If $X_t = (X^i_t)_{i \leq n}\in \mathbb{R}^n$ solves an SDE of the form
$$dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dB_t$$
Does it hold that each of the real valued random variables $X^i_t$ and $X^j_t$ are independent for each $t$ provided $i \neq j$? It's true for Brownian motion, and I'm mainly interested in the higher dimensional Ornstein-Uhlenbeck process, but I was curious if it just works in general.