Markov approximation continuous processes

At least two kinds of advantages come from the use of the semigroup operator approach. First, a mathematically justified approximation of discontinuous processes by continuous processes can be obtained. Second, not only models for purely temporal processes (e.g. pure chemical reactions) can be given, but stochastic models of spatio-temporal phenomena (e.g. chemical reactions with diffusion) can be defined in a mathematically concise manner. The celebrated monograph on the modern approach to Markov processes is still Dynkin s book (1965), and for chemical applications see Arnold Kotelenez (1981). 

Thus, the transitions are always from a state n to the state n + 1. The transitions are, of course, arrivals because they cause the count N to increase by 1. The probability of a transition in a short interval of time h is approximately Xh for any n by (26). This observation corresponds precisely with the description of the Poisson process in terms of coin tossing in Section 2. Moreover, the fact that the time tetween arrivals in a Poisson process is exponential may be seen now as a consequence of the fact, expressed in (33), that the holding times in any continuous-time Markov chain are exponentially distributed. 

Let a repair be performed when the coating ink = pn cells is damaged. Here p, with 0 < p < 1, is a fraction which is defined such that k = 1,2,... If the number of source processes n is sufficiently large, we may approximate the superposed process with a Poisson process with rate n/p, where p is the mean time between failures in the source processes. However, once a cell becomes damaged, the source process actually stops and the rate of the superposed process is reduced by /p. This means that the superposed process is a continuous-time Markov process which has a rate in state i equal to... 

In the fast-continuous region, species populations can be assumed to be continuous variables. Because the reactions are sufficiently fast in comparison to the rest of the system, it can be assumed that they have relaxed to a steady-state distribution. Furthermore, because of the frequency of reaction rates, and the population size, the population distributions can be assumed to have a Gaussian shape. The subset of fast reactions can then be approximated as a continuous time Markov process with chemical Langevin Equations (CLE). The CLE is an ltd stochastic differential equation with multiplicative noise, as discussed in Chapter 13. 

Salis and Kaznessis separated the system into slow and fast reactions and managed to overcome the inadequacies and achieve a substantial speed up compared to the SSA while retaining accuracy. Fast reactions are approximated as a continuous Markov process, through Chemical Langevin Equations (CLE), discussed in Chapter 13, and the slow subset is approximated through jump equations derived by extending the Next Reaction variant approach. 

Salis and Kaznessis proposed a hybrid stochastic algorithm that is based on a dynamical partitioning of the set of reactions into fast and slow subsets. The fast subset is treated as a continuous Markov process governed by a multidimensional Fokker-Planck equation, while the slow subset is considered to be a jump or discrete Markov process governed by a CME. The approximation of fast/continuous reactions as a continuous Markov process significantly reduces the computational intensity and introduces a marginal error when compared to the exact jump Markov simulation. This idea becomes very useful in systems where reactions with multiple reaction scales are constantly present. 

The method can be further sped up by allowing more than one zero crossing, i.e., more than one slow reaction, to occur in the time it takes the system of CLEs to advance by At. Though this is an additional approximation contributing to the error introduced by the approximation of the fast reactions as continuous Markov processes, it results in a significant decrease in simulation times. The accuracy depends on the number of slow reactions allowed within At and decreases as the number increases. 

---