Narrow Results

Any additive stationary and continuous Markovian process described by a Fokker–Planck equation can also be described in terms of a Schrödinger equation with an appropriate quantum potential. By using such analogy, it has been proved that a power-law correlated stationary Markovian process can stem from a quantum potential that (i) shows an x^-2 decay for large x values and (ii) whose eigenvalue spectrum admits a null eigenvalue and a continuum part of positive eigenvalues attached to it. In this paper we show that such two features are both necessary. Specifically, we show that a potential with tails decaying like x^-μ with μ < 2 gives rise to a stationary Markovian process which is not power-law autocorrelated, despite the fact that the process has an unbounded set of time scales. Moreover, we present an exactly solvable example where the potential decays as x^-2 but there is a gap between the continuum spectrum of eigenvalues and the null eigenvalue. We show that the process is not power law autocorrelated, but by decreasing the gap one can arbitrarily well approximate it. A crucial role in obtaining a power-law autocorrelated process is played by the weights giving the contribution of each time-scale contribute to the autocorrelation function. In fact, we will see that such weights must behave like a power-law for small energy values λ. This is only possible if the potential V_S(x) shows a x^-2 decay to zero for large x values.

From July 9 to 11, 2001, 50 researchers from the fields of climate dynamics and stochastic analysis met in Chorin, Germany, to discuss the idea of stochastic models of climate. The present issue of Stochastics and Dynamics collects several papers from this meeting. In this introduction to the volume, the idea of simple conceptual stochastic climate models is introduced amd recent results in the mathematically rigorous development and analysis of such models are reviewed. As well, a brief overview of the application of ideas from stochastic dynamics to simple models of the climate system is given.

In studies of perturbation dynamics in physical systems, certain specification of the governing perturbation dynamical system is generally lacking, either because the perturbation system is imperfectly known or because its specification is intrinsically uncertain, while a statistical characterization of the perturbation dynamical system is often available. In this report exact and asymptotically valid equations are derived for the ensemble mean and moment dynamics of uncertain systems. These results are used to extend the concept of optimal deterministic perturbation of certain systems to uncertain systems. Remarkably, the optimal perturbation problem has a simple solution: In uncertain systems there is a sure initial condition producing the greatest expected second moment perturbation growth.

A simple model of the thermohaline circulation of the World Ocean is considered, in which fluctuations in internal oceanic mixing and in freshwater forcing are represented by stochastic processes. The effects on the stationary probability density function of correlations between fluctuations in mixing and freshwater forcing, and of finite autocorrelation time in oceanic mixing, are determined using a mixture of analytical and numerical techniques. The quantitative behaviour of the system is found to depend on the strength and correlation character of the noise processes, quite sensitively so in some regions of parameter space. The results of this analysis suggest the importance of accurately modelling high-frequency variability in nonlinear models of the climate system.

In this work we analyze the convergence in distribution sense of the periodogram function (to the spectral density function) based on a time series of a stationary process X_t = (φ ◦ T^t)(X₀) obtained from the iterations of a continuous transformation T invariant for an ergodic probability μ and a continuous function φ taking values in ℝ. We only assume a certain rate of convergence to zero for the autocovariance coefficient of the stochastic process, i.e. we assume there exist C > 0 and β > 2 such that |γ_X(h)| ≤ C|h|^-β, for all h ∈ ℕ, where γ_X(h) = ∫(φ ◦ T^h)(x) φ(x)dμ(x) - (∫ φ(x)dμ(x))² is the h-autocovariance of the process. Our result applies to the case of exponential decay of correlation (or covariance), as it happens for a continuous expanding transformation T on the circle and a Holder potential φ. It can also be applied to the case when the transformation T has a fixed point with derivative equal to one.

We obtain asymptotic expansions for the large deviation principle (LDP) for continuous time stochastic processes with weakly-dependent increments. As a key example, we show that additive functionals of solutions of stochastic differential equations (SDEs) satisfying Hörmander condition on a $d$-dimensional compact manifold admit these asymptotic expansions of all orders.

We prove the existence of a pathwise weak solution to the single-phase, miscible displacement of one incompressible fluid by another in a porous medium with random forcing. Our system is described by a parabolic concentration equation driven by an additive noise coupled with an elliptic pressure equation. We use a pathwise argument combined with Schauder’s fixed point theorem.

The observable representation provides an embedding of a discrete space in a low dimensional continuous space. Typically, the discrete space is a model of a complex system. This graphical representation is known to highlight significant properties of the original space and can serendipitously reveal unanticipated relationships. We report on the current status of this technique and give examples of its applications and rationale.

Completely positive trace preserving (CPTP) maps are essential for the formulation of the second law of thermodynamics. The dynamics of quantum systems initially correlated with their environments are in general not described by such maps. We explore, how this issue can be fixed by describing the classical analogue of this problem. We consider initially correlated probability distributions, whose subsequent system dynamics is ill-described by stochastic maps, and prescribe the correct way to describe the dynamics. We use this prescription to discuss the classical version of the second law, valid for initially correlated probability distributions.

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Actin polymerization driven stochastic movement of the bacteria Listeria monocytogenes is often measured using single-particle tracking (SPT) methodology and analyzed in terms of statistics. Experimental results suggested a dynamic association between the growing actin filaments and the propelled bacteria. Based on an alternative mathematical formalism for a Brownian ratchet (BR), we introduce such an attractive interaction into the one-dimensional BR model and show that its effect is equivalent to an external resistant force on the bacterium. Such a force significantly reduces the Brownian motion of a driven bacterium, and accentuates the stepping due to polymerization. We then consider the growth, with and without a barrier, of a filamentous bundle consisting of N identical filaments. It is shown that the bundle grows with a similar rate as a single filament in the absence of a load, but can oppose N times the external force under the stalling condition. A set of relationships describing the velocity of the bacterium movement (V_z) and its apparent diffusivity (D_z) as functions of the resistant force (F) and the number of filaments in a bundle (N) are obtained. The theoretical study suggests methods for data analysis in future experiments with applied external resistant force.

This paper proposed a method to control the weighted average price of manufacturer sales. The study of the posed problem is necessary to hedge manufacturer profits from price slumps in commodity markets. The elaborated method is applied to managing the trading process on real world commodity exchanges to prove its effectiveness.

This paper proposed a method to control the weighted average price of manufacturer sales with the initially specified sales by the end of the trading interval on the commodity exchanges. The developed manufacturer sales strategy is proposed as a hedge against price slumps without the use of derivative financial instruments in commodity markets.

The quantum Liouville equation, which describes the phase space dynamics of a quantum system of fermions, is analyzed from stochastic point of view as a particular example of the Kramers–Moyal expansion. Quantum mechanics is extended to relativistic domain by generalizing the Wigner–Moyal equation. Thus, an expression is derived for the relativistic mass in the Wigner quantum phase space presentation. The diffusion with an imaginary diffusion coefficient is discussed. An imaginary stochastic process is proposed as the origin of quantum mechanics.

Myosin V, a two-headed motor protein, moves along actin filaments toward the positive end. Similar to other molecular motors, myosin V hydrolyzes Adenosine tri-phosphate (ATP) and releases its product to produce movement. This study proposes a multiple-paths model that considers the hydrolysis processes of catalytic sites at the two heads of myosin V as independent. The proposed model describes the myosin V transition process by seven states, with one load-dependent transition rate among these states. This model demonstrates how myosin V steps forward at different chemical reaction paths. This study also uses the enzyme kinetics to calculate the mean velocity. Stochastic processes are used to analyze the randomness as well. Analytical results reveal that the theoretical mean velocities, mean movement lengths, and randomness with various ATP concentrations under external load agree with the experimental data.

We construct diffusion processes associated with the Navier-Stokes system in R³ and use them to prove the existence and uniquenes of local solution of the Cauchy problem for this system in some functional space.

Independent of the other communities who have developed theories of wavelets over the last twenty years, we developed over the same period a view of wavelets seen as stochastic processes. That context arose naturally from our theory of Time operators in statistical mechanics. Essential ingredients in our theory included Kolmogorov dynamical systems and conditional expectations. The purpose of the present paper is to come up-to-date on the relationship of our theory to the general theory of wavelets.

We review a novel approach to evaluate the ground-state properties of many-body lattice systems based on an exact probabilistic representation of the dynamics and its long time approximation via a central limit theorem. The choice of the asymptotic density probability used in the calculation is discussed in detail.

Since the first works laying its foundations as a subfield of Complex Analysis, the theory of reproducing kernels has proved to be a powerful tool in many fields of Pure and Applied Mathematics. The aim of this paper is to give some idea of how and why this theory interacts with Probability and Statistics.

The well-scaled transition to the diffusion limit in the framework of the theory of continuous-time random walk (CTRW) is presented starting from its representation as an infinite series that points out the subordinated character of the CTRW itself. This formula allows us to treat the CTRW as a discrete-space discrete-time random walk that in the continuum limit tends towards a generalized diffusion process governed by a space-time fractional diffusion equation. The essential assumption is that the probabilities for waiting times and jump-widths behave asymptotically like powers with negative exponents related to the orders of the fractional derivatives. Plots of simulations for some case-studies are given in order to display the sample paths for the fractional diffusion processes, generally non Markovian, that are obtained by the composition of two Markovian processes.