Decomposition of self-similar stable mixed moving averages

 Let α? (1,2) and X _α be a symmetric α-stable (S α S) process with stationary increments given by the mixed moving average where is a standard Lebesgue space, is some measurable function and M _α is a SαS random measure on X ×ℝ with the control measure m _α (dx, du) = μ(dx)du. We show that if X _α is self-similar, then it is determined by a nonsingular flow, a related cocycle and a semi-additive functional. By using the Hopf decomposition of the flow into its dissipative and conservative components, we establish a unique decomposition in distribution of X _α into two independent processes where the process X _α ^D is determined by a nonsingular dissipative flow and the process X _α ^C is determined by a nonsingular conservative flow. In this decomposition, the linear fractional stable motion, for example, is determined by a conservative flow. Received: 20 June 2000 / Revised version: 6 September 2001 / Published online: 14 June 2002