 \sum_{\ell\in G_k^\ast} [P_n^{s-0}]_{i\ell} \cdot [P_{n,k}]_{\ell j}\cdot {\mathds{P}}(Y_{s}\in G_k^\ast \,|\, Y_{s-0} =\ell)\\ & \ \ \ \ + \sum_{\ell\not\in G_k^\ast}  {\mathds{P}}(Y_{s} = j\,|\, Y_{s-0} =\ell