扩散介导的表面反应、布朗泛函和Feynman-Kac公式
Diffusion-mediated surface reactions, Brownian functionals and the Feynman-Kac formula
摘要：
细胞生物学中的许多过程涉及在包含靶$\calU$的结构域$Ω$中的扩散，靶$\calU$的边界$\partial \calU$是化学反应性表面。这样的靶标可以代表单个反应性分子、细胞内区室或整个细胞。最近，研究扩散介导的表面反应的概率框架已经开发，认为联合概率密度或传播粒子的位置和所谓的边界本地时间。后者表征了布朗粒子在全反射边界上的点的邻域中花费的时间量。表面反应的影响，然后通过一个适当的停止条件的边界当地时间。在本文中，我们推广的扩散介导的表面反应的理论的情况下，整个内部目标域$\calU$作为一个部分吸收体，而不是目标边界$\partial \calU$。此时，颗粒可以自由进出$\calU$，并且只能在$\calU$内反应（被吸收）。因此，适当的布朗泛函是占据时间（粒子在$\calU$内花费的累积时间），而不是边界局部时间。我们表明，这两种情况下，可以考虑在一个统一的框架内使用Feynman-Kac公式推导出相应的布朗泛函的传播的边值问题（BVP），并引入相关的停止条件。我们通过计算位于球形域$Ω$中心的球形目标$\calU$的平均首次通过时间（MFPT）来说明该理论。这是通过直接求解传播算子BVP而不是使用谱方法来实现的。我们发现，如果停止时间密度的一阶矩是无限的，那么MFPT也是无限的，即球形目标是不充分吸收。
英文摘要：
Many processes in cell biology involve diffusion in a domain $Ω$ that contains a target $\calU$ whose boundary $\partial \calU$ is a chemically reactive surface. Such a target could represent a single reactive molecule, an intracellular compartment or a whole cell. Recently, a probabilistic framework for studying diffusion-mediated surface reactions has been developed that considers the joint probability density or propagator for the particle position and the so-called boundary local time. The latter characterizes the amount of time that a Brownian particle spends in the neighborhood of a point on a totally reflecting boundary. The effects of surface reactions are then incorporated via an appropriate stopping condition for the boundary local time. In this paper we generalize the theory of diffusion-mediated surface reactions to cases where the whole interior target domain $\calU$ acts as a partial absorber rather than the target boundary $\partial \calU$. Now the particle can freely enter and exit $\calU$, and is only able to react (be absorbed) within $\calU$. The appropriate Brownian functional is then the occupation time (accumulated time that the particle spends within $\calU$) rather than the boundary local time. We show that both cases can be considered within a unified framework by using a Feynman-Kac formula to derive a boundary value problem (BVP) for the propagator of the corresponding Brownian functional, and introducing an associated stopping condition. We illustrate the theory by calculating the mean first passage time (MFPT) for a spherical target $\calU$ located at the center of a spherical domain $Ω$. This is achieved by solving the propagator BVP directly, rather than using spectral methods. We find that if the first moment of the stopping time density is infinite, then the MFPT is also infinite, that is, the spherical target is not sufficiently absorbing.
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