Markovian stochastic processes

The process z(t) is called Markovian if the knowledge of the value of z (say zi) at a given time (say t ) fully determines the probability of observing z at any later time 

Markov processes have no memory of earlier information. Newton equations describe deterministic Markovian processes by this definition, since knowledge of system state (all positions and momenta) at a given time is sufficient in order to determine it at any later time. The random walk problem discussed in Section 7.3 is an example of a stochastic Markov process. 

F (z2t2 2lti zoto) = 7 (z2t2 Iziti)T (ziti zoto)  

P(z2t2 ziti Izoto) = /X 2t2 ziti)P(ziti Izoto) for to ty t2 (7A7) 

E(Z2t2 Izoto) = dziP(z2t2 ziti)P(ziti Izoto) This is the Chapman- Kolmogorov equation. 

P(Z2t2 Zoto) = j dz P z2t2 Z h)P Z t], zoto) This is the Chapman-Kolmogorov equation. 

The time evolution in a Markovian stochastic process is therefore fully described 

What is the significance of the Markovian property of a physical process Note that the Newton equations of motion as well as the time-dependent Schrodinger equation are Markovian in the sense that the future evolution of a system described by these equations is fully determined by the present ( initial ) state of the system. Non-Markovian dynamics results from reduction procedures used in order to focus on a relevant subsystem as discussed in Section 7.2, the same procedures that led us to consider stochastic time evolution. To see this consider a universe described by two variables, z and Z2, which satisfy the Markovian equations of motion 

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Under current treatment of statistical method a set of the states of the Markovian stochastic process describing the ensemble of macromolecules with labeled units can be not only discrete but also continuous. So, for instance, when the description of the products of living anionic copolymerization is performed within the framework of a terminal model the role of the label characterizing the state of a monomeric unit is played by the moment when this unit forms in the course of a macroradical growth [25]. 

An exhaustive statistical description of living copolymers is provided in the literature [25]. There, proceeding from kinetic equations of the ideal model, the type of stochastic process which describes the probability measure on the set of macromolecules has been rigorously established. To the state Sa(x) of this process monomeric unit Ma corresponds formed at the instant r by addition of monomer Ma to the macroradical. To the statistical ensemble of macromolecules marked by the label x there corresponds a Markovian stochastic process with discrete time but with the set of transient states Sa(x) constituting continuum. Here the fundamental distinction from the Markov chain (where the number of states is discrete) is quite evident. The role of the probability transition matrix in characterizing this chain is now played by the integral operator kernel ... 

The time evolution in a Markovian stochastic process is therefore fully described by the transition probability Pilyt z T). 

While Markovian stochastic processes play important role in modeling molecular dynamics in condensed phases, their applicability is limited to processes that involve relatively slow degrees of freedom. Most intramolecular degrees of freedom are characterized by timescales that are comparable or faster than characteristic environmental times, so that the inequality (7.53) often does not hold. Another class of stochastic processes that are amenable to analytic descriptions also in non-Markovian situations is discussed next. 

A stochastic process whose transition probability P(r, 11 ro, Zo) satisfies Eq, (8,3) is called a Wiener process. Another well-known Markovian stochastic process is the Orenstein-Uhlenbeckprocess, for which the transition probability satisfies the equation (in one-dimension)... 

As discussed in Section 8.2.1, the Langevin equation (8.13) describes a Markovian stochastic process The evolution of the stochastic system variable x(Z) is determined by the state of the system and the bath at the same time t. The instantaneous response of the bath is expressed by the appearance of a constant damping coefficient y and by the white-noise character of the random force 7 (Z). 

A series of probable transitions between states can be described with the Markov chain. A Markovian stochastic process is memoryless, and this is illustrated subsequently. We generate a sequence of random variables, (yo, yi, yi, ), so that each time t > 0, the next state yt+i would be sampled from a distribution P(y,+ily,), which would depend only on the current state of the chain, y,. Thus, given y, the next state y,+i would not depend additionally on the history of the chain (yo, yi, yi,---, y i). The name Markov chain is used to describe this sequence, and the transition kernel of the chain is i (.l.) does not depend on t if we assume that the chain is time homogeneous. A detail description of the Markov model is provided in Chapter 26. 

A series of probable transitions between states can be described with Markov modeling. The natural course of a disease, for example, can be viewed for an individual subject as a sequence of certain states of health (12). A Markovian stochastic process is memoryless. To predict what the future state will be, knowledge of the current state is sufficient and is independent of where the process has been in the past. This is termed the strong Markov property (13). 

The majority of all software reliability models are based on Markovian stochastic processes. This means that the future behavior after a time, say, t, is only dependent on the state of the process at time t and not on the history about how the state was reached. This assumption is a reasonable way to get a manageable model, and it is made in many other engineering fields. 

Cox, D.R. 1955. The Analysis of Non-Markovian Stochastic Processes by the Inclusion of Supplementary Variables. Proc. Cambridge Phil Soc.. 51, pp. 453-441... 

Hongler, M. O. (1979b). Exact time dependent probability density for a nonlinear non-Markovian stochastic process, Helv. Phys. Acta, 52, 280-... 

Equation [52] is also a Markovian stochastic process with zero mean and variance Af. The quantity X"(0,-Af) is correlated with X " (0,Af) through a bivariate Gaussian distribution. In the zero limit of the friction coefficient, this set of equations corresponds to the trajectories obtained with the Verlet algorithm. ... 

The original Langevin equation considers a Markovian stochastic process [6, 7] with simple constant friction H in the field of an external fluctuation force F f). For a harmonic oscillator this equation has the form ... 

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