An integral criterion for the existence of an invariant measure of an Itô process is developed. This new criterion is based on the probabilistic symbol of the Itô process. In contrast to the standard integral criterion for invariant measures of Markov processes based on the generator, no test functions and hence no information on the domain of the generator is needed.
"A criterion for invariant measures of Itô processes based on the symbol." Bernoulli 21 (3) 1697 - 1718, August 2015. https://doi.org/10.3150/14-BEJ618