Definition of a Markov process: What does $\mathbb P_x\{X_u\in \Gamma \mid \mathcal F_t\}=p(u-t,X_t,\Gamma)$ mean?

Question

I am reading the book Random perturbation of dynamical sustem of Freidlin and Wantzell (2nd edition). On page 20, they define a Markov process as follow:

Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space and $(X,\mathcal B)$ the state space. Let $(\mathcal F_t)$ a filtration. Let $(\mathbb P_x)_{x\in X}$ a familly of probability measure. Define the function $p$ as $$p(t,x,\Gamma )=\mathbb P_x\{X_t\in \Gamma \},\quad \Gamma \in \mathcal B, t\in [0,T],x\in X.$$ Then $X=(X_t)_{t\leq T}$ is a Markov process in $X$ if:
a) $X$ is adapted to the filtration.
b) $x\mapsto p(t,x,\Gamma )$ is measurable wrt $\mathcal B$.
c) $p(0,x, X\setminus \{x\})=0$.
d) $\mathbb P_x\{X_u\in Γ\mid \mathcal F_t\}=p(u-t,X_t,\Gamma )$ for all $t,u\in [0,T]$, $t\leq u$, $x\in X$ and $\Gamma \in \mathcal B$.

I am not sure how to interpret c) and d). Would these be correct?

Q1) For c), is it $\mathbb P_x\{X_0=x\}=1$?

Q2) For d), is it $$\mathbb P_{X_0=0}\{X_{t+h}\in \Gamma \mid X_t=k\}=\mathbb P_{X_0=k}\{X_h\in \Gamma \}?$$

But I don't really know how to interpret it.

Answer 1

$\def\Γ{{\mit Γ}}$For Q1, since$$ p(0, x, X \setminus \{x\}) = P_x(X_0 \in X \setminus \{x\}) = 1 - P_x(X_0 = x), $$ so $p(0, x, X \setminus \{x\}) = 0 \Leftrightarrow P_x(X_0 = x) = 1$. The process under $P_x$ can be regarded as starting from $x$.

For Q2, your identity is only a corollary of d) and not equivalent since $\mathscr{F}_t$ might be larger that $σ(X_t)$. To put d) in a clearer form, it is$$ P_x(X_u \in \Γ \mid \mathscr{F}_t) = P_{X_t}(X_{u - t} \in \Γ). $$ In other words, d) says that for any $0 \leqslant t < u$, if one knows the information of time $t$, which corresponds to expectation conditioning on $\mathscr{F}_t$, then the probability of an event in the future, i.e. $\{X_u \in \Γ\}$, with the process starting from $x$ is the same as the probability of $\{X_{u - t} \in \Γ\}$ with the process starting from $X_t$. This simply means that what happens before time $t$ does not matter to the evolution of the process as long as the information at time $t$, i.e. $\mathscr{F}_t$, is known.

Answer 2

To put it in simple words, Q1: You begin in state $x$ with probability 1.

Q2: (The Markov property: ) Given the entire history upto time $t$ (that's what $|\mathcal{F_t}$ means), the present sample $X_u$ is dependent only on the value of the most recent sample $X_t$.

Also in d: I think you wrote $\mathcal{G}$ instead of $\Gamma$.