We consider the stochastic ranking process with the jump times of the
particles determined by Poisson random measures. We prove that the joint
empirical distribution of scaled position and intensity measure converges
almost surely in the infinite particle limit. We give an explicit formula for
the limit distribution and show that the limit distribution function is a
unique global classical solution to an initial value problem for a system of a
first order non-linear partial differential equations with time dependent
coefficients
