First Exit Time Analysis for the Stochastic Reaction Diffusion Process in a One Dimensional Domain

Abstract

Recent advances in modeling stochastic reaction–diffusion (RD) process have focused on particle-based and master equation formulations. While these models offer strong theoretical foundation, a practical challenge remains: how does the choice of spatial discretization affect the accuracy and computational efficiency of simulation results, particularly when estimating first exit times. This thesis addresses this research gap by investigating the accuracy of first exit time estimates in one-dimensional stochastic RD systems. We design and analyze three simplified models using stochastic simulations: (1) model 1: pure diffusion, (2) model 2: diffusion with monomolecular reaction, and (3) model 3: diffusion with bimolecular reaction. We conduct theoretical study for the mean first exit times and evaluate them based on these models. Our results show that strictly following the Gillespie SSA is not necessary to obtain accurate results under certain conditions and a moderate discretizations size (e.g., K ≥ 5) already provides highly accurate estimates for first exit times. Our results can guide efficient and accurate simulation of RD systems.