State-Constrained Chemical Reactions: Discrete-to-Continuous Hamilton--Jacobi Equations and Large Deviations

Abstract: We study the macroscopic behavior of chemical reactions modeled as random time-changed Poisson processes on discrete state spaces. Using the WKB reformulation, the backward equation of the rescaled process leads to a discrete Hamilton--Jacobi equation with state constraints. As the grid size tends to zero, the limiting solution and its associated variational representation are closely connected to the good rate function of the large deviation principle for state-constrained chemical reactions in the thermodynamic limit. In this work, we focus on the limiting behavior of discrete Hamilton--Jacobi equations defined on bounded domains with state-constraint boundary conditions. For a single chemical reaction, we show that, under a suitable reparametrization, the solution of the discrete Hamilton--Jacobi equation converges to the solution of a continuous Hamilton--Jacobi equation with a Neumann boundary condition. Building on this convergence result and the associated variational representation, we establish the large deviation principle for the rescaled chemical reaction process in bounded domains.