Stochastic stationary Markov process

If one chooses Pi(Vi, 0) = fi ) a non-stationary Markov process is defined, called the Wiener process or Wiener-Levy process. ) It is usually considered for f >0 alone and was originally invented for describing the stochastic behavior of the position of a Brownian particle (see VIII.3). The probability density for t > 0 is according to (2.2)... 

The extraction of a homogeneous process from a stationary Markov process is a familiar procedure in the theory of linear response. As an example take a sample of a paramagnetic material placed in a constant external magnetic field B. The magnetization Y in the direction of the field is a stationary stochastic process with a macroscopic average value and small fluctuations around it. For the moment we assume that it is a Markov process. The function Px (y) is given by the canonical distribution... 

Mixing has been described as a stochastic process by means of stationary and non-stationary MARKOV chains in which the probabilities of particle movement from place to place in the bed are determined. 

Recall that a Levy process X (t) is a continuous-time stochastic process that has independent and stationary increments. It represents a natural generalization of a simple random walk defined as a sum of independent identically distributed random variables. The independence of increments ensures that Levy processes are Markov processes. The main feature of a Levy process is that it is infinitely divisible for... 

This set of assumptions on the statistical properties of f(t) determines the statistical properties of the solution v(0 of the stochastic differaitial equation in Equation 1.1, which are summarized saying that v(0 is a Gaussian stationary Markov stochastic process, that is, it is generally not delta-correlated. The specific results that follow from this simple mathanatical model regarding propo ties such as the velocity autocorrelation function, msd, and so on, are reviewed in standard statistical physics textbooks [48]. 

In order to obtain the expression for the components of the vector of instantaneous copolymer composition it is necessary, according to general algorithm, to firstly determine the stationary vector ji of the extended Markov chain with the matrix of transitions (13) which describes the stochastic process of conventional movement along macromolecules with labeled units and then to erase the labels. In this particular case such a procedure reduces to the summation ... 

For many synthetic copolymers, it becomes possible to calculate all desired statistical characteristics of their primary structure, provided the sequence is described by a Markov chain. Although stochastic process 31 in the case of proteinlike copolymers is not a Markov chain, an exhaustive statistic description of their chemical structure can be performed by means of an auxiliary stochastic process 3iib whose states correspond to labeled monomeric units. As a label for unit M , it was suggested [23] to use its distance r from the center of the globule. The state of this stationary stochastic process 31 is a pair of numbers, (a, r), the first of which belongs to a discrete set while the second one corresponds to a continuous set. Stochastic process ib is remarkable for being stationary and Markovian. The probability of the transition from state a, r ) to state (/i, r") for the process of conventional movement along a heteropolymer macromolecule is described by the matrix-function of transition intensities... 

Sources of disturbances considered in this example are categorized in three classes. First, the production plants are stochastic transformers, i.e. the transformation processes are modelled by stationary time series models with normally distributed errors. The plants states are modelled by Markov models as introduced before. The corresponding transition matrices are provided in the appendix in Table A.15 and Table A.16. Additionally, normally distributed errors are added to simulate the inflovj rates with e N (O, ) where oj is the current state of the plant. 

The use of the embedded Discrete Time Markov Chain in a continuous stochastic process for deter-mining the events probability makes assumption that the system is in a stationary state characterizing by stationary distribution probabihties over its states. But the embedded DTMC is not limited to Continuous Time Markov Chain a DTMC can also be defined from semi-Markov or under some hypothesis from more generally stochastic processes. Another advantage to use the DTMC to obtain the events probability is that the probability of an event is not the same during the system evolution, but can depends on the state where it occurs (in other words the same event can be characterized by different occurrence probabilities). The use of the Arden lemma permits to formally determine the whole set of events sequences, without model exploring. Finally, the probability occurrence for relevant or critical events sequences and for a sublanguage is determined. 

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