Restricting isotropic a-stable Levy processes from Rn to fractal sets

The restriction of an isotropic a-stable Levy process X-alpha in IRn on the fractal support F of certain finite Borel measures mu is introduced in three different ways: by means of the trace of its Dirichlet form on F and related Besov spaces, by its mu-local time on F and the corresponding time changed process, and by restricting on the fine support of mu via balayage spaces and potentials. Moreover, for compact upper d-regular sets F with d > n - alpha an approximation by means of restrictions of X-alpha to small epsilon-neighborhoods of F is given. 2000 Mathematics Subject Classification.