We study the asymptotic behavior of the tail probability of integrated stable processes exceeding power barriers. In the first part of the paper the limiting behavior of the integrals of stable processes generated by ergodic dissipative flows is established. In the second part an example with the integral of a stable process generated by a conservative flow is analyzed. Finally, the difference in the order of magnitude of the exceedance probability in the two cases is related to the dependence structure of the underlying stable process.
