Several questions, but related:
- How do you prove that a stationary process defined by a SDE is also ergodic?
- Anywhere where I can find any reference examples (for simplicity, the OU process) of the proof?
- Could you provide some reference (book/paper/notes) that links stochastic processes (with the Ito formalism) with ergodic theory?
It impossible to reply to your first question in such generality!
Overall there is really nothing special about a SDE from the point of view of ergodic theory: its solutions define a measurable cocycle over a flow (people in the "West" would use the expression "measurable sew-product flow") and what you are asking has to do with the ergodicity of the flow on the base.
The reason why I say that it is impossible to reply to your first question in such generality is that the same happens in ergodic theory, unless some specifics of the particular system can help. Together with references there you will be able to find the better points, but really the theory is huge.
My best recommendation for a general source is Ludwig Arnold's book Random Dynamical Systems (forget about the part on the multiplicative ergodic theorem, he got carried away with 120 pages when it is possible to make it much simpler). You will be able to read about what I say above about measurable cocycles and some very basic discussion of ergodic theory.