Action diffusion for symplectic maps with a noisy linear frequency

We consider an area preserving map in the neighbourhood of an elliptic fixed point, whose linear frequency is stochastically perturbed. The nonlinearity couples the random motion in the phase with the action which exhibits a diffusive behaviour. If the unperturbed dynamics is almost integrable and no macroscopic resonant structures are present in the phase space, a Fokker-Planck equation for the action diffusion is derived and its solution shows an excellent agreement with the simulation of the process. The key points are the description of the unperturbed motion by using the normal forms and the derivation of a stochastically perturbed interpolating Hamiltonian for which the action diffusion coefficient is analytically computed. The angle averaging is justified by the much faster time scale on which the angle relaxes to a uniform distribution.