Abstract
One way of analysing complicated Markov population processes is to approximate them by a diffusion about the deterministic path. This approximation alone may not, however, answer all the questions which might reasonably be asked. Many processes have phases, for example near boundaries, where a different approximation is required; such processes are better described by a succession of diffusion and special approximations alternately.This paper looks at the special treatment required near a point where the deterministic equations are in equilibrium. When the equilibrium is unstable, the process will eventually wander off, and possibly follow a diffusion around another deterministic path. If the equilibrium is stable, the process will behave as if in stable equilibrium about it for a very much longer time, but may at last be trapped away from it, for instance in an absorbing state. The results presented describe the distributions of the time and place of leaving neighbourhoods of the equilibrium point. The neighbourhoods considered are large enough, in the unstable case, to make possible the link with the next phase of motion. In the stable case, exit times are shown to be so long that the possibility of exit can often be ignored in practice, and the quasi-equilibrium distribution treated as a true equilibrium. A more detailed result, showing how closely the normal approximation holds in this situation, is also provided.
Publisher
Cambridge University Press (CUP)
Subject
Applied Mathematics,Statistics and Probability
Reference14 articles.
1. Ridler-Rowe C. J. (1964) Some Two-Dimensional Markov Processes. Ph.D. thesis, University of Durham.
2. The Asymptotic Analysis of a Stochastic Model of an Epidemic
3. Barbour A. D. (1973) Limit Theorems for Markov Population Processes. Ph.D. thesis, University of Cambridge.
Cited by
67 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献