On the error of stochastic solutions of Boltzmann-type equations: Sharp upper bounds

Abstract

Boltzmann-type nonlinear equations, which describe the time evolution of a system involving many “particles” and pair interactions, are model equations in many areas of natural science, such as the dynamics of rarefied gases, coagulation theory, quantum physics, etc. In many cases, the only method for solving such equations is a Monte Carlo method simulating, to some extent, the corresponding physical process. At the same time, the question on the error of this method in particular situations cannot be qualified as completely answered. The paper considers homogeneous Boltzmann-type equations with constant scattering cross-section and one of the stochastic methods for solving them by using so-called (n, 1)-particle random processes. For the solution error the distance between the corresponding distributions in the variation norm is taken. The main result is a bound of a special form for the error, which is sharp in the class of all Boltzmann-type equations under consideration. In other words, an upper bound for the error is obtained which is valid for all equations under consideration and sharp for at least one of them.