one in a high-dimensional anisotropic shell illustrate the theoretical rates, computation time, and the mesh-free character of the method.
