I'm a beginner of Markov processes and I have some basic questions. Consider two sequences of real-valued random variables $\{X_t\}_t, \{Y_t\}_t$ where $t$ is a discrete time index, $t=0,1,...$, all defined on the same probability space $(\Omega, \mathcal{F},\mathbb{P})$.
Assume $X_t,Y_t$ follow a Markov Process i.e. $$\mathbb{P}(X_{t+1},Y_{t+1}|\{X_s,Y_s\}_{s=0}^t)=\mathbb{P}(X_{t+1},Y_{t+1}|X_t, Y_t)\hspace{3cm} (\star)$$
(1) I believe that stating that all random variables are defined on the same probability space does not imply that the Markov Chain is homogeneous (i.e. $\mathbb{P}(X_{t+1},Y_{t+1}|X_t, Y_t)$ invariant over $t$). Is this correct?
(2) Does $(\star)$ imply $\mathbb{P}(X_{t+1}|\{X_s\}_{s=0}^t)=\mathbb{P}(X_{t+1}|X_t) $ and $\mathbb{P}(Y_{t+1}|\{Y_s\}_{s=0}^t)=\mathbb{P}(Y_{t+1}|Y_t) $, i.e. also the marginal processes are Markov?