# Cross Covariance Matrix of Multidimensional Ornstein-Uhlenbeck Processes

The multivariate Ornstein–Uhlenbeck process is defined as the following $$$$dX(t) = - I_p X(t) dt + \sqrt{2}I_p dW(t)$$$$ where $$I_p$$ is an $$p \times p$$ identity matrix, and $$W(t)$$ is an $$p$$-dimensional standard Wiener process. Let $$X(0)$$ be standard $$p$$-dimensional Gaussian.

I am looking for the unconditional cross covariance matrix $$\text{E}(X(0)X(t))$$. Could anyone please help me with this? As of now, I have only found cross-covariance matrix conditioning on $$X(0)$$, $$\text{E}(X(s)X(t)|X(0)=x)$$, on Page 16 in "Review of Statistical Arbitrage, Cointegration, and Multivariate Ornstein-Uhlenbeck" or in Theorem 2 in "Alternative way to derive the distribution of the multivariate Ornstein–Uhlenbeck process".

I am not sure if unconditional cross covariance follows from these. In Theorem 2, $$\text{E}(X(s)X(t)|X(0)=x)$$ gets $$0$$ if one of $$s$$ or $$t$$ is $$0$$. "Mathematical properties" in Ornstein–Uhlenbeck process Wikipedia gives the solution for the one-dimensional case. Thanks a lot!