For $x,y \in R^d$ let $p_t(x,y)=(2\pi t)^{-d/2} e^{-\frac{|x-y|^2}{2t}}$. Prove that $P_t(x,dy)=p_t(x,dy)$ satisfies the Chapman Kolmogorov equation. Below is the solution to this problem. My questions are : How do we come up with the below decomposition of the squares? I can't see the intuition behind coming up with such a completion of squares.
Second, what justifies the second equality at the end? When we make the change of variable to $\tilde{y}$, $\tilde{y}$ depends on $z$ so how is it fine to separate the iterated integral into two different integrals? I would greatly appreciate any help.
They are simply completing the square with respect to the $y$ variable. Nothing more, nothing less.