Convergence rate analysis of time discretization scheme for confined Lagrangian processes

In this paper, we propose and analyze the convergence of a time-discretization scheme for the motion of a particle when its instantaneous velocity is drifted by the known velocity of the carrying flow, and when the motion is taking into account the collision event with a boundary wall. We propose a symetrized version of the Euler scheme and prove a convergence of order one for the weak error. The regularity analysis of the associated Kolmogorov PDE is obtained by mixed variational and stochastic flow techniques for PDE problem with specular condition