A Stochastic Hybrid Framework for Obtaining Statistics of Many Random Walkers in a Switching Environment

Abstract

We analyze a population of randomly walking particles in a stochastically switching environment by formulating the model as a stochastic hybrid system. The latter describes the evolution of the probability distribution of the particles, which is a random variable depending on realizations of the random environment. We derive a hierarchy of moment equations for the probability distribution, which allows us to extract statistics of the multiparticle system. As a specific example, we consider a population of particles walking on a one-dimensional lattice with a dynamic gate at some unknown location, which stochastically switches between an open and closed state according to a two-state Markov process. This type of model has two levels of stochasticity: one due to the jump process describing the evolution of each particle on the lattice, and the other due to the switching of the gate. By solving the moment equations for the stochastic hybrid system, we extract statistical information about the location and dynamics of the gate in terms of how the mean and variance of site occupancies varies with distance of a given site from the gate. This has potential applications in the analysis of time series data obtained from biophysical experiments on the diffusion of particles in the plasma membrane of cells.