Ornstein Uhlenbeck with drift
Hi: This is a statistics viewpoint and not a DSP one. I just say that because there are termimology issues that have caused confusion with my answers in the past.  The autocorrelation is only defined for constant mean processes. A process with drift doesn't have a constant mean so you would need to detrend the process first which would bring you back to the zero mean case. One way to do that would be run a regression with time as your "x", in order to estimate $\alpha$. Then, once you have the estimate, you can subtract the trend off. I'm not sure if this is the answer you wanted to hear but the trend makes it not possible to estimate correlations because they can't be calculated because the mean of the process is not constant.