For a Markov process $X_t$ derive the Chapman Kolmogorov Equation $$P(X_{t_3} | X_{t_1}) = \int_{-\infty}^{\infty} P(X_{t_3} | X_{t_2} = x) P(X_{t_2} = x | X_{t_1}) dx$$ with $t_3 > t_2 > t_1$
From my lecture notes, I have the following which I struggling to understand (and I might have it incorrect):
Note that $$P(X_{t_3}, X_{t_2}, X_{t_1}) = P(X_{t_3}| X_{t_2})P(X_{t_2}| X_{t_1})P(X_{t_1})$$ Then integrate over $X_2$ to get $$P(X_{t_3}, X_{t_1}) = P(X_{t_3} | X_{t_1})P(X_{t_1})$$ $$= \int dx \hspace{1mm} P(X_{t_3} | X_{t_2} = x)P(X_{t_2} = x | X_{t_1})$$
Now it is the last line I am not sure about. Where has the $P(X_1)$ gone? Maybe I omitted it from my notes at the time of writing? However, in that case, this would not align with the identity.