通过非线性半群研究化学主方程的大偏差原理与热力学极限
Large deviation principle and thermodynamic limit of chemical master equation via nonlinear semigroup
AI总结：
本文利用 Varadhan 离散非线性半群与单调格式证明化学主方程的适定性、HJE 粘性解收敛及大偏差原理，并恢复平均场反应速率方程。
摘要：
化学反应可以通过可数状态上的随机时间变换 Poisson 过程来建模。诸如大波动之类的宏观行为可以借助 WKB 重构进行研究。反向方程的 WKB 重构是 Varadhan 离散非线性半群，同时也是一种单调格式，用于逼近极限的一阶 Hamilton–Jacobi 方程（HJE）。离散 Hamiltonian 是一个 m-增生算子，它在可数网格上生成一个非线性半群，并证明了在“无反应”边界条件下化学主方程（CME）与反向方程的适定性。通过构造障碍函数以克服 Hamiltonian 中系数的多项式增长，证明了单调格式向 HJE 粘性解的收敛。这意味着 Varadhan 离散非线性半群收敛到连续的 Lax–Oleinik 半群，并导出化学反应过程在任意单一时刻的大偏差原理。由此恢复了宏观平均场极限反应速率方程，并给出集中速率估计。此外，还建立了从可逆不变测度到稳态 HJE 的上半连续粘性解的收敛性。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
