Diffusion-type Processes Research Articles

A stochastic equation for the distribution law of diffusion type processes

A stochastic equation for the distribution law of diffusion type processes

Absolute continuity of measures in the class of Markov and semi-Markov processes of diffusion type

The property of absolute continuity of measures in the class of one-dimensional semi-Markov processes of diffusion type is investigated. The measure of such a process can be composed of two measures. The first one is a distribution of a random track, and the second one is a conditional distribution of a time run along the track. The desired density is represented in the form of product of two corresponding densities.

A stochastic equation for the distribution law of diffusion type processes

We study distribution laws of diffusion type processes and corresponding generalized functions. The stochastic integral equation satisfed by these generalized functions is derived.

EXTENSION OF THE METHOD OF EIGENFUNCTIONS TO THE BOUNDARY-VALUE PROBLEMS OF MECHANICAL DIFFUSION FOR MULTILAYER BODIES WITH INTERLAYERS

A method of eigenfunctions is developed for nonclassical axisymmetrical problems of heat and substance diffusion with generalized contact conditions in multilayer bodies. Based on a scalar product of a nonclassical form, the solutions of the above problems are constructed in the form of expansions in terms of eigenvalues. The solution of the problem of finding stresses caused by diffusion-type processes in multilayer bodies with interlayers is found.

Stochastic models of exotic transport

Non-typical transport phenomena may arise when randomly driven particles remain in an active relationship with the environment instead of being passive. If we attribute to Brownian particles an ability to induce alterations of the environment on suitable space–time scales, those in turn must influence their further movement. In that case a general feedback mechanism needs to be respected. By resorting to a specific choice of the particle-bath coupling, an enhanced (super-diffusion) or non-dispersive diffusion-type processes are found to exist in generically non-equilibrium contexts.

Quantifying tillage translocation and deposition rates due to moldboard plowing in the Palouse region of the Pacific Northwest, USA

Most of the erosion research in the Palouse region of eastern Washington State, USA has focused on quantifying the rates and patterns of water erosion for purposes of conservation planing. Tillage translocation, however, has largely been overlooked as a significant geomorphic process on Palouse hillslopes. Tillage translocation and tillage deposition together have resulted in severe soil degradation in many steep croplands of the Palouse region. Few controlled experiments have heretofore been conducted to model these important geomorphic processes on Palouse hillslopes. The overarching purpose of this investigation, therefore, was to model tillage translocation and deposition due to moldboard plowing in the Palouse region. Soil movement by moldboard plowing was measured using 480-steel flat washers. Washers were buried in silt loam soils on convex–convex shoulder, linear-convex backslope, and linear-concave footslope landform components, and then displaced from their original burial locations by a moldboard plow pulled by a wheel tractor traveling parallel to the contour at ca. 1.0 m s −1. Displaced washers were located using a metal detector, and the distance and azimuth of the resultant displacement of each washer from its original burial location was measured using compass and tape. Resultant displacement distances were then resolved into their component vectors of displacement parallel and perpendicular to the contour. A linear regression equation was developed expressing mean soil displacement distance as a function of slope gradient. Tillage translocation and deposition were modeled as diffusion-type geomorphic processes, and their rates were described in terms of the diffusion constant ( k). A multivariate statistical model was developed expressing mean soil displacement distance as a function of gravimetric moisture content, soil bulk density, slope gradient, and direction of furrow slice displacement. Analysis of variance (ANOVA) revealed a weak correlation between soil displacement and both bulk density and moisture content. Soil displacement was, however, significantly correlated with direction of furrow slice displacement. Tillage translocation rates were expressed in terms of the diffusion constant ( k) and ranged from 105 to 113 kg m −1 per tillage operation. Tillage deposition rates ranged from 54 to 148 kg m −1 per tillage operation. With respect to tillage deposition, the diffusion constant calculated from volumetric measurements of tillage deposits equals ca. 150 kg/m. The rates of tillage translocation and deposition are not completely in balance; however, these rates do suggest that soil tillage is a significant geomorphic process on Palouse hillslopes and could account for the some of the variations in soil physical properties and crop yield potential at the hillslope and farm-field scale in the Palouse region.

Markov Dilation of Diffusion Type Processes and its Application to the Financial Mathematics

Abstract The Markov dilation of diffusion type processes is defined. Infinitesimal operators and stochastic differential equations for the obtained Markov processes are described. Some applications to the integral representation for functionals of diffusion type processes and to the construction of a replicating portfolio for a non-terminal contingent claim are considered.

Noise perturbations in the Brownian motion and quantum dynamics

The third Newton law for mean velocity fields is utilised to generate anomalous (enhanced) or non-dispersive diffusion-type processes which, in particular, can be interpreted as a probabilistic counterpart of the Schrödinger picture quantum dynamics.

Perturbations of noise: Origins of isothermal flows

We make a detailed analysis of both phenomenological and analytic background for the "Brownian recoil principle" hypothesis (Phys. Rev. A 46, (1992), 4634). A corresponding theory of the isothermal Brownian motion of particle ensembles (Smoluchowski diffusion process approximation), gives account of the environmental recoil effects due to locally induced tiny heat flows. By means of local expectation values we elevate the individually negligible phenomena to a non-negligible (accumulated) recoil effect on the ensemble average. The main technical input is a consequent exploitation of the Hamilton-Jacobi equation as a natural substitute for the local momentum conservation law. Together with the continuity equation (alternatively, Fokker-Planck), it forms a closed system of partial differential equations which uniquely determines an associated Markovian diffusion process. The third Newton law in the mean is utilised to generate diffusion-type processes which are either anomalous (enhanced), or generically non-dispersive.

Markov dilation of Diffusion Type Processes and Its Application to the Financial Mathematics

The Markov dilation of diffusion type processes is defined. Infinitesimal operators and stochastic differential equations for the obtained Markov processes are described. Some applications to the integral representation for functionals of diffusion type processes and to the construction of a replicating portfolio for a non-terminal contingent claim are considered.

Consistency issues for numerical methods for variance control, with applications to optimization in finance

The paper is concerned with numerical algorithms for the optimal control of diffusion-type processes when the noise variance also depends on the control. This problem is of increasing importance in applications, particularly in financial mathematics. We discuss the construction of numerical algorithms guaranteed to converge to the true minimum as the discretization level decreases and with acceptable numerical properties. The algorithms are based on the popular Markov chain approximation method. The basic criterion the algorithms must satisfy is a weak "local consistency" condition, which is essential for convergence to the true optimal cost function. This condition is often hard to satisfy by simple algorithms (with essentially only local transitions) when the variance is also controlled. Numerical "noise" can be introduced by the more convenient approximations. This question of "numerical noise" (also called "numerical viscosity") is dealt with in detail, and methods for eliminating or greatly reducing it are discussed.

On the absolute continuity of distributions of occupation times

Some results about the structure of distributions for occupation times $$\tau = \int\limits_T {II_G (t,\xi (t))dt}$$ , where G is a subset of T × ℝ1 and ξ is a Brownian motion or a process of diffusion type, are proved. Bibliography: 10 titles.

On Controlled Diffusion Processes in a Bounded Domain

The paper studies an optimization problem for controlled diffusion processes in a smooth bounded domain. The existence of an optimal control with “good” properties is demonstrated for controlled (in the weak sense) processes of diffusion type.

Controlled and optimally controlled multiplexing systems: A numerical exploration

Large controlled multiplexing systems are approximated by diffusion type processes yielding a very efficient way of approximation and good numerical methods. The “limit” equations are an efficient aggregation of the original system, and provide the basis of the actual numerical approximation to the control problem. The numerical approximations have the structure of the original problem, but are generally much simpler. The control can occur in a variety of places; e.g., “leaky bucket” controllers, control of “marked cells” at the transmitter buffer, or control of the transmitter speed. From the point of view of the limit equations, those are equivalent. Various forms of the optimal control problem are explored, where the main aim is to control or balance the losses at the control with those due to buffer overflow. It is shown that much can be saved via the use of optimal controls or reasonable approximations to them. We discuss systems with one to three classes of sources, various aggregation methods and control approximation schemes. There are qualitative comparisons of various systems with and without control and a discussion of the variations of control and performance as the systems data and control bounds vary. The approach is a very useful tool for providing both qualitative and quantitative information which would be hard to get otherwise. The results have applications to various forms of the ATM and broadband integrated data networks.

Analysis of controlled multiplexing systems via numerical stochastic control methods

The paper demonstrates the usefulness of numerical methods of stochastic control theory for the design, analysis and control of multiplexing type systems and networks, as well as ATM type systems. The sources are of the Markov-modulated type, although the final results hold for other types (e.g. low-order AR schemes). Control problems arise when we wish to control cell loss due to buffer overflows by regulating the sources. The basic control mechanism is the deletion of selected low-priority cells, according to an appropriate state dependent rule. But the same results hold for many other schemes (e.g. purchasing incremental bandwidth). By exploiting the large size of the system (large number of users), the systems can be efficiently approximated by diffusion type processes, whether there is a control term or not, and for many types of control mechanisms. The basic controls are of the "low-priority cell deletion" type, and various extensions. They might be state dependent, and we can obtain optimal controls for cost functions that weigh buffer overflow, controller cell deletion loss as well as queue length. The limit equations are an effective aggregation of the original system. It is shown that there are substantial savings in losses with the use of optimal control techniques. The numerical methods can be used to balance the losses at the control with those due to buffer overflow, to minimize losses at the controller subject to constraints on buffer overflow, and to explore various approximations, systems aggregations, the interaction of multiple source classes of different priorities.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The finite element method: Is weighted volume integration essential?

In developing finite element equations for steady state and transient diffusion-type processes, weighted volume integration is generally assumed to be an intrinsic requirement. It is shown that such finite element equations can be developed directly and with ease on the basis of the elementary notion of a surface integral. Although weighted volume integration is mathematically correct, the algebraic equations stemming from it are no more informative than those derived directly on the basis of a surface integral. An interesting upshot is that the derivation based on surface integration does not require knowledge of a partial differential equation but yet is logically rigorous. It is commonly stated that weighted volume integration of the differential equation helps one carry out analyses of errors, convergence and existence, and therefore, weighted volume integration is preferable. It is suggested that because the direct derivation is logically consistent, numerical solutions emanating from it must be testable for accuracy and internal consistency in ways that the style of which may differ from the classical procedures of error- and convergence-analysis. In addition to simplifying the teaching of the finite element method, the thoughts presented in this paper may lead to establishing the finite element method independently in its own right, rather than it being a surrogate of the differential equation. The purpose of this paper is not to espouse any one particular way of formulating the finite element equations. Rather, it is one of introspection. The desire is to critically examine our traditional way of doing things and inquire whether alternate approaches may reveal to us new and interesting insights.

Nonconsistent estimation by diffusion type observations

We consider the properties of the maximum likelihood, Bayes and minimum distance estimates of the finite-dimensional parameter constructed by the observations of the process of diffusion type with the small coefficient of diffusion in the situation when the trend coefficients corresponding to the different values of the parameter are the same and the consistent estimation is impossible. It is proved that these estimates converge to the different random variables.

Some considerations on mass-transfer resistances in solution—diffusion-type membrane processes

The influence of feed-side concentration polarization and the resistance of the porous support on membrane processes where solution—diffusion-type composite membranes are involved is discussed in this paper. Reverse osmosis, nanofiltration, gas permeation, pervaporation and vapor permeation represent the complete variation of liquid and gaseous state at the feed and permeate side and cover cases where the driving force is realized by increasing the feed-side potential and cases where the permeateside potential is decreased. In the calculations, the major parameters, especially feed-side concentration of the preferentially permeating component is varied over a wide range - from “traces” to “fairly high”. The discussion of the results shows clearly the importance or unimportance of the resistances relative to each other for every process. Finally, the effect of transport resistances on module performance is discussed.

Limit theorems for diffusion-type processes inR m

For a sequence of stochastic equations of diffusion type, conditions that are close to necessary and sufficient ones are established for the weak convergence of measures $$\mu _{(\xi ^{(n)} ,w)} $$ ,n=1, ... , associated with the solutions, to the limiting measure $$ \mu _{(\xi ^{(0)} ,w)} $$ . The conditions under which the weak convergence of the solutions of stochastic equations implies their strong convergence are established as well.

Stochastic version of the averaging principle for diffusion type processes

We consider the family of stochastic processes where x ε is the solution of the Ito stochastic differential equation whose drift coefficient depends also on the strictly stationary and ergodic process . Using a transformation which is similar to the classical averaging principle we find explicitly first and second order approximations of (in the sense of weak convergence). The approximations are diffusion Markov processes.