An averaging approach to the Smoluchowski-Kramers approximation in the presence of a varying magnetic field

We study the small mass limit of the equation describing planar motion of a charged particle of a small mass µ in a force field, containing a magnetic component, perturbed by a stochastic term. We regularize the problem by adding a small friction of intensity ǫ > 0. We show that for all small but fixed frictions the small mass limit of q_µ,ǫ gives the solution q_ǫ to a stochastic first order equation, containing a noise-induced drift term. Then, by using a generalization of the classical averaging theorem for Hamiltonian systems by Freidlin and Wentzell, we take the limit of the slow component of the motion q_ǫ and we prove that it converges weakly to a Markov process on the graph obtained by identifying all points in the same connected components of the level sets of the magnetic field intensity function.