# Markov property definition, for (continuous) markov processes

We have defined markov processes (in continuous time) as a collection of random variables $$(X(t))_{t\in \mathbb{R_+}}$$ such that in particular we have the property :

$$P(X(t+s)=j|X(u), 0\leq u \leq t)=P(X(t+s)=j|X(t))$$

So my question is : is this really the conditional probability on a random variable, $$P(A|X)$$ (I know this concept exists). Or does it really mean

$$P(X(t+s)=j|X(t)=i, X(u)=x(u), 0\leq u < t)=P(X(t+s)=j|X(t)=i)$$?

The process $$X$$ is called a (continuou-time) Markov chain if it satisfies the Markov property: $$P(X(t_n)=j|X(t_1)=i_1,\cdots, X(t_{n-1})=i_{n-1})= P(X(t_n)=j|X(t_{n-1})=t_{n-1})$$ for all $$j,i_1,\cdots, i_{n-1} \in S$$ and any sequence $$t_1 of times.