Here I am considering a simple random walk on the vertices of a cube: at each time, an ant jumps from one vertex to one of its neighbours, each with probably 1/3.
$v_0$ is a fixed vertex of the cube and the chain also starts here. We let $T_{v_0} = min\{n\geq 1 : X_n = v_0\}$ and $\mathbb{E}[T_{v_0} | X_0 = v_i] = m_i $
My goal here is to show:
for $i\neq0$ $$\mathbb{E}[T_{v_0}|X_1=v_i] = m_i + 1 $$
However, I don't know really where to start here. Any assistance would be greatly appreciated. Thanks in advance.