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.

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    $\begingroup$ I believe that you will find Yvan's Velenik Book's Appendix B very helpful: unige.ch/math/folks/velenik/smbook/Mathematical_Appendix.pdf on page 513, they address that. $\endgroup$ – Kernel Mar 21 at 12:50
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    $\begingroup$ The idea is basically that the integral taken from $[-\pi,\pi]^d$ makes the Fourier basis orthogonal. $\endgroup$ – Kernel Mar 21 at 12:51

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