I try to gain an intuition what a difference between Markov property, strong Markov property and stationarity of a random process is. I'm physicist with no strong background in stochastic theory. I have found many of the questions here concerning strong Markov property, but I still didn't get it. In particular I don't understand:
why the time-continuity play a role by Markov<=>strong Markov?
how the Markov process can at all not be stationary (for example by fixing the initial condition)?
By some examples of non-strong Markov but Markov processes for me it is a difference between stationary/not stationary process (what is false if I get it right).
I would be happy if somebody could explain me the matter.