Large deviation principle and thermodynamic limit of chemical master equation via nonlinear semigroup

TL;DR

This paper establishes a large deviation principle and thermodynamic limit for the chemical master equation using nonlinear semigroup theory, connecting microscopic stochastic models to macroscopic deterministic equations.

Contribution

It introduces a novel approach using Varadhan's discrete nonlinear semigroup and Hamilton-Jacobi equations to analyze the CME and its large deviations.

Findings

Proves convergence of discrete schemes to viscosity solutions of HJE.

Derives large deviation principle for chemical reaction processes.

Recovers mean-field reaction rate equations with convergence rates.

Abstract

Chemical reactions can be modeled by a random time-changed Poisson process on countable states. The macroscopic behaviors, such as large fluctuations, can be studied via the WKB reformulation. The WKB reformulation for the backward equation is Varadhan's discrete nonlinear semigroup and is also a monotone scheme that approximates the limiting first-order Hamilton-Jacobi equations (HJE). The discrete Hamiltonian is an m-accretive operator, which generates a nonlinear semigroup on countable grids and justifies the well-posedness of the chemical master equation (CME) and the backward equation with 'no reaction' boundary conditions. The convergence from the monotone schemes to the viscosity solution of HJE is proved by constructing barriers to overcome the polynomial growth coefficients in the Hamiltonian. This implies the convergence of Varadhan's discrete nonlinear semigroup to the continuous Lax-Oleinik semigroup and leads to the large deviation principle for the chemical reaction process at any single time. Consequently, the macroscopic mean-field limit reaction rate equation is recovered with a concentration rate estimate. Furthermore, we establish the convergence from a reversible invariant measure to an upper semicontinuous viscosity solution of the stationary HJE.