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

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.

$$\def\Γ{{\mit Γ}}$$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.
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 $$\Gamma$$.