I was just reading Statistical Field Theory of Itzykson and Drouffe and saw that they wrote the inverse Fourier transform $$P(\vec{x},t,\vec{x}_0,t_0)=\int_{[-\pi,\pi]^d}\frac{\text{d}^d\vec{k}}{(2\pi)^d}e^{i\vec{k}\cdot\vec{x}}\tilde{P}(\vec{k},t).$$ In here $P(\vec{x},t,\vec{x}_0,t_0)$ is the probability of a random walker to be at the lattice site $\vec{x}$ at time $t$ if it started from $\vec{x}_0$ at time $t_0$. Why isn't the integral taken over all of $\mathbb{R}^d$? I am not sure what the assumptions are but I believe that they've taken a continuum limit. Moreover, I think that $P(\vec{x},t,\vec{x_0},t_0)$ only dependes on $\vec{x}-\vec{x}_0$. What is the analogous thing on a uniform lattice?
BTW: I am asking in the mathstack exchange becaus I believe that a mathematician's perspective may be more illuminating.