• Corpus ID: 248887649

Thermodynamic limit of chemical master equation via nonlinear semigroup

@inproceedings{Gao2022ThermodynamicLO,
  title={Thermodynamic limit of chemical master equation via nonlinear semigroup},
  author={Yuan Gao and Jian‐Guo Liu},
  year={2022}
}

• Yuan Gao, Jian‐Guo Liu
• Published 19 May 2022
• Mathematics

. Chemical reactions, at a mesoscopic scale, can be modeled by a random time-changed Poisson process on countable states. The macroscopic behaviors in the large size limit, particularly the estimates for the large fluctuations, can be studied via the WKB reformulation, aka. nonlinear semigroup for the chemical master equation(CME) and the backward equation. The WKB reformulation for the backward equation is Varadhan’s discrete nonlinear semigroup and is also a monotone scheme which approximates the limiting first order Hamiltonian-Jacobi equations(HJE). Then the maximality and monotonicity of the discrete Hamiltonian leads to a m-accretive operator, which generates a nonlinear semigroup on countable grids. From this, we obtain the wellposedness of CME and the backward equation with ‘no reaction’ boundary conditions. The convergence from the monotone schemes to the viscosity solution of HJE is proved via the upper/lower semicontinuous envelopes of the discrete resolvent problem and nonlinear semigroup theory. Our construction of a solution to HJE automatically inherits the boundary conditions of CME and the backward equation. Thanks to the Lax-Oleinik representation for the viscosity solution of first order HJE, the above convergence from the monotone scheme to HJE solution, i.e., the convergence of Varadhan’s discrete nonlinear semigroup to the continuous Lax-Oleinik semigroup, indeed leads to the large deviation principle for the chemical reaction process at any single time. Here the required exponential tightness at any time is proved for either compact supported initial distribution or with the help of an exponentially tight reversible invariant measure to CME. As a consequence, the macroscopic mean-field limit reaction rate equation is recovered. The convergence from a reversible invariant measure to an upper semicontinuous viscosity solution to the stationary HJE is also proved.