=+13.8 Problems 353 17. Consider n equally spaced points on the boundary of a circle. Turing suggested a simple model for the diffusion of a morphogen, a chemical important in development, that involves the migration of the morphogen from point to point. In the stochastic version of his model, we follow a single morphogen particle. At any time the particle has the same intensity λ of jumping to the neighboring points on its right and left. Let pj (t) be the probability that the morphogen occupies point j at time t, where j = 0,...,n − 1. One can solve for these n probabilities using the finite Fourier transform on periodic sequences cj of period n.

(a) Show that p j (t) = λ[pj−1(t) − 2pj(t) + pj+1(t)].

(b) If a ∗ bk = n−1 j=0 ak−j bj denotes the convolution of two periodic sequences of period n, then express the differential equation in part

(a) as p j (t) = p(t) ∗ dj for p(t)=[pj (t)] and an appropriate sequence d = (dj ) of period n.