# Fourier transform of a random walk

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.

• 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. – Kernel Mar 21 at 12:50
• The idea is basically that the integral taken from $[-\pi,\pi]^d$ makes the Fourier basis orthogonal. – Kernel Mar 21 at 12:51