Theory Of Diffusion Processes Research Articles

Diffusion processes in linear spaces and pseudotopologies

The theory of diffusion processes with a nonnormable phase space (a nuclear Frechet space) is developed and the Cauchy problem for parabolic equations relative to functions on this space is solved by probabilistic methods. A series of examples are given, demonstrating a significant difference between the theory of stochastic differential equations and parabolic equations in the case of locally convex spaces, on one hand, and the analogous theory in the case of Banach spaces, on the other hand. The difficulty which arises, when passing from a Banach space to a Frechet space, involves basically a functional rather than a probabilistic character. There appears a sufficiently complex intertwinement of the theory of locally convex and pseudotopological spaces with probability theory.

Time reversal and stationarity of infinite-dimensional Markov birth-and-death processes

By using stochastic calculus for pure jump martingales, we study a class of infinite-dimensional birth-and-death processes. A technique, based on a relative entropy condition, which is adopted from diffusion process theory, enables us to also handle the corresponding processes obtained by reversing the direction of time. The duality between the processes of forward and backward time, respectively, is considered for Markov processes, defined by a prescribed family of upward and downward jump rates. A new characterization is obtained of the probability measures, which are invariant for the stochastic evolution associated with a specific set of jump rates. It leads to the conclusion that, if phase transition occurs, then all measures, with a given set of local conditional distributions in common, are invariant provided one of them is.

On the Spectral Properties of a Class of Elliptic Functional Differential Operators Arising in Feedback Control Theory for Diffusion Processes

In this paper Dirichlet boundary value problems are considered for certain operators of the form $L + \Pi $, where L is a 2nd order, elliptic, formally self-adjoint $PDO$ and $\Pi $ is a feedback operator with finite-dimensional range.>The results concern mainly the resolvent $(L + \Pi - \lambda )^{ - 1} $, the analyticity of the semigroup generated by $L + \Pi $, the location of the spectrum $\sigma $$\sigma (L + \Pi )$ and the completeness of the eigenspaces and the eigenprojections associated to a $(L + \Pi )$.As a consequence of the completeness of the eigenprojections it is possible to derive quite a number of interesting formulas of the “resolution of the identity” type.

A STOCHASTIC LOGISTIC MODEL WITH TIME DELAY

SummaryThe stochastic version of the logistic model for population growth is generalized to take account of continuously distributed time delay with an exponentially decaying kernel. The theory of diffusion processes is used to analyse the probability density function of the population size. The explicit expression for the stationary distribution is worked out and the effect of time delay on various statistics is discussed.

Stochastic continuous-time model reference adaptive systems with decreasing gain

Recursive estimation is considered for parameters of certain continuous stochastic models. Several optimality properties are shown to hold for the resulting recursive estimator, where a stochastic approximation viewpoint is taken when deriving statistical properties, like strong consistency and convergence in distribution. Applications are considered throughout, where for example explosion theory for diffusion processes is used as a modeling guide, in a particular application.

Stochastic continuous-time model reference adaptive systems with decreasing gain

Recursive estimation is considered for parameters of certain continuous stochastic models. Several optimality properties are shown to hold for the resulting recursive estimator, where a stochastic approximation viewpoint is taken when deriving statistical properties, like strong consistency and convergence in distribution. Applications are considered throughout, where for example explosion theory for diffusion processes is used as a modeling guide, in a particular application.

Method of moments in problems of dynamics of systems with randomly varying parameters

Derivation of general equations of the method of moments for linear systems with random variation of parameters is presented on the basis of Markovian theory of diffusion processes. As an example, a system with a single degree of freedom with random variation of its natural frequency with periodic or random external action, and a system with two degree of freedom under conditions of stochastic combination resonance are considered. These examples are used for demonstrating the method of moments in conjunction with that of averaging.

New approach to the semiclassical limit of quantum mechanics

We propose a new approach for the estimate of the rate of degeneracy of the lowest eigenvalues of the Schrodinger operator in the presence of tunneling based on the theory of diffusion processes. Our method provides lower and upper bounds for the energy splittings and the rates of localization of the wave functions and enables us to discuss cases which, as far as we know, have never been treated rigorously in the literature. In particular we give an analysis of the effect on eigenvalues and eigenfunctions of localized deformations of 1) symmetric double well potentials 2) potentials periodic and symmetric over a finite interval. Theses situations are characterized by a remarkable dependence on such deformations. Our probabilistic techniques are inspired by the theory of small random perturbations of dynamical systems.

Etude du mecanisme d'oxydo-reduction du cuivre dans les solutions chlorurees acides—I. Systeme CuCu Cl −2

In the area of the curves lof I = f( E) situated on either side of the potential of net zero current, it can be established that the redox system is CuCu Cl − 2. Applying the theory of diffusion processes, the following equation can be calculated ▪ which coincides with experimental curves and includes an anodic straight line with the equation ▪ If we assume the existence of the absorbed species Cu Cl ads on the electrode, it is possible to describe a multiple step process which comprises successively: • the charge transfer step ▪ • the absorption desorption step ▪ when the degree of coverage of the surface with Cu Cl ads is small, the net current density I may be expressed in the form ▪ which is congruent with the experimental results.

Intermediate equations of the theory of diffusion processes with two-point boundary conditions

Intermediate equations of the theory of diffusion processes with two-point boundary conditions

Solution of terephthalic acid in the reaction system terephthalic acid-ethylene glycol-esterification products

UDC 678.674'524'420.03 An investigation of the solution of terephthalic acid (TPA) in ethylene glycol (EG) and their reaction products is of considerable interest because the solution is a necessary precondition for the reaction of direct esterification. It has been established [1] that at high temperatures (->250°C) the esterification proceeds in the diffusion region, i.e. the process rate is limited by the solution process of the TPA. The published information about the influence of the reaction medium on the solubility and esterification of TPA is highly contradictory. Some workers researching the solubility of TPA in organic solvents do not relate the solubility index to the concrete properties of the solvent [2] or relate it to its proton-acceptanc e capacity [ 3]. Krumpole et al. [4] cite data about the solubility of TPA in EG and draw the conclusion that the solubility of TPA does not depend on the presence of reaction products. Other workers [ 5; 6] offer evidence that the presence of diglycol terephthalate (DGT) in the system TPA-EG stimulates the reaction rate but fail to disclose the nature of this effect of the medium. Kodaira et al. [7] also report the favorable influence of DGT on the esterification rate and ascribe it to the entropy effect which increases with the DGT concentration. Other workers [ 8] state, however, that the solubility of TPA is higher in EG than in a mixture of mono- and diesters and show that DGT has an adverse effect on the reaction rate because its basicity characteristics enable it to combine with the TPA over hydrogen bonds resulting in the formation of inactive associated products. In view of these facts it was deemed relevant to investigate the solution of TPA in EG and in their reaction products and the effect of the chemical reaction on the solution process, and to determine the temperature boundary between the purely physical solution process and the onset of the chemical reaction. The experiments were carried out with TPA with a specific surface Ssp of 0.5, 1.5, and 2.9 m2/g (Ssp was determined by the BlOT adsorption method). All experiments were carried out with an EG/TPA ratio of 10 by weight. The COOH groups were determined in specimens from which the solid TPA had first been filtered out. The content of the products of the chemical reaction in the specimens was determined by polarography and thin-film chromatography [9]. To begin with, the influence of the speed of stirring on the rate of TPA solution in EG was investigated. It was found that the solution rate increases with an increase in the stirrer speed to 700 rpm after which it remains unchanged. The influence of the other diffusion factor, viz. Ss_, on the solution rate was therefore investigated at a stirrer speed of 700 rpm. The results are set out in Table 1. According to the theory of diffusion processes, the rate of the solution process described by Shchukarev's equation [ 10] always increases with the specific surface of the substance being dissolved:

Almost sure asymptotic likelihood theory for diffusion processes

We consider maximum likelihood estimators for parameters of diffusion processes that are generated by nth-order Ito equations. We establish asymptotic consistency as well as convergence in distribution to normality for the estimators. Examples are presented and discussed.

Almost sure asymptotic likelihood theory for diffusion processes

We consider maximum likelihood estimators for parameters of diffusion processes that are generated by nth-order Ito equations. We establish asymptotic consistency as well as convergence in distribution to normality for the estimators. Examples are presented and discussed.

Studies on the Release of Ammonium Nitrogen from the Bottom Sediments in Freshwater Bodies-I

In an aquatic environment, bottom mud plays an important role in the determination of the quality of water, especially in shallow freshwater bodies. A considerable amount of nutrients, originating from organic matter, are often accumulated in bottom mud and released into the water, which may cause eutrophication. From this point of view, a quantitative study has been carried out as to the release of ammonium nitrogen into the water from the bottom mud of Akanoi Bay in Lake Biwa by the use of an annular channel. The authors applied diffusion process theory in this study and obtained the diffusion coefficient of 5×10-6cm2/sec when overlying water was still and 2×10-5cm2/sec when it was moving with current velocity 1040 cm/sec. Ammonium nitrogen in mud must be considered to consist of diffusive and non-diffusive parts in order to explain the change in the vertical distribution of ammonium nitrogen in mud. The diffusive part of ammonium nitrogen in each mud sample with various ammonia concentrations was determined. The increasing rate of ammonium nitrogen with the lapse of time agrees well with the release rate estimated from the diffusion process theory.

Asymptotic likelihood theory for diffusion processes

We investigate the large-sample behaviour of maximum likelihood estimates (MLE's) of the parameters of a diffusion process, which is observed throughout continuous time. The results (limit normal distribution for the MLE and an asymptotic chi-squared likelihood ratio test) correspond exactly to classical asymptotic likelihood results, and follow easily from a central limit theorem for stochastic integrals.

Asymptotic likelihood theory for diffusion processes

We investigate the large-sample behaviour of maximum likelihood estimates (MLE's) of the parameters of a diffusion process, which is observed throughout continuous time. The results (limit normal distribution for the MLE and an asymptotic chi-squared likelihood ratio test) correspond exactly to classical asymptotic likelihood results, and follow easily from a central limit theorem for stochastic integrals.

On the Convergence of Random Search Algorithms In Continuous Time with Applications to Adaptive Control

The use of a gradient search algorithm for the computation of the minimum of a function is well understood. In particular, in continuous time, the algorithm can be formulated as a differential equation whose equilibrium solution is a local minimum of the function. Khas'minskii has proposed using the continuous gradient algorithm with a white-noise driving term. He shows, using an argument on the convergence of the probability density function, that the equilibrium solution of this differential equation is the global minimum of the function. This paper reviews that result from the point of view of the theory of diffusion processes. It is shown that the conclusion of Khas'minskii is not correct, and that in fact the operation of the stochastic gradient search is the same as the operation of the noise-free gradient search. In fact, the search with noise converges with probability one to any of the local minima in the same way as the noise-free search converges to a local minimum. Several stochastic adaptive control systems use random search algorithms for their operation. One of these, due to Barron, is analyzed to show that it meets the conditions for convergence imposed by the results which have been derived here.

A Diffusion Method for Reliability Prediction

Elements of the theory of diffusion processes are utilized to develop methods for predicting the reliability and the moments of the time to first failure of systems that have nonconstant failure rates and exhibit degradation failure. Specifically, systems characterizable by k independent parameters are considered, each of which independently exhibits degradation failure. The methods presented for obtaining a probabilistic description of such systems rest on the assumption that the time behavior of each of the system parameters can be characterized by a Brownian process (a special type of diffusion process). The methods then allow predictions of reliability and the moments of the time to first failure to be made from data taken early in life tests. In particular, the first moment is the mean time to first failure of the system.

On the solution of the equation of diffusion processes during the uptake of excess calcium by calcium fluoride

This paper presents an iteration method for the solution of a generalized diffusion equation derived from the theory of diffusion process during the uptake of excess calcium by calcium fluoride. The iteration process can be carried out by a digital computer to obtain a numerical integration of the diffusion equation with estimated accuracy. By using a GE-225 computer it takes, on the average, no more than 2 min to obtain a solution of the diffusion equation to within 1 per cent error. To demonstrate the general applicability of the method numerical solutions for a wide range of the parameter have been worked out.

Die Hydrographie der Nordsee. Ein Überblick über unsere Kenntnisse in Beziehung zu Problemen der Gewässerverunreinigung

1. The development of the various theories of diffusion processes in the open sea are reviewed with particular reference to work in the North Sea. 2. The stages by which our present knowledge of the residual current system of the region has been built up are discussed. 3. A detailed discussion follows of the methods by which current measurements of one kind or another have been made at various points in the water column and the use to which these measurements have been put. 4. An outline is given of the way in which we feel future work in this field must develop.