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In this paper, we propose numerical methods for computing the boundary localtime of reflecting Brownian motion (RBM) in R3 and its use in the probabilisticrepresentation of the solution of the Laplace equation with the Neumannboundary condition. Approximations of the RBM based on a walk-on-spheres (WOS)and random walk on lattices are discussed and tested for sampling the RBM pathsand their applicability in finding accurate approximation of the local time anddiscretization of the probabilistic formula. Numerical tests for several typesof domains (cube, sphere, and ellipsoid) have shown the convergence of thenumerical methods as the length of the RBM path and number of paths sampledincrease.

Computation of Local Time of Reflecting Brownian Motion and Probabilistic Representation of the Neumann Problem