Recurrence-transience for self-similar additive processes associated with stable distributions

For any stable distribution mu on the line, recurrence-transience of the se lfsimilar additive process {X-t, t greater than or equal to0} with L(X-1)=m u is determined. Comparison with the stable Levy process {Y-t, t greater th an or equal to0} with L(Y-1)=mu is made: if mu is not strictly stable, then {Y-t} is transient but {X-t} is recurrent except the obviously transient c ase of monotone sample functions.