In a paper I'm reading it's written:
Let $(\zeta _t,P_{(i,j)})$ be the direct product process taking value in $\mathbb Z^d\times \mathbb Z^d$ of two copie of the Markov process $(\xi_t,P_i)$ which has probability transition $$P_{t}^{(2)}((i,j),(k,l))=P_t(i,k)P_t(j,l).$$ What does it mean ? That $$\zeta _t=\xi^1_t\xi^2_t$$ s.t. $\xi_i$ is a Markov process refer to $P_i$ ?