Say we have two i.i.d. continuous-time Markov chains $X(t)$ and $Y(t)$ whose state space is multidimensional and discrete. Also, $X(0) = Y(0)$. I am trying to understand the following: $$P(X(t) = Y(t)),\,t> 0$$ Or alternatively, $P(X(t) \neq Y(t))$. Is there are name for that probability? Any terms I should search for?
I thought that maybe I should consider the Markov chain $(X(t),Y(t))$. Then I could write down Kolmogorov's forward/backward equations and see where that takes me. However, this seems like a concept that has likely been well-studied.
Thanks.