Can I know if I have white noise given a conditional probability functional on the noise?

Question

If I have a SDE of the form $$ \dot{x} = f_1(x,z,t) + g_1(x,z,t)\eta(t) $$ $$ \dot{z} = f_2(x,z,t) + g_2(x,z,t)\eta(t). $$ And I know how to work out the probability of some noise realization given state parameters ($x(t),y(t)$) $$ P(\eta(t) | x(t),y(t)) = \exp \left(\int\frac{ \left(2 \eta(t) x(t) \sin \left(\text{$\theta$}(t)\right)+2 \eta(t) z(t) \cos \left(\text{$\theta $}(t)\right)-\eta(t)^2-1\right)}{2 } dt\right)$$ Where $\theta(t)$ is some time dependent parameter. Is there some way from here to work out if $\eta(t)$ is white noise? $$ < \eta(t) > = 0 $$ $$ < \eta(t)\eta(t') > = \delta(t - t')$$