This is an important result too. The denseness of the complex-diagonalizable matrices (in either the space or real square matrices or complex square matrices) gives the easiest proof of the the Cayley–Hamilton Theorem (since the the theorem is almost trivial for the diagonalizable set).

What do you mean by "coupling"? Because your condition on $\operatorname{Pr}(X_t\neq Y_t)$ doesn't seem to make sense; that probability seems like it is unavoidably going to be 1.

The quoted section says that that is the definition of the nonlocal operator $G(-\hbar^{2}\Delta)$; it is defined by its action in Fourier space, and that is standard in mathematical physics. I suppose that you would want to show that, e.g. $\Delta\left[\left(\frac{1}{\Delta}\right)\psi\right]=\psi$, but that is completely straightforward.