We develop walk-on-sphere for fractional Poisson equations with Dirichilet boundary conditions in high dimensions. The walk-on-sphere method is based on probabilistic represen tation of the fractional Poisson equation. We propose effcient quadrature rules to evaluate integral representation in the ball and apply rejection sampling method to drawing from the computed probabilities in general domains. Moreover, we provide an estimate of the number of walks in the mean value for the method when the domain is a ball. We show that the number of walks is increasing in the fractional order and the distance of the starting point to the origin. We also give the relationship between the Green function of fractional Laplace equation and that of the classical Laplace equation. Numerical results for problems in 2-10 dimensions verify our theory and the effciency of the modified walk-on-sphere method.

翻译：我们开发了具有高维分偏差边界条件的Poisson单方程式。 以分偏差线对视方方程式的概率反射法为基础。 我们提出精准的二次曲线规则, 以评价球的整体代表性, 并应用拒绝采样法从一般域的计算概率中提取。 此外, 我们提供了当域为球时该方法的平均值中行走次数的估计值。 我们显示行走次数在分位顺序和起点距离上正在增加。 我们还给出了分差拉普尔方程式的绿色功能与古典拉普尔方程式之间的关系。 2- 10 度问题的数字结果验证了我们的理论以及修改的行走方式的有效性 。