An Asymptotic Analysis of the Spatially Inhomogeneous Velocity-Jump Process

We analyze the one-dimensional velocity-jump process, where a particle moves at a constant velocity determined by the particle’s internal velocity state that randomly fluctuates with exponentially distributed waiting times. The transition rates between the internal velocity states depend on the location of the particle, leading to a spatially inhomogeneous random process. An asymptotic analysis is applied to obtain the stationary distribution of the random process. The result is compared to the often-used quasi–steady-state diffusion approximation, and it is found that the diffusion approximation breaks down in the presence of a turning point, where the average velocity of the particle changes sign. We extend the analysis to approximate the first-exit time density for the particle to escape the confining effect of the turning point, and we find the diffusion approximation also fails to accurately describe the long-time behavior of the process. The accuracy of the two approximations is explored for a simple model of molecular-motor transport by comparing results to Monte Carlo simulations.