 Solution

Number the vertices $A,B,C$ by $0,1,2$. The continuous-time Markov chain moves one step forward cyclically at rate one, as shown in the left diagram.

Write $P$ for the cyclic shift matrix, so $Q=P-I$. With $\omega =e^{2\pi i/3}$, the eigenvalues of $P$ are $1,\omega ,{\omega }^2$ and those of $Q$ areThe stationary distribution is uniform. A jump count is a rate-one Poisson process, and the displacement is that count modulo three. The roots-of-unity filter $1_{n\equiv d(3)}=\frac{1}{3}{\sum }_{j=0}^2{\omega }^{j(n-d)}$ and $E(s^{N_t})=e^{t(s-1)}$ therefore give, for $d=(y-x)\mathrm{mod}3$,These are also the entries of $e^{tQ}$ obtained by diagonalizing $Q$. In particular,