A Monte Carlo solution at heat conduction and Poisson equations

A Monte Carlo method is developed for solving the heat conduction. Poisson, and Laplace equations. The method is based on properties of Brownian motion and Ito processes, the Ito formula for differentiable functions of these processes, and the similarities between the generator of Ito processes and the differential operators of these equations. The proposed method is similar to current Monte Carlo solutions, such as the fixed random walk, exodus, and floating walk methods, in the sense that it is local, that is, it determines the solution at a single point or a small set of points of the domain of definition of the heat conduction equation directly. However, the proposed and the current Monte Carlo solutions are based on different theoretical considerations. The proposed Monte Carlo method has some attractive features. The method does not require to discretize the domain of definition of the differential equation, can be applied to domains of any dimension and geometry, works for both Dirichlet and Neumann boundary conditions, and provides simple solutions for the steady-state and transient heat equations. Several examples are presented to illustrate the application of the proposed method and demonstrate its accuracy.