Definition of a Markov process

Question

I found 2 Definitions for a Markov process and I am trying to understand how they are connected.

1. Let $X=\left(X_t\right)_{t\geq 0}$ be a $\mathcal{F}_t$ adapted Process. We say $X$ is a Markov process, if for all bounded and measurable $f$ and $t\geq s$ \begin{equation} \mathbb{E}\left[f\left(X_t\right)|\mathcal{F}_s\right]= \mathbb{E}\left[f\left(X_t\right)|X_s\right] \end{equation} holds.
2. Let $Q=\left(Q_t\right)_{t\geq 0} $ a transition Semigroup. Let $X=\left(X_t\right)_{t\geq 0}$ be a $\mathcal{F}_t$ adapted Process. We say X is a Markov process if for every bounded and measurable function $f$ \begin{equation} \mathbb{E}\left[f\left(X_{s+t}\right)|\mathcal{F}_s\right]=\int Q_t\left(X_s,dy\right)f\left(y\right) \end{equation} holds.

How is the relation between these Definitions ? Is one of them more general ? And how would I construct the transition semigroup for a Levy Process ?

Answer 1

Obviously, the first definition is the most general one. It says that, given the present, the future and the past are independent. In particular, it contains inhomogeneous and homogeneous Markov processes. And sometimes, Markov processes don't even possess transition kernels like the second definition unless the state spaces are nice.

The second definition is for homogeneous Markov processes. In this case, it is still true that "given the present, the future and the past are independent". But there is more to this: if you are at the present, say at time $s$, and you look into the future at time $t+s$, then $X_{t+s}$ given $X_s$ is similar to see $X_t$ given $X_0$. Most of the times, probabilists prefer to work with this definition because it is nicer and general enough for their investigations.

The last question on Levy Process should be posted in a separate post.