Thermodynamic limit of chemical master equation via nonlinear semigroup
Abstract
Chemical reactions can be modeled by a random timechanged Poisson process on countable states. The macroscopic behaviors such as the 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 which approximates the limiting first order HamiltonianJacobi equations(HJE). The discrete Hamiltonian is a maccretive operator, which generates a nonlinear semigroup on countable grids and justifies 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 constructing barriers to overcome the polynomial growth coefficient in Hamiltonian. This means the convergence of Varadhan's discrete nonlinear semigroup to the continuous LaxOleinik semigroup, and 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. Consequently, the macroscopic meanfield 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2022
 arXiv:
 arXiv:2205.09313
 Bibcode:
 2022arXiv220509313G
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Probability;
 Physics  Chemical Physics
 EPrint:
 31pages