Showing papers on "Stochastic process published in 1980"
TL;DR: In this article, a generic stochastic finite-element method for modeling structures is proposed as a means to analyze and design structures in a probabilistic framework, which is applied in structures discretized with the finite element methodology, and an estimate of the probability of failure based on known and established procedures in second-moment reliability analysis is made with the aid of a transformation to gaussian space of the random variables that define structural reliability.
648 citations
TL;DR: In this article, the authors proposed a method to determine the velocities of simulated molecules after a small time increment was derived from the Boltzmann equation, which was shown to give an exact solution of the Boltzman equation.
Abstract: The stochastic law that prescribes the velocities of simulated molecules after a small time increment was derived from the Boltzmann equation. The scheme to determine the velocity of a molecule after a small time increment is divided into three steps. The first step gives the collision probability of the molecule without specifying its collision partner. The second step gives a conditional probability distribution. If the molecule is accepted in the first step as a colliding molecule, its collision partner is sampled from this probability distribution. The last step gives a probability density from which the direction of the relative velocity after collision is sampled, and hence the step gives the post-collision velocity of the molecule. It is shown that the use of the present simulation scheme gives an exact solution of the Boltzmann equation.
360 citations
TL;DR: In this article, a new definition of concentration fluctuations in turbulent flows is proposed, which implicitly incorporates smearing effects of molecular diffusion and instrumental averaging, and a stochastic model of two-particle dispersion, consistent with this definition, is formulated.
Abstract: A new definition of concentration fluctuations in turbulent flows is proposed. The definition implicitly incorporates smearing effects of molecular diffusion and instrumental averaging. A stochastic model of two-particle dispersion, consistent with this definition, is formulated. The stochastic model is an extension of Taylor's (1921) model and is consistent with Richardson's represents net destruction of fluctuations by relative dispersion. Only the first term is included in the usual one-particle model (Corrsin 1952).
260 citations
TL;DR: An algorithm that generates an MRF on a finite toroidal square lattice from an independent identically distributed (i.i.d.) array of random variables and a given set of independent real-valued statistical parameters is presented.
209 citations
TL;DR: In this paper, the authors proposed new criteria by which to gauge the extent of quantum intramolecular randomization in isolated molecules, and found that it is very important to tailor the criteria to the specific experimental situation, with the consequence that a given molecule can be labeled both stochastic and nonstochastic, even in the same general energy regime, depending on the experiment.
Abstract: This paper proposes new criteria by which to gauge the extent of quantum intramolecular randomization in isolated molecules. Several hallmarks of stochastic and nonstochastic behavior are identified, some of which are readily available from spectral data. We find that it is very important to tailor the criteria to the specific experimental situation, with the consequence that a given molecule can be labeled both stochastic and nonstochastic, even in the same general energy regime, depending on the experiment. This unsettling feature arises as a quantum analog of the necessity, in classical mechanics, of specifying the a priori known integrals of the motion before ergodic or stochastic behavior can be defined. In quantum mechanics, it is not possible to have flow or measure local properties (analog of trajectories and phase space measure) without some uncertainty in the integrals of the motion (most often the energy). This paper addresses the problems this creates for the definition of stochastic flow. Sev...
117 citations
TL;DR: In this article, the Navier-Stokes invariant measures for a time evolution in a finite-dimensional space 7 f have been studied and the notions of strange attractor, sensitive dependence on initial condition, and characteristic exponents have been useful in this respect.
Abstract: INTRODUCTION The motion of a fluid in a region R of R' or R' is defined by a function t : , ( t ) , where u ( t ) belongs to some functional space, %, of velocity fields in R. In a turbulent regime, one expects r , ( t ) to be distributed according to some probability law. This probability law is defined by a measure p on 7f . invariant under the deterministic time evolution of the system. We have reached some understanding of the invariant measures for a time evolution in a finite-dimensional space 7 f . The notions of strange attractor, sensitive dependence on initial condition, and characteristic exponents have been useful in this respect. Also, it is possible to define stable and unstable manifolds almost everywhere, and one can, in a number of cases, identify those measures which are stable under small stochastic perturbations. In this paper, we discuss the extension of results obtained for finite-dimensional dynamical systems to the more realistic case of the time evolution defined in a Hilbert space by the Navier-Stokes equation.
93 citations
TL;DR: In this article, the authors consider unsaturated and generally unsteady flow in heterogeneous systems and review the mathematical nature of the flow equation, the concept of scale-heterogeneity, analytical and quasi-analytical solutions.
Abstract: Present-day soil-water physics enables useful quantitative predictions in the laboratory and in simple field situations. Difficulties, however, frequently arise for areas of appreciable size in the field. Known and unknown heterogeneities, on many scales, may vitiate predictions based on theory for homogeneous, or very simple heterogeneous, systems. Two types of heterogeneity are distinguished, deterministic and stochastic. The first often demands an extension of established analyses and may involve important phenomena absent from the analogous homogeneous problem. Stochastic heterogeneity may involve many scales and is imperfectly known. The statistical properties may be stationary, but in more complicated cases, randomness may be embedded in (either known or unknown) systematic trends. Some aspects of unsaturated and generally unsteady flow in heterogeneous systems are reviewed: the mathematical nature of the flow equation; the concept of scale-heterogeneity; analytical and quasi-analytical solutions. The enormity of the total problem of unsaturated unsteady flows in stochastic heterogeneous systems is illustrated through a dialectic of eight successive stages of simplification. The concept of the autocorrelation function governing λ, the internal characteristic length, is introduced; and the problem posed in terms involving the distribution and autocorrelation functions of λ, the reduced potential and conductivity functions, and the initial and boundary conditions as the data, from which it is required to establish distribution functions of various descriptors of the flow. The solution to a grossly simplified example of horizontal absorption is presented. Mean apparent sorptivity decreases rapidly to about one fifth of the mean (and about half the minimum) sorptivity of the component soils. Variation about the mean is very great but decreases as absorption proceeds. The example epitomizes the failure of additivity of properties in stochastic heterogeneous media, which arises because spatial textural changes in either sense tend to reduce unsaturated flow rates.
89 citations
TL;DR: In this article, an approach to study the exponential stability of linear difference equations with random coefficients through the use of Lyapunov stability techniques is presented. But the approach is restricted to the case where the coefficients of the difference equations are random.
Abstract: We consider an approach to studying the exponential stability of linear difference equations with random coefficients through the use of Lyapunov stability techniques. The equations we study are of a form familiar from adaptive estimation algorithms, which motivates the examination. It is necessary to define the almost sure exponential convergence of a random process, and then to derive sufficient conditions on the coefficients of the difference equations to ensure the almost sure exponential convergence of the state. We consider, in particular, two very reasonable types of random coefficients-ergodic and stationary and φ-mixing and nonstationary-which would appear to encompass many engineering situations. An example of the power of the theory is given, where it is applied to a common adaptive filtering algorithm to derive mild conditions for exponential convergence with dependent random inputs.
82 citations
TL;DR: A general convergence result is given for stochastic approximation schemes with (or without) equality constraints, both classical and nonclassical ones, to illustrate the applicability of the convergence theorem.
Abstract: A general convergence result is given for stochastic approximation schemes with (or without) equality constraints. The following features are taken into account. The forcing term is a strongly dependent sequence and may be discontinuous. Many examples are given to illustrate the applicability of the convergence theorem, both classical (recursive least squares scheme) and nonclassical ones (arising in the theory of self-adaptive eqnalizers).
81 citations
TL;DR: In this paper, a stochastic trajectory simulation of iodine recombination in dense liquid solvents is presented, utilizing a mean force potential which contains direct I-I interactions as well as solvent structure effects.
Abstract: A stochastic trajectory simulation of iodine recombination in dense liquid solvents is presented. The calculations utilize a mean force potential which contains direct I–I interactions as well as solvent structure effects. Dynamical solvent effects are accounted for by a random force and friction coefficient. The time dependent probability of reaction for two initially separated radicals is determined. The choice of initial separations and atomic velocity distributions is appropriate for secondary recombination. The results of this study show the importance of including the strong direct chemical forces between the I atoms; the validity of simple diffusion equation approaches can thus be assessed. Effects due to solvent structure are quantitatively examined and are interpreted in terms of ’’caging’’ in dense fluids. The computer simulation results are also compared with the solution of the Smoluchowski equation for this problem and effects due to friction coefficient variation are discussed.
TL;DR: In this article, the authors studied the asymptotic behavior of the solution of a bilinear stochastic evolution equation for large times and established conditions for the existence of a stationary random field.
TL;DR: In this article, the effects of the initial preparation of the generalized Fokker-Planck and Langevin equations are taken into account explicitly, which allows for the construction of uniquely determined projection operator.
Abstract: Starting from classical Hamiltonian mechanics, we derive for the dynamics of gross variables in nonequilibrium systems exact nonlinear generalized Fokker-Planck and Langevin equations in which the effect of the initial preparation is taken into account explicitly. This latter concept allows for the construction of a uniquely determined projection operator. The memory functions occurring in the Langevin equations are related to the random forces by a fluctuation-dissipation theorem of the second kind. We discuss the connection with the generalized Fokker-Planck equation. The known results for equilibrium fluctuations are recovered as a special case.
TL;DR: In this article, an almost sure approximation of R(s,t) by a Kiefer process was given for stationary sequences of random vectors with continuous distribution function and satisfying a strong mixing condition.
Abstract: Let R(s,t) be the empirical process of a sequence of independent random vectors with common but arbitrary distribution function. In this paper we give an almost sure approximation of R(s,t) by a Kiefer process. The result continues to hold for stationary sequences of random vectors with continuous distribution function and satisfying a strong mixing condition.
TL;DR: The construction is accomplished for semi-Markov processes for which all subprobability transition rates are absolutely continuous with failure rates uniformly bounded over finite intervals by representing the two semi- Markov processes as compositions of discrete-time stochastic processes with a sequence of Poisson processes.
Abstract: Sufficient conditions are found for two semi-Markov processes to be stochastically ordered, i.e., for which two new semi-Markov processes can be constructed on a common probability space so that the new processes individually have the same distributions as the original processes and every sample path of the first new process lies below the corresponding sample path of the second new process. This ordering has recently been shown by Kamae, Krengel, and O'Brien Kamae, T., V. Krengel, G. L. O'Brien. 1977. Stochastic inequalities on partially ordered spaces. Ann. Probab.5 899--912. to be equivalent to the definition of stochastic order usually seen in the literature; see Veinott Veinott, A. F. Jr. 1965. Optimal policy in a dynamic, single product, nonstationary inventory model with several demand classes. Oper. Res.13 761--778.. The conditions are: i the initial distribution of the first process is stochastically smaller than that of the second; ii for each ordered triple of states j ≤ k ≤ 1 and unordered holding times s and t, the instantaneous transition rate of the first process from j to the set of states exceeding 1 given its holding time in j is s is not less than the corresponding transition rate of the second process where j and s are replaced by k and t respectively; and iii the dual of condition ii obtained by reversing the order of the states. The construction is accomplished for semi-Markov processes for which all subprobability transition rates are absolutely continuous with failure rates uniformly bounded over finite intervals by representing the two semi-Markov processes as compositions of discrete-time stochastic processes with a sequence of Poisson processes. This permits application of recent comparison results for discrete-time stochastic processes by O'Brien O'Brien, G. L. 1975a. The comparison method for stochastic processes. Ann. Probab.3 80--88; O'Brien, G. L. 1975b. Inequalities for queues with dependent interarrival and service times. J. Appl. Probab.12 653--656..
TL;DR: Operational analysis, an alternative to stochastic analysis based on measurable variables rather than abstract parameters, simplifies proofs of formulas for computing response times and queue lengths.
Abstract: Operational analysis, an alternative to stochastic analysis based on measurable variables rather than abstract parameters, simplifies proofs of formulas for computing response times and queue lengths.
TL;DR: It is shown that the “equivalent deterministic demand” is always greater than the ”expected demand,” where the latter is defined by the expected time to first reach various levels of demand, and is not the expected number of customers.
Abstract: When demand is assumed to be a birth-death process, and capacity expansion costs are assumed to occur instantaneously at the time of expansion, it is shown that an “equivalent” deterministic-demand problem can readily be generated. The derived problem is equivalent in the sense that its solution by ordinary deterministic capacity expansion methods would also yield the solution of the stochastic problem. It is shown that the “equivalent deterministic demand” is always greater than the “expected demand,” where the latter is defined by the expected time to first reach various levels of demand, and is not the expected number of customers. In addition to general formulas for the discrete-customer case, equations are also derived for the “equivalent deterministic demand” when demand is based on a diffusion process.
TL;DR: In this paper, the effect of finite memory time cannot be ignored in many dispersion problems: it is proposed that a random flight model will be more satisfactory than the diffusion approximation in such situations.
Abstract: The effect of finite memory time cannot be ignored in many dispersion problems: It is proposed that a random flight model will be more satisfactory than the diffusion approximation in such situations. A calculation of dispersion in open channel flow is presented.
TL;DR: In this article, a model based on stochastic processes is proposed for predicting the spread of fire in a building, and an outline of such a model is given specifying the data required for estimating and validating the parameters of the model.
TL;DR: In this paper, a microscopic transport theory for stochastic and correlated hopping on ordered and random lattices that contain a small fraction of supertraps and a small number of "hoppers" (i.e., excitons).
Abstract: A microscopic transport theory is developed for stochastic and correlated hopping on ordered and random lattices that contain a small fraction of supertraps and a small number of ’’hoppers’’ (i.e., excitons). It includes short‐time (’’transient’’) behavior, which is of interest for both time‐resolved and steady‐state experiments. The relations with diffusion, percolation, random walk, and rate equations are exhibited and applications to energy transport in disordered molecular aggregates illustrate the approach, which is a combination of a rigorous analytical method and simple computer simulations of general validity. Simple analytical results, derived for special (limiting) cases, are compared with other methods, thus emphasizing the roles of time, dimensionality, anisotropy, clusterization, correlation of hops, and the order parameter of the lattice as well as the suitability of various approaches for dealing with these factors.
TL;DR: In this paper, a new theory of random surface scattering based on the group theoretic consideration of the stochastic homogeneity of the infinite random surface is described, and the wave solution is then written in terms of a stationary random function that is approximately solved as a stochiastic functional of the random surface.
Abstract: This paper describes a new theory of random surface scattering based on the group theoretic consideration of the stochastic homogeneity of the infinite random surface. To show the basic idea of the theory, we only discuss scalar wave scattering for one-dimensional random surface that is described by a reactance boundary condition having a random reactance of a stationary random function. A form of the stochastic wave solution associated with the random boundary condition is determined by the group theoretic consideration. The wave solution is then written in terms of a stationary random function that is approximately solved as a stochastic functional of the random surface. The optical theorem, the angle dependence of the incoherent scattering, and the energy flow of the surface wave are calculated and are shown in the figures.
TL;DR: In this paper, an adiabatic elimination from the Langevin equations is proposed for a stochastic Haken-Zwanzig model for non-equilibrium phase transitions, which is applied to a laser model for illustration.
Abstract: 1)iJ(t-t'). · With the aid of stochastic processes of this new type, an adiabatic elimination from the Langevin equations is proposed for a stochastic Haken-Zwanzig model for non-equilibrium phase transitions. A projector elimination from the Langevin equations and an adiabatic elimination from the Fokker-Planck equation are also explored. Calculation is carried out up to second order in the slowness parameter. Three different methods are thus developed with consistent results and are applied to a laser model for illustration.
TL;DR: In this article, it was shown that such processes consist of a random sequence of delta functions with random coefficients, whose solutions of the differential equation are Markov processes, whose master equation can be constructed, from which closed equations for the successive moments may be obtained, and the auto-correlation is determined.
Abstract: In a recent paper1) a differential equation was studied which involves a stochastic process having the property that all its cumulants are delta-correlated. It is here shown that such processes consist of a random sequence of delta functions with random coefficients. As a consequence the solutions of the differential equation are Markov processes, whose master equation can be constructed. From it closed equations for the successive moments may be obtained, and the auto-correlation is determined, in agreement with the results of reference 1. Some generalizations are given in Appendices B and C.
TL;DR: It is shown here that threshold crossing probabilities using the mixture density can be reliably approximated by integrations of an equivalent continuous Gaussian density.
Abstract: The output voltage of an optical receiver is statistically a mixture random variable, composed of the sum of a discrete count variable and a continuous Gaussian thermal noise variable. Based on some computer analyses, it is shown here that threshold crossing probabilities using the mixture density can be reliably approximated by integrations of an equivalent continuous Gaussian density. The conditions for this approximation and an accurate assessment of its error are derived.
TL;DR: In this article, a theory of stochastic calculus of variations is presented which generalizes the ordinary calculus of variation to stochastically processes and generalizations of the Euler equation and Noether's theorem are obtained.
Abstract: A theory of stochastic calculus of variations is presented which generalizes the ordinary calculus of variations to stochastic processes. Generalizations of the Euler equation and Noether's theorem are obtained and several conservation laws are discussed. An application to Nelson's probabilistic framework of quantum mechanics is also given.
TL;DR: In this article, the response of a lightly damped linear structure to a broad-band nonstationary random process with evolutionary spectral density is considered, and a first-order stochastic differential equation governing the time evolution of the structural energy is derived.
Abstract: The response of a lightly damped linear structure to a broad-band nonstationary random process with evolutionary spectral density is considered. A first-order stochastic differential equation governing the time evolution of the structural energy is derived. Utilizing this equation a readily applicable equation for the determination of the mean energy is obtained. Nonstationary random processes proposed in the literature for the simulation of earthquakes are examined in detail, and equations for the construction of probabilistic energy spectra are presented.
TL;DR: In this paper, a relatively simple method is developed whereby the manybody features of a typical generalized Fokker-planck equation (GFPE) for a diffusing molecule are first replaced by stochastic bath variables that are assumed to be Markovian.
Abstract: A relatively simple method is developed whereby the many‐body features of a typical generalized Fokker–Planck equation (GFPE) for a diffusing molecule are first replaced by stochastic bath variables that are assumed to be Markovian. Then the combined molecular and bath variables are characterized as a multidimensional Markov process obeying a stochastic–Liouville equation, which is, in general, incomplete, because it ignores the back reaction of the molecule on the bath variables. In the final step, the equation is completed by subjecting it to the appropriate constraints required for detailed balance. In this form the augmented Fokker–Planck equation (AFPE) properly describes relaxation to thermal equilibrium, and, for the appropriate limiting conditions, it reduces to the classical Fokker–Planck equation. This procedure for stochastic modeling of GFPE is both an improvement on and a generalization of a method previously outlined by Hwang, Mason, Hwang, and Freed (HMHF). Detailed illustrations of AFPE’s ...
TL;DR: An unusual definition of mutual information rate for continuous-alphabet processes is used, but it is shown to be operationally appropriate and more useful mathematically and it allows generalizations of some fundamental results of ergodic theory that are useful for information theory.
Abstract: Several new properties as well as simplified proofs of known properties are developed for the mutual information rate between discrete-time random processes whose alphabets are Borel subsets of complete separable metric spaces. In particular, the asymptotic properties of quantizers for such spaces provide a fink with finite-alphabet processes and yield the ergodic decomposition of mutual information rate. This result is used to prove the equality of stationary and ergodic process distortion-rate functions with the usual distortion-rate function. An unusual definition of mutual information rate for continuous-alphabet processes is used, but it is shown to be operationally appropriate and more useful mathematically; it provides an intuitive link between continuous-alphabet and finite-alphabet processes, and it allows generalizations of some fundamental results of ergodic theory that are useful for information theory.
TL;DR: In this paper, a method for proving weak convergence of a sequence of non-Markovian processes to a jump-diffusion process is proved. But it is not shown that the limit solves the martingale problem of Strook and Varadhan.
Abstract: A convenient method for proving weak convergence of a sequence of non-Markovian processesxe(·) to a jump-diffusion process is proved. Basically, it is shown that the limit solves the martingale problem of Strook and Varadhan. The proofs are relatively simple, and the conditions apparently weaker than required by other current methods (in particular, for limit theorems for a sequence of ordinary differential equations with random right hand sides). In order to illustrate the relative ease of applicability in many cases, a simpler proof of a known result on averaging is given.