  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.
