Process Markov

While the Smoliichowski equation is necessary for a Markov process, in general it is not sufficient, but known counter-examples are always non-Gaiissian as well. 

Markov process [12], Denoting tire inverse of R.j by and using the definition... 

A proposal based on Onsager s theory was made by Landau and Lifshitz [27] for the fluctuations that should be added to the Navier-Stokes hydrodynamic equations. Fluctuating stress tensor and heat flux temis were postulated in analogy with the Onsager theory. Flowever, since this is a case where the variables are of mixed time reversal character, tlie derivation was not fiilly rigorous. This situation was remedied by tlie derivation by Fox and Ulilenbeck [13, H, 18] based on general stationary Gaussian-Markov processes [12]. The precise fomi of the Landau proposal is confimied by this approach [14]. 

To conclude this section it should be pointed out again that the friction coefficient has been considered to be frequency independent as implied in assuming a Markov process, and that zero-frequency friction as represented by solvent viscosity is an adequate parameter to describe the effect of friction on observed reaction rates. 

Perikinetic motion of small particles (known as colloids ) in a liquid is easily observed under the optical microscope or in a shaft of sunlight through a dusty room - the particles moving in a somewhat jerky and chaotic manner known as the random walk caused by particle bombardment by the fluid molecules reflecting their thermal energy. Einstein propounded the essential physics of perikinetic or Brownian motion (Furth, 1956). Brownian motion is stochastic in the sense that any earlier movements do not affect each successive displacement. This is thus a type of Markov process and the trajectory is an archetypal fractal object of dimension 2 (Mandlebroot, 1982). 

The simplest random process is completely stochastic so that one may write, for example, Pjivih yih) = d [(y h)P y2h)- However, here we are concerned with a slightly more complex process known as the Markov process, characterized by... 

While static Monte Carlo methods generate a sequence of statistically independent configurations, dynamic MC methods are always based on some stochastic Markov process, where subsequent configurations X of the system are generated from the previous configuration X —X —X" — > with some transition probability IF(X —> X ). Since to a large extent the choice of the basic move X —X is arbitrary, various methods differ in the choice of the basic unit of motion . Also, the choice of transition probability IF(X — > X ) is not unique the only requirement is that the principle... 

As our first example we shall define what is known as a gaussian Markov process. This process, as we shall see later, is a good model for thermal noise or vacuum-tube-generated noise that has been passed through an RC filter with time constant a 1. We begin by defining two functions / and Q as follows... 

This result can now be used to verify our earlier statement that the gaussian Markov process defined by Eq. (3-218) is a good model for RC filtered vacuum tube noise. We have already seen that vacuum tube noise is essentially gaussian (as long as n is large) and that its spectrum is essentially white .70 A reasonable model for RC filtered... 

Equation (3-325), along with the fact that Y(t) has zero mean and is gaussian, completely specifies Y(t) as a random process. Detailed expressions for the characteristic function of the finite order distributions of Y(t) can be calculated by means of Eq. (3-271). A straightforward, although somewhat tedious, calculation of the characteristic function of the finite-order distributions of the gaussian Markov process defined by Eq. (3-218) now shows that these two processes are in fact identical, thus proving our assertion. 

Beck J, Pauker S. The Markov process in medical prognosis. Med Decis Making 1983 3 419-58. 

Nuitjen MJC, Hardens M, Souetre E (1995). A Markov Process Analysis comparing the cost effectiveness of maintenance therapy with citalopram versus standard therapy in major depression. Pharmacoeconomics, 159-68. 

Barucha-Reid AT (1960) Elements of theory of Markov processes and their applications. McGraw-Hill, New York... 

The last two results are rather similar to the quadratic forms given by Fox and Uhlenbeck for the transition probability for a stationary Gaussian-Markov process, their Eqs. (20) and (22) [82]. Although they did not identify the parity relationships of the matrices or obtain their time dependence explicitly, the Langevin equation that emerges from their analysis and the Doob formula, their Eq. (25), is essentially equivalent to the most likely terminal position in the intermediate regime obtained next. 

Gaussian approximation, heat flow, 60 Gaussian distribution, transition state trajectory, white noise, 206-207 Gaussian-Markov process, linear... 

See, for instance [20] for a detailed discussion of the Markov property and of Markov processes. 

A. Probability to Reach a Boundary by One-Dimensional Markov Processes... 

The aim of this chapter is to describe approaches of obtaining exact time characteristics of diffusion stochastic processes (Markov processes) that are in fact a generalization of FPT approach and are based on the definition of characteristic timescale of evolution of an observable as integral relaxation time [5,6,30—41]. These approaches allow us to express the required timescales and to obtain almost exactly the evolution of probability and averages of stochastic processes in really wide range of parameters. We will not present the comparison of these methods because all of them lead to the same result due to the utilization of the same basic definition of the characteristic timescales, but we will describe these approaches in detail and outline their advantages in comparison with the FPT approach. 

Markov process. In the next few sections we will briefly introduce properties of Markov processes as well as equations describing Markov processes. 

If we will consider arbitrary random process, then for this process the conditional probability density W xn,tn x, t, ... x i,f i) depends on x1 X2,..., x . This leads to definite temporal connexity of the process, to existence of strong aftereffect, and, finally, to more precise reflection of peculiarities of real smooth processes. However, mathematical analysis of such processes becomes significantly sophisticated, up to complete impossibility of their deep and detailed analysis. Because of this reason, some tradeoff models of random processes are of interest, which are simple in analysis and at the same time correctly and satisfactory describe real processes. Such processes, having wide dissemination and recognition, are Markov processes. Markov process is a mathematical idealization. It utilizes the assumption that noise affecting the system is white (i.e., has constant spectrum for all frequencies). Real processes may be substituted by a Markov process when the spectrum of real noise is much wider than all characteristic frequencies of the system. 

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 ... 

Markov processes are processes without aftereffect. Thus, the -dimensional probability density of Markov process may be written as... 

Formula (2.2) contains only one-dimensional probability density W(xi, t ) and the conditional probability density. The conditional probability density of Markov process is also called the transition probability density because the present state comprehensively determines the probabilities of next transitions. Characteristic property of Markov process is that the initial one-dimensional probability density and the transition probability density completely determine Markov random process. Therefore, in the following we will often call different temporal characteristics of Markov processes the transition times, implying that these characteristics primarily describe change of the evolution of the Markov process from one state to another one. 

In the most general case the diffusive Markov process (which in physical interpretation corresponds to Brownian motion in a field of force) is described by simple dynamic equation with noise source ... 

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

---