Thouvenot's Isomorphism Problem for Tensor Powers of Ergodic Flows

Let $S$ and $T$ be automorphisms of a probability space whose powers $S \otimes S$ and $T \otimes T$ isomorphic. Are the automorphisms $S$ and $T$ isomorphic? This question of Thouvenot is well known in ergodic theory. We answer this question and generalize a result of Kulaga concerning isomorphism in the case of flows. We show that if weakly mixing flows $S_t \otimes S_t$ and $T_t \otimes T_t$ are isomorphic, then so are the flows $S_t$ and $T_t$, provided that one of them has a weak integral limit.