6 ms·
If you find it GP, I'd love to see where the integral constructed comes from, since that's the clever part rather than the evaluation.
by Ntrails 11y ago
If you find it GP, I'd love to see where the integral constructed comes from, since that's the clever part rather than the evaluation.
- emfree 11y agoHere's a reference I found for one way to do it: http://www.math.nus.edu.sg/~matsr/ProbII/Lec6.pdf http://www.math.nus.edu.sg/~matsr/ProbII/Lec6.pdf (Theorem 2.1). You define the Green's function G(x, y) = \sum_n Pr_x(S_n=y), where x and y are 3-vectors and Pr_x(S_n=y) is the probability that an n-step random walk starting at x ends up at y. If you have an infinite random walk starting at 0, then G(0, 0) is the expected number of times that the walk returns to 0. That's what the mathworld link calls u(3). You can use Fourier inversion to compute G(0, 0) -- the link gives the gnarly details. It's pretty cool.
- Ntrails 11y agoYou're a scholar and a gentleman, merci buckets