An Introduction to Stochastic Processes with Applications to Biology.
Linda Allen
Consider a random walk on the vertices of a triangle. (a) let the moves be from one vertex to another with prob 1/2 (p12=p21=p13=p23=p31=p32=1/2) Find the prob that in n steps wou return to the vertex you started from; is every state recurrent? ; if so, compute the stationary distribution. (b) Same thing but now, p12=p23=p31=2/3, p21=p32=p13=1/3. (Hint: The transition matrix P is a circulant matrix, so it can be diagonalized by the matrix S:
S = (1/sqrt(3)) [1 1 1; 1 w wbar;1 wbar,w]
where w=exp(2 pi i/3). Note that S^(-1)=S^*. P^n = S D^n S^* where D is the matrix of eigenvalues.)