A polynomial preserving process (PPP) is a strong Markov process whose (extended) generator maps polynomials to polynomials of at most the same degree. They somewhat generalise the popular class of affine process. The key feature of PPP are the tractable formulas how polynomials are mapped to polynomials by their transition semigroups. One advantage of the PPP-class over affine processes is the greater flexibility. Additionally, it seemed to be believed that affine process, unlike PPP, with compact state space are deterministic -- which, however, is not the case. In this talk we classify all affine processes with compact state spaces.
If the state space is compact, then the power moments of the marginals determine the law of the PPP. In case of unbounded state spaces it is still unknown which properties a PPP needs in addition to the polynomial structure -- i.e. how polynomials transition to polynomials -- so that those determine the law of the PPP. In this talk we provide examples that show that the law of a PPP on the real line can fail to be determined by their polynomial structure.