Stochastic process definition

A stochastic process is a family of random variables X( depending on a parameter t (e.g., time). This definition may be extended to include... 

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. 

See L. Takacs, Stochastic Processes, Methuen, London, 1966, for the definitions of the symbols and of the nomenclature employed in this work. 

For a general definition of multidimensional Markov chains see, for example, N. Bailey, Elements of Stochastic Processes, Wiley, New York, 1964, Chap. 10. 

The expansion in eigenfunctions leads to expressions of the various quantities pertaining to the stochastic process - as in equations (7.13) through (7.16). It also simplifies some of the derivations, in particular the proof of the approach to equilibrium. In fact, according to (7.13) it is sufficient to prove that all X other than X = 0 are positive, i.e., that W is negative semi-definite. In the same notation as in V.5 one has for any vector pn = x pl in the Hilbert space... 

It is a random walk over the integers n = 0,1,2,... with steps to the right alone, but at random times. The relation to chapter II becomes more clear by the following alternative definition. Every random set of events can be treated in terms of a stochastic process Y by defining Y(t) to be the number of events between some initial time t = 0 and t. Each sample function consists of unit steps and takes only integral values n = 0,1, 2,... (fig. 5). In general this Y is not Markovian, but if the events are independent (in the sense of II.2) there is a probability q(t) dt for a step to occur between t and t + dt, regardless of what happened before. If, moreover, q does not depend on time, Y is a Poisson process. 

In his interesting paper Professor Nicolis raises the question whether models can be envisioned which lead to a spontaneous spatial symmetry breaking in a chemical system, leading, for example, to the production of a polymer of definite chirality. It would be even more interesting if such a model would arise as a result of a measure preserving process that could mimic a Hamiltonian flow. Although we do not have such an example of a chiral process, which imbeds an axial vector into the polymer chain, several years ago we came across a stochastic process that appears to imbed a polar vector into a growing infinite chain. 

Along with the isomerism of linear copolymers due to various distributions of different monomeric units in their chains, other kinds of isomerisms are known. They can appear even in homopolymer molecules, provided several fashions exist for a monomer to enter in the polymer chain in the course of the synthesis. So, asymmetric monomeric units can be coupled in macromolecules according to "head-to-tail" or "head-to-head"—"tail-to-tail" type of arrangement. Apart from such a constitutional isomerism, stereoisomerism can be also inherent to some of the polymers. Isomers can sometimes substantially vary in performance properties that should be taken into account when choosing the kinetic model. The principal types of such an account are analogous to those considered in the foregoing. The only distinction consists in more extended definition of possible states of a stochastic process of conventional movement along a polymer chain. 

For a coherent state oto), p = a0)(oto, and the quasiprobability distribution P a) = 8 (a — oto) giving ((a+)man) = (a )man). When P(a) is a well-behaved, positive definite function, it can be considered as a probability distribution function of a classical stochastic process, and the field with such a P function is said to have classical analog. However, the P function can be highly singular or can take negative values, in which case it does not satisfy requirements for the probability distribution, and the field states with such a P function are referred to as nonclassical states. 

The derivation of the error variance requires some theory from different fields. For the convenience of the reader a very short overview will be given, including some basic principles and definitions of the required theory. Another reason to give some textbook theory is that the definition of several quantities can differ literature is not very consistent in that respect. A detailed description of signal theory, system theory, stochastic processes and of course mathematics can be found in several textbooks (1-5). 

A formal definition of a stochastic process is given in Aj ndix A. 

We begin with some definitions. A random process is usually referred to as a stochastic process. This is a collection of random variables X t) and the process may be either discrete or continuous. We write X t),t T and a sample x(f),0 < f < fmax of the random process X(f),f > 0 is known as the realisation or path of the process. 

This definition shows that Brownian motion is closely linked to the Gaussian/normal distribution. The formalism of Weiner processes opens stochastic processes to rigorous mathematical analysis and has enabled the use of Weiner processes in the field of stochastic differential equations. Stochastic differential equations are analogs of classical differential equations where... 

New methodology for exact reliability quantification of highly reliable systems with maintenance was introduced in (Bris 2008a). It assumes that the system structure is mathematically represented by the use of directed acyclic graph (AG), see more details in (Bris 2008b). Terminal nodes of the AG that represent system components are established by the definition of deterministic or stochastic process, to which they are subordinate. From them we can compute a time dependent unavailability function, of individual terminal nodes. Finally a correspondent time dependent imavailability function U(x,t) of the highest node (SS node or top event in classic PRA model) which represents rehabdity behaviour of the whole system may be found. It is clear that U(x,t) < Us(x). 

Markoff processes, I should like to stress the fact that many of the ideas developed below are already weU known to specialists in the theory of stochastic processes [5]. By definition, the stochastic case is the case that P is a so-caUed stochastic matrix satisfying = 1 (whence Q satisfies... 

Because the rupture event is a stochastic process, the rupture forces are distributed in a certain range. Therefore, as in the experimental studies, we performed a large number of simulations and analyzed the rupture force distributions and the rejoin force distributions. The mean values as a function of the pulling velocity represent the so-called force spectmm. Experimentally, often a logarithmic dependence of the mean rupture force on v is observed, but the data collected by Schlesier et al. [98] allow no definite conclusion regarding this dependence. 

Fatigue tests are both costly and time consuming because of the number of samples that must be tested in an expensive test machine for days or weeks until fracture. Also, since fracture is a stochastic process by nature, repeated tests should be conducted at each stress level considered. Finally, the number of cycles to failure, Ni, gives no clear indication of the fatigue limits for a given application of a polymeric material because in many cases, the definition of failure is ambiguous. For example, failure may be defined as a decrease in specimen stiffness or other irreversible changes rather than actual specimen fracture. 

The cybernetic description of systems of different types is characterized by concepts and definitions like feedback, delay time, stochastic processes, and stability. These aspects will, for the time being, be demonstrated on the control loop as an example of a simple cybernetic system, but one that contains all typical properties. Thereby the importance of the probability calculus and communication in cybernetics can be clearly explained. This is then followed by a general representation of cybernetic systems. 

Leira, B.J.A. Comparison of stochastic process models for definition of design contours. Structural Safety (30) 493-505, 2008. 

The angular brackets indicate averaging over a sufficiently long period of time or equivalently an equilibrium average. h, t) and g(f, t) are functions corresponding to the specific stochastic process. With the definition of the correlation function given in Equation 6.52, the noise strength is absorbed into g(, t). 

It is well known that the earthquake excitation can be represented as a nmistatimiaiy process, and a correct definition has to be addressed through the theory of stochastic process (Chopra 1995). 

In the analysis of stochastic processes, an important role is played by the so-called spectral moments (SM), introduced by Vanmarcke (1972), important for the definition of some characteristics of stochastic processes and in reliability analysis (Michaelov et al. (1999)). These quantities are defined as the moments of the one-sided PSD with respect to the frequency origin. Let Sy (co) be the PSD of the earthquake acceleration with 5y(ffl) = 5y(—co). Let Gy(co) be the one-sided PSD defined... 

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