  This paper considers differential problems with random switching, with
specific applications to the motion of cells and centrally coordinated motion.
Starting with a differential-equation model of cell motion that was proposed
previously, we set the relaxation time to zero and consider the simpler model
that results. We prove that this model is well-posed, in the sense that it
corresponds to a pure jump-type continuous-time Markov process (without
explosion).We then describe the model's long-time behavior, first by specifying
an attracting steady-state distribution for a projection of the model, then by
examining the expected location of the cell center when the initial data is
compatible with that steady-state. Under such conditions, we present a formula
for the expected velocity and give a rigorous proof of that formula's validity.
We conclude the paper with a comparison between these theoretical results and
the results of numerical simulations.
