论文标题
化学主方程的大偏差原理和热力学极限通过非线性半群
Large deviation principle and thermodynamic limit of chemical master equation via nonlinear semigroup
论文作者
论文摘要
化学反应可以通过可数状态的随机时间变化的泊松过程进行建模。宏观行为，例如大波动，可以通过WKB重新制定研究。向后方程的WKB重新印象是Varadhan的离散非线性半群，也是一种单调方案，可近似限制的一阶Hamilton-Jacobi方程（HJE）。离散的哈密顿量是M型操作员，它会在可数的网格上产生非线性半群，并证明化学主方程（CME）的适当性和以“无反应”边界条件的向后方程合理。通过构建障碍以克服哈密顿量中的多项式生长系数，证明了从单调方案到HJE粘度解的收敛性。这意味着Varadhan离散的非线性半群与连续的Lax-Oleinik半群的收敛，并在任何时候都导致化学反应过程的较大偏差原理。因此，宏观平均场极限反应速率方程以浓度速率估计回收。此外，我们建立了从可逆的不变测度到固定HJE的上半连续粘度解的收敛性。
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.
