In this note we show that, in the context of stochastic control systems, the uniform existence of a limit of Cesàro averages implies the existence of uniform limits for averages with respect to a wide class of measures dominated by the Lebesgue measure and satisfying some asymptotic condition. It gives a partial answer to the problem mentioned in Renault (2014) and it provides an alternative method for the approach in Li, Quincampoix, Renault (2015) (in the deterministic control setting). Finally, we mention that the arguments rely essentially on integration-by-parts and is applicable to general deterministic or stochastic control problems.
"A note on general Tauberian-type results for controlled stochastic dynamics." Electron. Commun. Probab. 20 1 - 12, 2015. https://doi.org/10.1214/ECP.v20-4142