随机混合反应扩散过程的路径积分
Path integrals for stochastic hybrid reaction-diffusion processes
摘要：
本文构造了随机混合反应扩散（RD）过程的路径积分，其中反应项依赖于随机切换环境的离散状态。我们继续通过空间离散化一个给定的RD系统，并使用旋量表示的环境状态，以获得晶格模型的路径积分。在大分子数的情况下，相应的连续路径积分作用量表示在一个有效的哈密顿量，它涉及一个浓度场$u(\x,t)$，$\x\in \R^d$，共轭场$v(\x,t)$，和$M$辅助共轭对$(c_m(t),ϕ_m(t))$，其中$M$是离散的环境状态的数量。变量$c_m(t)$确定样本路径在时间$t$暴露于第$m$环境状态的有效概率，其中$\sum_{m=1}^Mc_m(t)=1$。然后，我们考虑半经典（绝热）的限制$ε\rightarrow 0$，其中$ε^{-1}$确定的环境状态之间的切换率。我们展示了如何可以消除辅助变量，以产生一个单独的字段$u$和$v$的行动功能。相关的哈密顿量是扩散项和Perron或主特征值的函数线性算子，涉及的反应项和矩阵生成器的开关过程的总和。减少路径积分，然后用于导出一个功能的Hamilton-Jacobi方程的最小作用路径，并获得高斯噪声近似的随机混合RD系统的绝热极限。最后，通过考虑一个相应的RD主方程，构造了低分子数情况下的路径积分。现在有必要考虑两个噪声源，一个是由于开关环境，另一个是由于分子数的波动。
英文摘要：
We construct path integrals for stochastic hybrid reaction-diffusion (RD) processes, in which the reaction terms depend on the discrete state of a randomly switching environment. We proceed by spatially discretizing a given RD system and using a spinor representation of the environmental states to derive a path integral for the lattice model. In the case of large molecular numbers, the corresponding continuum path integral action is expressed in terms of an effective Hamiltonian, which involves a concentration field $u(\x,t)$, $\x\in \R^d$, a conjugate field $v(\x,t)$, and $M$ auxiliary conjugate pairs $(c_m(t),ϕ_m(t))$, where $M$ is the number of discrete environmental states. The variable $c_m(t)$ determines the effective probability that a sample path is exposed to the $m$-th environmental state at time $t$, with $\sum_{m=1}^Mc_m(t)=1$. We then consider the semi-classical (adiabatic) limit $ε\rightarrow 0$, where $ε^{-1}$ determines the rate of switching between the environmental states. We show how the auxiliary variables can be eliminated to yield an action functional for the fields $u$ and $v$ alone. The associated Hamiltonian is the sum of a diffusion term and the Perron or principal eigenvalue of a functional linear operator involving the reaction terms and the matrix generator of the switching process. The reduced path integral is then used to derive a functional Hamilton-Jacobi equation for least action paths and to obtain a Gaussian noise approximation of the stochastic hybrid RD system in the adiabatic limit. Finally, the path integral in the case of low molecular numbers is constructed by considering a corresponding RD master equation. It is now necessary to take into account two sources of noise, one due to the switching environment and the other due to fluctuations in molecular numbers.
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