Local Diffusion Properties of the Standard Map

We consider the diiusion problem for the standard map by examining the distribution functions both in the angle and the action. For small perturbations the diiusion occurs only in the presence of an external noise of small amplitude and it is well described by a local isochronous approximation which allows to compute analytically the diiusion coeecient. The angle diiusion is very fast since the correlation grows cubically rather than linearly with the iteration number. For large nonlinearities the phase space is no longer foliated with invariant curves and the analytic description fails. In this case there is diiusion without noise; the phase space has invariant stochastic regions together with KAM curves. Their presence is clearly reeected on the structure of the distribution function; the stochastic region gives an almost gaussian background, the invariant curves a sequence of sharp peaks.