A cross-diffusion system for two compoments with a Laplacian structure is
analyzed on the multi-dimensional torus. This system, which was recently
suggested by P.-L. Lions, is formally derived from a Fokker-Planck equation for
the probability density associated to a multi-dimensional It\={o} process,
assuming that the diffusion coefficients depend on partial averages of the
probability density with exponential weights. A main feature is that the
diffusion matrix of the limiting cross-diffusion system is generally neither
symmetric nor positive definite, but its structure allows for the use of
entropy methods. The global-in-time existence of positive weak solutions is
proved and, under a simplifying assumption, the large-time asymptotics is
investigated
