  In this paper, we study the statistical evolution of the large-scale
structure (LSS), focusing on the joint probability distribution function (PDF)
of the coarse-grained cosmic field and its role in constructing effective
dynamics. As the most comprehensive statistics, this PDF encodes all
cosmological information of large-scale modes, therefore, could serve as the
basis in the LSS modelling. Following the so-called PDF-based method from
turbulence, we write down this PDF's evolution equation, which describes the
probability conservation. We show that this conservation equation's
characteristic curves follow the same PDF history and could be considered as an
effective dynamics of the coarse-grained field. Unlike the EFT of LSS, which
conceptually would work at both realization and statistics level, this
effective dynamics is valid only statistically. However, this `statistical
equivalence' also provides valuable insight into scale interactions at the
statistical level. It also enables predicting a wide variety of statistics
beyond the typical N-point polyspectra, including, e.g. topologies, density PDF
and non-linear covariance matrices etc. Our formula expresses the small-scale
effect as the ensemble average of their interactions conditional on the
large-scale modes. This suggests an interesting way to measure effective terms
directly from simulation. By applying the Gram-Charlier expansion, we
demonstrate a different structure of these effective terms. This formalism is a
natural framework for discussing the evolution of statistical properties of
large-scale modes, and provides an alternative view for understanding the
relationship between general effective dynamics and standard perturbation
theory.
