Reconstruction of a persistent random walk from exit time distributions

In this paper, we study the inverse problem of reconstructing the spatially dependent transition rate F(x) of a 1D Broadwell process from exit time distributions. In such a process, an advecting particle is assumed to undergo transitions between states with constant positive (+v) and negative (−v) velocities. The goal is to reconstruct the transition rate function F(x) from the exit time distributions out of a finite interval. Using the associated backward equation, we compute the distribution of exit times and its Laplace transform, given a fixed starting position and velocity. We propose two methods (called ‘t’ and ‘s’) for finding F(x). In both methods, we represent F(x) as a linear combination of polynomials and repeatedly solve the backward equation to minimize the difference between its solution and given first exit time data. In the t-method, we work in the time domain, using exit times directly and leveraging a novel series solution for the exit time distribution. In the s-method, we work with the Laplace-transformed equation and Laplace-transformed exit times. Noisy data are generated using a custom-designed algorithm to simulate the trajectories of a Broadwell process. In most cases, we can find four coefficients to within O(10
⁻¹
) accuracy from O(10
⁴
) exit times, with the t-method slightly outperforming the s-method. We also explore the effectiveness of our algorithms for a fixed number of exit times under different advection speeds and find that optimal reconstruction occurs when v=O(1).