化学反应网络的耦合扩散过程的样本轨道构造及数值算法

Construction of sampling trajectories of coupled diffusion processes and its simulation algorithm

耦合扩散过程是一个耦合了扩散运动和随机跳过程的混合系统.为解决这类系统的样本轨道模拟问题,本文在一个由[0,1]~∞上的Lebesgue测度空间与C(R_+,R^r)上的Wiener测度空间所形成的乘积空间中对耦合扩散过程的轨道进行了构造,并证明了构造得到的过程具有Markov性,给出了剩余寿命及下次反应序号的分布函数,进一步考察了过程的无穷小生成元与鞅问题,在分布意义下证明了鞅问题的唯一性.最后,根据构造得到的样本轨道,给出了耦合扩散过程的数值模拟方法.

Coupled diffusion process is a system in which a diffusion process is coupled with a Markov jumping process. To simulate such a hybrid process, we consider a product space formed by a Lebesgue space on [0, 1]∞and a Wiener space on C（R＋, Rr）, and intuitively construct a coupled diffusion process on this probability space. We prove that the constructed process is a Markov process, and present the probability distribution of the residual lifetime as well as the joint probability distribution of the next reaction index and the residual lifetime. We further consider the infinitesimal generator and the martingale problem of the process, and prove the solution to the martingale problem is unique in distribution. Based on the sampling process we constructed, we propose an algorithm to simulate the coupled diffusion process.