Nonlinear Monte Carlo Estimators for Parabolic Equation

We consider the parabolic type equation with a source-sink term and construct the Monte Carlo estimator for it. The procedure is based on the Hopf-Cole transformation and a Monte Carlo estimator for the correspondent Burgers equation. Monte Carlo estimators for the heat and diffusion equations are very well known since the first steps of the method. The algorithms are based usually on the intimate relation between the diffusion processes and partial differential equations of the parabolic type. The natural sciences problems that may be described by such equations are numerous, and the diversity of coefficients’ properties, domains and boundaries, boundary value problems, etc. makes it necessary to take into account the individual features of the problem in every particular case. Thus, random walk on spheres, random walk on boundary, different versions of random walks on the grid and Lagrangian algorithms simulating real diffusing particles may be applied, depending on the problem under consideration. In this paper we consider u(x, t) to be a solution of the (onedimensional for simplicity) parabolic type equation