Wave-induced transport can produce diffusion both in coordinate space and in velocity space with transport rates which are much closer than the corresponding rates for classical transport. In such a case the quasi-linear diffusion in velocity space should be solved in conjunction with the diffusion in coordinate space. This leads to processes hitherto unknown, such as the generation of a large number of accelerated particles near the plasma-vacuum boundary. As an example, the theory of a double diffusion caused by kinetic Alfv¿n wave turbulence is presented, and generation of ion beams is discussed. ¿ American Geophysical Union 1987