Reconstructing the Transition Rate Function of a Broadwell Random Walk from Exit Times

Abstract

In this paper, we utilize the layer stripping method, a method used in seismology by geophysicists, to study a stochastic inverse problem arising from a one-dimensional Broadwell process. The Broadwell process can be described as a random walk of a particle that transitions, with a spatially dependent flip rate, between two states associated with the same speed but with different signs of velocities. Our goal is to reconstruct the spatially dependent flip rate function of a one-dimensional Broadwell process from exit time distributions out of a finite interval. In principle, we are able to reconstruct flip rate functions with error within $O(10^{-2})$ from $O(10^5)$ exit times when the particle speed is unity. For smaller particle speeds, the noise increases for a fixed number of exit times. This method is less time consuming compared with traditional projection methods which involve optimization.