Diffusion of the random Lorentz process in a magnetic field

Consider the motion of a charged, point particle moving in the complement of a Poisson distribution of hard sphere scatterers in two dimensions under the effect of a fixed magnetic field. Building on, and extending a coupling method established by the authors [Lutsko and Tóth, Commun. Math. Phys. 379, 589–632 (2020)], we show that this “magnetic Lorentz gas” satisfies an invariance principle in an intermediate scaling limit. That is, we apply the low-density (Boltzmann–Grad) limit and simultaneously take the limit as time goes to infinity, then prove convergence of the rescaled trajectory to a Brownian motion in this limit.