I have a linear stochastic differential equation: $$ dx = Axdt+dv $$ for some constant matrix $A$. Moreover $x(t_0)$ is normally distributed with mean $m_0$ and covariance $P_0$. $v(t)$ is a Wiener process with incremental covariance $Qdt$. If I have a sequence of instants $t_0<t_1<t_2<\dots$ with $t_{k+1}-t_{k}=h$, it is known that one can write $$ x(t_{k+1}) = \exp(Ah)x(t_k) + v_k $$ where $v_k$ is normally distributed with zero mean and covariance $$ Q_h = \int_0^h\exp(Ah)Q\exp(A^Th)dt $$
My question is: are the random variables $v_1,v_2,\dots$ independent from each other?
EDIT Another way to write $v_k$ is as $$ v_k = \int_{t_{k}}^{t_{k+1}}\exp(A(t_{k+1}-\tau))dv(\tau) $$ following equation (19) here.