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

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