in regular discrete-time (1 dimensional) kalman filter, it is assumed that we have white gaussian noise affecting the transitions and the observations:
$x(t+1) = Ax + w$
$y(t) = Cx(t) + v$
assuming $w$ follows Gaussian(0, $\sigma_1$) and $v$ follows Gaussian(0, $\sigma_2$) distributions.
will the standard kalman filter not work or not be optimal if the gaussian noise variables are time varying, meaning $w$ and $v$ become $w(t)$ and $v(t)$ and somehow change with time, becoming systematically bigger or smaller?
related: the kalman filter generally has the behavior that its variance is decreasing with time. will that be true even if $w(t)$ and $v(t)$ are are time-varying even if they are assumed not to be? will the kalman filter's variance still get smaller even if the data are sampled from time-varying noise process?