论文标题
使用Malliavin微积分求解化学扩散主方程
Using Malliavin calculus to solve a chemical diffusion master equation
论文作者
论文摘要
我们提出了一种新的方法，以解决化学扩散的大师的出生和死亡类型方程。这是一个无限的fokker-planck方程系统，其中不同的组件与化学主方程相似的反应动力学耦合。该系统是在[3]中提出的，用于建模与单个颗粒的空间扩散相关的化学反应动力学的概率演化。使用一些基本工具和思想来自无限维度高斯分析，我们能够重新将上述的fokker-planck方程的无限系统重新制定为通过通用随机过程求解的单个进化方程，并用Malliavin衍生物和差异二量化操作员编写。通过这种替代表示，我们将原始问题的解的某些有限尺寸投影与Ornstein-Uhlenbeck类型的单个部分微分方程的解联系，该方程包含与上述投影空间的维数一样多的变量。
We propose a novel method to solve a chemical diffusion master equation of birth and death type. This is an infinite system of Fokker-Planck equations where the different components are coupled by reaction dynamics similar in form to a chemical master equation. This system was proposed in [3] for modelling the probabilistic evolution of chemical reaction kinetics associated with spatial diffusion of individual particles. Using some basic tools and ideas from infinite dimensional Gaussian analysis we are able to reformulate the aforementioned infinite system of Fokker-Planck equations as a single evolution equation solved by a generalized stochastic process and written in terms of Malliavin derivatives and differential second quantization operators. Via this alternative representation we link certain finite dimensional projections of the solution of the original problem to the solution of a single partial differential equations of Ornstein-Uhlenbeck type containing as many variables as the dimension of the aforementioned projection space.
