Let $(\Omega,F,(F_t)_{t\geq 0},P)$ be a filtered probability space with the standard condition.
Let $W_t$ be the $F_t$-Wiener process, and
let $(X_t)_{t\geq 0}\subset {\mathbb{R}}$ be the (strong) solution of the SDE
$$
X_t = X_0 + \int _0^t b(X_s,s)ds + \int_0^t \sigma(X_s,s)dW_s,
$$
where the initial condition $X_0$ is $F_0$-measurable.
Is it true that $X_t$ ($t>0$) and $X_0$ are independent? The argument in Independence of Solution of SDE $S^{(x_0, \sigma, \mu)}_t$ of Initial Information $\mathcal{G}_0$ seems to say that they are.