This paper presents a stochastic approach to the clustering evolution of dark
matter haloes in the Universe. Haloes, identified by a Press-Schechter-type
algorithm in Lagrangian space, are described in terms of `counting fields',
acting as non-linear operators on the underlying Gaussian density fluctuations.
By ensemble averaging these counting fields, the standard Press-Schechter mass
function as well as analytic expressions for the halo correlation function and
corresponding bias factors of linear theory are obtained, thereby extending the
recent results by Mo and White. The non-linear evolution of our halo population
is then followed by solving the continuity equation, under the sole hypothesis
that haloes move by the action of gravity. This leads to an exact and general
formula for the bias field of dark matter haloes, defined as the local ratio
between their number density contrast and the mass density fluctuation. Besides
being a function of position and `observation' redshift, this random field
depends upon the mass and formation epoch of the objects and is both non-linear
and non-local. The latter features are expected to leave a detectable imprint
on the spatial clustering of galaxies, as described, for instance, by
statistics like bispectrum and skewness. Our algorithm may have several
interesting applications, among which the possibility of generating mock halo
catalogues from low-resolution N-body simulations.Comment: 23 pages, LaTeX (included psfig.tex), 4 figures. Few comments and
references have been added, and minor typos and errors corrected. This
version matches the refereed one, in press in MNRA
