  We consider the stochastic ranking process with space-time dependent jump
rates for the particles. The process is a simplified model of the time
evolution of the rankings such as sales ranks at online bookstores. We prove
that the joint empirical distribution of jump rate and scaled position
converges almost surely to a deterministic distribution, and also the tagged
particle processes converge almost surely, in the infinite particle limit. The
limit distribution is characterized by a system of inviscid Burgers-like
integral-partial differential equations with evaporation terms, and the limit
process of a tagged particle is a motion along a characteristic curve of the
differential equations except at its Poisson times of jumps to the origin.
