  We study systems of Brownian particles on the real line, which interact by
splitting the local times of collisions among themselves in an asymmetric
manner. We prove the strong existence and uniqueness of such processes and
identify them with the collections of ordered processes in a Brownian particle
system, in which the drift coefficients, the diffusion coefficients, and the
collision local times for the individual particles are assigned according to
their ranks. These Brownian systems can be viewed as generalizations of those
arising in first-order models for equity markets in the context of stochastic
portfolio theory, and are able to correct for several shortcomings of such
models while being equally amenable to computations. We also show that, in
addition to being of interest in their own right, such systems of Brownian
particles arise as universal scaling limits of systems of jump processes on the
integer lattice with local interactions. A key step in the proof is the
analysis of a generalization of Skorokhod maps which include `local times' at
the intersection of faces of the nonnegative orthant. The result extends the
convergence of TASEP to its continuous analogue. Finally, we identify those
among the Brownian particle systems which have a probabilistic structure of
determinantal type.
