non-Markovian Processes Research Articles

Computer modeling of non-Markovian processes based on the principle of balance of "complex probabilities"

Computer modeling of non-Markovian processes based on the principle of balance of "complex probabilities"

Investigation of Markovian and Non-Markovian Search Processes of Monomers of a Rouse Chain Confined in a Spherical Cavity

We study the process of monomers of a Rouse chain confined in a spherical cavity of radius Rb to search for a small opening of radius a on the cavity surface by combining scaling analysis and Langevin dynamics simulations. We show that the search process is nearly Markovian when the radius of gyration Rg of the chain is much smaller than Rb and becomes non-Markovian for Rg ≥ Rb. For the non-Markovian search process, we first derive a general scaling relation for the mean search time ⟨τ∞⟩ in the long chain limit and then explicitly determine the scaling exponents based on simulation results, ⟨τ∞⟩ ∼ Rb4a–1DH–3/2 and ⟨τ∞⟩ ∼ Rb5a–1DH–2 for the end and middle monomers, respectively, where DH is a generalized diffusion coefficient associated with the monomer motion. We further find that the Hurst exponent, which characterizes the persistence of the dynamics, decreases exponentially from approximately 1/3 to about 1/4 as the monomer position changes from the end to the middle. This result implies that the subdiffusion of monomers depends considerably on their positions in the chain under confinement, providing insight into the anomalous dynamics of confined polymeric systems.

Systems of differential equations modeling non-Markov processes

The work deals with non-Markov processes and the construction of systems of differential equations with delay that describe the probability vectors of such processes. The generating stochastic operator and properties of stochastic operators are used to construct systems that define non-Markov processes.

Persistence, extinction and practical exponential stability of impulsive stochastic competition models with varying delays

<abstract><p>This paper studies the persistence, extinction and practical exponential stability of impulsive stochastic competition models with time-varying delays. The existence of the global positive solutions is investigated by the relationship between the solutions of the original system and the equivalent system, and the sufficient conditions of system persistence and extinction are given. Moreover, our study shows the following facts: (1) The impulsive perturbation does not affect the practical exponential stability under the condition of bounded pulse intensity. (2) In solving the stability of non-Markovian processes, it can be transformed into solving the stability of Markovian processes by applying Razumikhin inequality. (3) In some cases, a non-Markovian process can produce Markovian effects. Finally, numerical simulations obtained the importance and validity of the theoretical results for the existence of practical exponential stability through the relationship between parameters, pulse intensity and noise intensity.</p></abstract>

A Wiener Path Integral Formalism for Treating Nonlinear Systems with Non-Markovian Response Processes

A Wiener Path Integral Formalism for Treating Nonlinear Systems with Non-Markovian Response Processes

The “complex probabilities” balance principle for non-Markov processes modeling

This paper presents the principle of “complex probabilities” balance, based on the description of the stochastic process not in the time, but in the complex domain, which allows developing models of non-stationary queuing systems with arbitrary probabilities distributions of the requests and their servicing time, taking into account random or deterministic time delays. A system of balance equations in the complex domain was compiled and solved using Laplace images. Performed an inverse Laplace transform to move from images to probabilities in the time domain. Presented models for acyclic and cyclic stochastic processes with arbitrary distributions of time points between incoming requests and their service. Given calculation examples. Recommendations for the further application of the principle of balance of “complex probabilities” are given.

Occupation time statistics of the fractional Brownian motion in a finite domain.

We study statistics of occupation times for a fractional Brownian motion (fBm), which is a typical model of a non-Markov process. Due to the non-Markovian nature, recurrence times to the origin depend on the history. Numerical simulations indicate that dependence on the sum of successive recurrence times becomes weak. As a result, the distribution of the occupation time in a finite domain follows the Mittag-Leffler distribution when the Hurst exponent of the fBm is close to 1/2. We show this distributional behavior of a time-averaged observable by renewal theory. This result is an extension of the distributional limit theorem known as the Darling-Kac theorem in general Markov processes to non-Markov processes.

Effects of contact tracing and non-Markovian awareness process on the spread of air-borne diseases

Contact tracing is an important tool to contain the spread of many airborne diseases. We develop an approximated pairwise model to investigate the impact of non-Markovian awareness process among infectious individuals represented by pairwise endemic models. We derive the basic reproduction number and the final epidemic size, which are dependent on the tracing rate and the distribution of awareness process. The model analysis provides the threshold of contact tracing rate. When the contact tracing rate is greater than the threshold, the basic reproduction number will be less than one, then the epidemic will eventually die out. The analysis further shows that, higher variance in the awareness process generates smaller basic reproduction number, lower tracing threshold, and larger final epidemic size, when the mean awareness period is fixed. Extensive numerical analysis show the comprehensive effects of tracing rates and non-Markovian awareness processes on human behavior and the transmissibility ability of epidemic. It turns out that large tracing rates and high variances in awareness process lead to obvious reductions in contacts between susceptible and infectious individuals, so as to curb the transmission of infectious diseases. Moreover, contact tracing is more effective in reducing the number of infected individuals and the contacts between susceptible and infected individuals when the awareness process has a larger variance.

Field theory of survival probabilities, extreme values, first-passage times, and mean span of non-Markovian stochastic processes

We provide a perturbative framework to calculate extreme events of non-Markovian processes, by mapping the stochastic process to a two-species reaction-diffusion process in a Doi-Peliti field theory combined with the Martin-Siggia-Rose formalism. This field theory treats interactions and the effect of external, possibly self-correlated, noise in a perturbation about a Markovian process, thereby providing a systematic, diagrammatic approach to extreme events. We apply the formalism to Brownian motion and calculate its survival probability distribution subject to self-correlated noise.

On the classicality of quantum dephasing processes

We analyze the multitime statistics associated with pure dephasing systems repeatedly probed with sharp measurements, and search for measurement protocols whose statistics satisfy the Kolmogorov consistency conditions possibly up to a finite order. We find a rich phenomenology of quantum dephasing processes which can be interpreted in classical terms. In particular, if the underlying dephasing process is Markovian, we find sufficient conditions under which classicality at every order can be found: this can be reached by choosing the dephasing and measurement basis to be fully compatible or fully incompatible, that is, mutually unbiased bases (MUBs). For non-Markovian processes, classicality can only be proven in the fully compatible case, thus revealing a key difference between Markovian and non-Markovian pure dephasing processes.

Path integrals for fractional Brownian motion and fractional Gaussian noise.

Wiener's path integral plays a central role in the study of Brownian motion. Here we derive exact path-integral representations for the more general fractional Brownian motion (FBM) and for its time derivative process, fractional Gaussian noise (FGN). These paradigmatic non-Markovian stochastic processes, introduced by Kolmogorov, Mandelbrot, and van Ness, found numerous applications across the disciplines, ranging from anomalous diffusion in cellular environments to mathematical finance. Their exact path-integral representations were previously unknown. Our formalism exploits the Gaussianity of the FBM and FGN, relies on the theory of singular integral equations, and overcomes some technical difficulties by representing the action functional for the FBM in terms of the FGN for the subdiffusive FBM and in terms of the derivative of the FGN for the super-diffusive FBM. We also extend the formalism to include external forcing. The exact and explicit path-integral representations make inroads in the study of the FBM and FGN.

Stochastic thermodynamics of fractional Brownian motion

This paper is concerned with the stochastic thermodynamics of nonequilibrium Gaussian processes that can exhibit anomalous diffusion. In the systems considered, the noise correlation function is not necessarily related to friction. Thus there is no conventional fluctuation-dissipation relation (FDR) of the second kind and no unique way to define a temperature. We start from a Markovian process with time-dependent diffusivity (an example being scaled Brownian motion). It turns out that standard stochastic thermodynamic notions can be applied rather straightforwardly by introducing a time-dependent temperature, yielding the integral fluctuation relation. We then proceed to our focal system, that is, a particle undergoing fractional Brownian motion (FBM). In this case, the noise is still Gaussian, but the noise correlation function is nonlocal in time, defining a non-Markovian process. We analyze in detail the consequences when using the conventional notions of stochastic thermodynamics with a constant medium temperature. In particular, the heat calculated from dissipation into the medium differs from the log ratio of path probabilities of forward and backward motion, yielding a deviation from the standard integral fluctuation relation for the total entropy production if the latter is defined via system entropy and heat exchange. These apparent inconsistencies can be circumvented by formally defining a time-nonlocal temperature that fulfills a generalized FDR. To shed light on the rather abstract quantities resulting from the latter approach, we perform a perturbation expansion in terms of $\ensuremath{\epsilon}=H\ensuremath{-}1/2$, where $H$ is the Hurst parameter of FBM and $1/2$ corresponds to the Brownian case. This allows us to calculate analytically, up to linear order in $\ensuremath{\epsilon}$, the generalized temperature and the corresponding heat exchange. By this, we provide explicit expressions and a physical interpretation for the leading corrections induced by non-Markovianity.

Path integral description and direct interaction approximation for elastic plate turbulence

In this work, we apply the Martin–Siggia–Rose path integral formalism to the equations of a thin elastic plate. Using a diagrammatic technique, we obtain the direct interaction approximation (DIA) equations to describe the evolutions of the correlation function and the response function of the fields. Consistent with previous results, we show that DIA equations for elastic plates can be derived from a non-markovian stochastic process and that in the weakly nonlinear limit, the DIA equations lead to the kinetic equation of wave turbulence theory. We expect that this approach will allow a better understanding of the statistical properties of wave turbulence and that DIA equations can open new avenues for understanding the breakdown of weakly nonlinear turbulence for elastic plates.

Towards a general framework of Randomized Benchmarking incorporating non-Markovian Noise

The rapid progress in the development of quantum devices is in large part due to the availability of a wide range of characterization techniques allowing to probe, test and adjust them. Nevertheless, these methods often make use of approximations that hold in rather simplistic circumstances. In particular, assuming that error mechanisms stay constant in time and have no dependence in the past, is something that will be impossible to do as quantum processors continue scaling up in depth and size. We establish a theoretical framework for the Randomized Benchmarking protocol encompassing temporally-correlated, so-called non-Markovian noise, at the gate level, for any gate set belonging to a wide class of finite groups. We obtain a general expression for the Average Sequence Fidelity (ASF) and propose a way to obtain average gate fidelities of full non-Markovian noise processes. Moreover, we obtain conditions that are fulfilled when an ASF displays authentic non-Markovian deviations. Finally, we show that even though gate-dependence does not translate into a perturbative term within the ASF, as in the Markovian case, the non-Markovian sequence fidelity nevertheless remains stable under small gate-dependent perturbations.

SRRS: A blockchain fast propagation protocol based on non-Markovian process

The blockchain technology that underlies many cryptocurrencies, such as Bitcoin and Ethereum, is demanded to raise for more efficient transactions. Thus, in the last decade, many solutions have been proposed to improve the propagation of blocks. Motivated by SRRS (social reinforcement rumor spreading), we propose a blockchain fast propagation protocol named SRRS protocol based on a non-Markovian process, where the verification and transmission of a block depend on its propagation history. The simulated experimental results show that our protocol can significantly accelerate the propagation of blocks. Compared with existing major solutions, our protocol does not change the main procedure of Bitcoin and the block data structure, which is a huge advantage for practical applications. Moreover, employing the theories of complex networks and percolation, we develop a method of theoretical analysis, which can find the optimal setting of our protocol.

Everlasting impact of initial perturbations on first-passage times of non-Markovian random walks

Persistence, defined as the probability that a signal has not reached a threshold up to a given observation time, plays a crucial role in the theory of random processes. Often, persistence decays algebraically with time with non trivial exponents. However, general analytical methods to calculate persistence exponents cannot be applied to the ubiquitous case of non-Markovian systems relaxing transiently after an imposed initial perturbation. Here, we introduce a theoretical framework that enables the non-perturbative determination of persistence exponents of Gaussian non-Markovian processes with non stationary dynamics relaxing to a steady state after an initial perturbation. Two situations are analyzed: either the system is subjected to a temperature quench at initial time, or its past trajectory is assumed to have been observed and thus known. Our theory covers the case of spatial dimension higher than one, opening the way to characterize non-trivial reaction kinetics for complex systems with non-equilibrium initial conditions.

Fractional Brownian motion with random Hurst exponent: Accelerating diffusion and persistence transitions.

Fractional Brownian motion, a Gaussian non-Markovian self-similar process with stationary long-correlated increments, has been identified to give rise to the anomalous diffusion behavior in a great variety of physical systems. The correlation and diffusion properties of this random motion are fully characterized by its index of self-similarity or the Hurst exponent. However, recent single-particle tracking experiments in biological cells revealed highly complicated anomalous diffusion phenomena that cannot be attributed to a class of self-similar random processes. Inspired by these observations, we here study the process that preserves the properties of the fractional Brownian motion at a single trajectory level; however, the Hurst index randomly changes from trajectory to trajectory. We provide a general mathematical framework for analytical, numerical, and statistical analysis of the fractional Brownian motion with the random Hurst exponent. The explicit formulas for probability density function, mean-squared displacement, and autocovariance function of the increments are presented for three generic distributions of the Hurst exponent, namely, two-point, uniform, and beta distributions. The important features of the process studied here are accelerating diffusion and persistence transition, which we demonstrate analytically and numerically.

Resemblance of the power-law scaling behavior of a non-Markovian and nonlinear point processes

We analyze the statistical properties of a temporal point process driven by a confined fractional Brownian motion. The event count distribution and power spectral density of this non-Markovian point process exhibit power-law scaling. We show that a nonlinear Markovian point process can reproduce the same scaling behavior. This result indicates a possible link between nonlinearity and apparent non-Markovian behavior.

STRATEGIC BRAND PORTFOLIO MANAGEMENT

In the past twenty years or so, three approaches to brand portfolio management strategies have emerged. The first approach is marketing. This approach is associated with building a corporate brand portfolio. The goal is to increase diversified cash flows by entering new market segments. The second approach is related to the competitive strategy of the enterprise. A false portfolio of intellectual property applications is being created. Competitors are expected to spend resources in retaliation. The third approach is the formation of a dynamic strategy for investment portfolio management. Due to the complex structure of the modern global financial market, the heterogeneous structure of available financial instruments and traders using different approaches and time horizons, forecasts, as a rule, require a large number of observations, work poorly in the vicinity of bifurcations and do not have a computer model that could build forecasts in real time. In such structures, slow diffusion-type processes with the phenomenon of memory arise, that is, non-Markov processes. Moreover, such structures can have fractal properties. In this work, it seems to us, the first step has been taken to build a "synthetic" model of dynamic asset portfolio management. By analyzing the data available in the scientific literature, a mathematical model of strategic brand portfolio management is proposed. In view of the above, the model has the form of a differential equation in fractional derivatives. In connection with the risk analysis, two models of fractional entropy are also considered - fractional Kolmogorov-Sinai entropy and fractional Shannon entropy.

Estimation of drift and diffusion functions from unevenly sampled time-series data.

Complex systems can often be modeled as stochastic processes. However, physical observations of such systems are often irregularly spaced in time, leading to difficulties in estimating appropriate models from data. Here we present extensions of two methods for estimating drift and diffusion functions from irregularly sampled time-series data. Our methods are flexible and applicable to a variety of stochastic systems, including non-Markov processes or systems contaminated with measurement noise. To demonstrate applicability, we use this approach to analyze an irregularly sampled paleoclimatological isotope record, giving insights into underlying physical processes.