Let $(X_t)_{t\in\mathbb{N}_0}$ be a stationary Markov process valued in $\mathbb{R}$ and $c\in\mathbb{R}$. Is the process $(Y_t)_{t\in\mathbb{N}_0}$ defined by $$ Y_t={\bf 1}{(X_t<c)} $$ again a Markov process (valued in $\{0,1\}$)? Here ${\bf 1}$ denotes the characteristic function, i.e. it is $1$ if $X_t<c$ and $0$ otherwise. The main example I'm interested in is $X_t$ being an $\rm{AR}(1)$ process, so I would also appreciate answers for this special case.
I tried, for example, to show that $$ \mathbb{P}(Y_t=0 \ \mid \ Y_{t-1}=0,\dots, Y_0=0)=\mathbb{P}(Y_t=0 \ \mid \ Y_{t-1}=0) $$ using the Markov property of $Y_t$ and marginalization, but I was not able to show the equality of the resulting integrals, which lead me to believe that $Y_t$ is actually not a Markov process.
I would really appreciate any help or a reference for these type of results/proofs/counterexamples.