Let us consider a controlled Markov process with the transition kernel $p(dy|x,\theta)$ ($\theta$ being the control parameter. Now, consider the map $\theta \to I(\theta)$ where $I(\theta)$ is the space of invariant probability measures for of the Markov process when we fix the control to $\theta$. I am interested in the properties of the map. If for every $x$, the map $\theta \to p(dy|x,\theta)$ is continuous and the Markov process takes value in a comapct metric space, it is easy to check that $\theta \to I(\theta)$ is set valued upper semi continuous map.
I am interested in the properties of the map when the Markov process takes value in more general, say Polish space. Are there standard results already ?
