Markov processes continuous

A continuous Markov process (also known as a diffusive process) is characterized by the fact that during any small period of time At some small (of the order of %/At) variation of state takes place. The process x(t) is called a Markov process if for any ordered n moments of time t < < t < conditional probability density depends only on the last fixed value ... 

The transition probability density of continuous Markov process satisfies to the following partial differential equations (WXo(x, t) = W(x, t xo, to)) ... 

Constrained Brownian motion may be represented as a continuous Markov process of the / soft variables <7, ...,. A standard result of the theory of... 

Use of the stochastic differential equation (2.2.2) as the equation of motion instead of equation (2.1.1) results in the treatment of the reaction kinetics as a continuous Markov process. Calculations of stochastic differentials, perfectly presented by Gardiner [26], allow us to solve equation (2.2.2). On the other hand, an averaged concentration given by this equation could be obtained making use of the distribution function / = f(c, ..., cs t). The latter is nothing but solution of the Fokker-Planck equation [26, 34]... 

Therefore, the simplest procedure to get the stochastic description of the reaction leads to the rather complicated set of equations containing phenomenological parameters / (equation (2.2.17)) with non-transparent physical meaning. Fluctuations are still considered as a result of the external perturbation. An advantage of this approach is a useful analogy of reaction kinetics and the physics of equilibrium critical phenomena. As is well known, because of their nonlinearity, equations (2.1.40) reveal non-equilibrium bifurcations [78, 113]. A description of diffusion-controlled reactions in terms of continuous Markov process - equation (2.2.15) - makes our problem very similar to the static and dynamic theory of critical phenomena [63, 87]. When approaching the bifurcation points, the systems with reactions become very sensitive to the environment fluctuations, which can even produce new nonequilibrium transitions [18, 67, 68, 90, 108]. The language developed in the physics of critical phenomena can be directly applied to the processes in spatially extended systems. 

For a continuous Markov process, the master equation is of the form... 

Horsthemke and Brenig [ 3 ] stress a very important point about this analysis since a continuous Markov process is completely characterized by its first two differential moments it is unnecessary to consider the asjnnptotic behavior of the higher order differential moments. 

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. 

Propagation of the fast subsystem - chemical Langevin equations The fast subset dynamics are assumed to follow a continuous Markov process description and therefore a multidimensional Fokker-Planck equation describes their time evolution. The multidimensional Fokker-Plank equation more accurately describes the evolution of the probability distribution of only the fast reactions. The solution is a distribution depicting the state occupancies. If the interest is in obtaining one of the possible trajectories of the solution, the proper course of action is to solve a system of chemical Langevin equations (CLEs). 

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. 

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