  The dynamics of a Markov process are often specified by its infinitesimal
generator or, equivalently, its symbol. This paper contains examples of
analytic symbols which do not determine the law of the corresponding Markov
process uniquely. These examples also show that the law of a polynomial process
is not necessarily determined by its generator. On the other hand, we show that
a combination of smoothness of the symbol and ellipticity warrants uniqueness
in law. The proof of this result is based on proving stability of univariate
marginals relative to some properly chosen distance.
