  We present a general framework to treat the evolution of one-point
probability distribution function (PDF) for cosmic density $\delta$ and
velocity-divergence fields $\theta$. In particular, we derive an evolution
equation for the one-point PDFs and consider the stochastic nature associated
with these quantities. Under the local approximation that the evolution of
cosmic fluid fields can be characterized by the Lagrangian local dynamics with
finite degrees of freedom, evolution equation for PDFs becomes a closed form
and consistent formal solutions are constructed. Adopting this local
approximation, we explicitly evaluate the one-point PDFs $P(\delta)$ and
$P(\theta)$ from the spherical and the ellipsoidal collapse models as the
representative Lagrangian local dynamics. In a Gaussian initial condition,
while the local density PDF from the ellipsoidal model almost coincides with
the that of the spherical model, differences between spherical and ellipsoidal
collapse model are found in the velocity-divergence PDF. Importantly, the joint
PDF of local density, $P(\delta,t;\delta',t')$, evaluated at the same
Lagrangian position but at the different times $t$ and $t'$ from the
ellipsoidal collapse model exhibits a large amount of scatter. The mean
relation between $\delta$ and $\delta'$ does fail to match the one-to-one
mapping obtained from spherical collapse model. Moreover, the joint PDF
$P(\delta;\theta)$ from the ellipsoidal collapse model shows a similar
stochastic feature, both of which are indeed consistent with the recent result
from N-body simulations.
