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I am quite confident about this, but do have my doubts, as I have failed to find proof in the literature in the past hour; so would very much like some clarification. My understanding is that if the process $Y_t$ is a Markov process to which time-varying constants are added (say $c_t$), then the process $\widetilde{Y}_t=Y_t+c_t$ remains a Markov process and is invariant to these time-varying constants. I am sure I have come across this in the past and was looking at some notes on ergodic theory to find a rigorous proof, but was unable to accomplish this. If someone could elaborate on this, I would be ever so grateful.

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