  We consider conservative cross-diffusion systems for two species where
individual motion rates depend linearly on the local density of the other
species. We develop duality estimates and obtain stability and approximation
results. We first control the time evolution of the gap between two bounded
solutions by means of its initial value. As a by product, we obtain a
uniqueness result for bounded solutions valid for any space dimension, under a
non-perturbative smallness assumption. Using a discrete counterpart of our
duality estimates, we prove the convergence of random walks with local
repulsion in one dimensional discrete space to cross-diffusion systems. More
precisely, we prove quantitative estimates for the gap between the stochastic
process and the cross-diffusion system. We give first rough but general
estimates; then we use the duality approach to obtain fine estimates under less
general conditions.
