  We develop perturbation theory approaches to model the marked correlation
function constructed to up-weight low density regions of the Universe, which
might help distinguish modified gravity models from the standard cosmology
model based on general relativity. Working within Convolution Lagrangian
Perturbation Theory, we obtain that weighted correlation functions are
expressible as double convolution integrals that we approximate using a
combination of Eulerian and Lagrangian schemes. We find that different
approaches agree within 1$\%$ on quasi non-linear scales. Compared with {\it
N}-body simulations, the perturbation theory is found to provide accurate
predictions for the marked correlation function of dark matter fields, dark
matter halos as well as Halo Occupation Distribution galaxies down to $30$
Mpc/h. These analytic approaches help to understand the degeneracy between the
mark and the galaxy bias and find a way to maximize the differences among
various cosmological models.
