In this paper, we present numerical methods to implement the probabilistic representation of third kind (Robin) boundary problem for the Laplace equations. The solution is based on a Feynman-Kac formula for the Robin problem which employs the standard reflecting Brownian motion (SRBM) and its boundary local time arising from the Skorohod problem. By simulating SRBM paths through Brownian motion using Walk on Spheres (WOS) method, approximation of the boundary local time is obtained and the Feynman-Kac formula is calculated by evaluating the average of all path integrals over the boundary under a measure defined through the local time. Numerical results demonstrate the accuracy and efficiency of the proposed method for finding a local solution of the Laplace equations with Robin boundary conditions.
Numerical algorithms for the forward and backward fractional Feynman-Kac equations
School of Mathematics and Statistics, Lanzhou University（兰州大学数学与统计学院）；Department of Physics and Advanced Materials and Nanotechnology Institute, Bar-Ilan University（巴伊兰大学物理系及先进材料纳米技术研究所）
Numerical algorithms of the two-dimensional Feynman-Kac equation for reaction and diffusion processes
School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University（兰州大学数学与统计学院，甘肃省应用数学与复杂系统重点实验室）
Kernel Methods for Stochastic Dynamical Systems with Application to Koopman Eigenfunctions: Feynman-Kac Representations and RKHS Approximation
Department of Computing and Mathematical Sciences, Caltech（加州理工学院计算与数学科学系）；The Alan Turing Institute（艾伦·图灵研究所）；Department of Mechanical Engineering, Clemson University（克莱姆森大学机械工程系）
