Poissonian Actions of Polish Groups

The paper’s Theorem 2 proves that for Poissonian actions of Polish groups, ergodicity, weak mixing, and the absence of finite-measure invariant sets in the base are equivalent. The candidate solution establishes the same equivalence by a representation-theoretic route (first chaos intertwining and second quantization), whereas the paper proceeds via chaos-preserving projections and a Parreau–Roy theorem, plus a double-ergodicity argument to get weak mixing. The model’s argument is essentially correct; one step (stability of the absence of finite-dimensional subrepresentations under tensor/symmetric powers) should be explicitly justified, but does not affect the final equivalence.