3.9. Brownian motion on a ring Consider a Brownian motion on a ring of N sites, with a transition rate to next-neighbour sites equal to 1/2. Let Pn(s) be the probability to find the walker at the site s at the time n

(s = 1, 2, . . . ,N). Show that ifN is an odd number, there is a unique stationary probability distribution n→∞. Vice versa, if N is an even number, for n→∞, the distribution probability can oscillate between two different probability distributions.