non-Markovian Stochastic Process Research Articles

A unifying representation of path integrals for fractional Brownian motions

Fractional Brownian motion (fBm) is an experimentally-relevant, non-Markovian Gaussian stochastic process with long-ranged correlations between the increments, parametrised by the so-called Hurst exponent H; depending on its value the process can be sub-diffusive (0<H<1/2) , diffusive (H=1/2) or super-diffusive (1/2<H<1) . There exist three alternative equally often used definitions of fBm—due to Lévy and due to Mandelbrot and van Ness (MvN), which differ by the interval on which the time variable is formally defined. Respectively, the covariance functions of these fBms have different functional forms. Moreover, the MvN fBms have stationary increments, while for the Lévy fBm this is not the case. One may therefore be tempted to conclude that these are, in fact, different processes which only accidentally bear the same name. Recently determined explicit path integral representations also appear to have very different functional forms, which only reinforces the latter conclusion. Here we develop a unifying equivalent path integral representation of all three fBms in terms of Riemann–Liouville fractional integrals, which links the fBms and proves that they indeed belong to the same family. We show that the action in such a representation involves the fractional integral of the same form and order (dependent on whether H<1/2 or H>1/2 ) for all three cases, and differs only by the integration limits.

Machine Learning Memory Kernels as Closure for Non-Markovian Stochastic Processes.

Finding the dynamical law of observable quantities lies at the core of physics. Within the particular field of statistical mechanics, the generalized Langevin equation (GLE) comprises a general model for the evolution of observables covering a great deal of physical systems with many degrees of freedom and an inherently stochastic nature. Although formally exact, GLE brings its own great challenges. It depends on the complete history of the observables under scrutiny, as well as the microscopic degrees of freedom, all of which are often inaccessible. We show that these drawbacks can be overcome by adopting elements of machine learning from empirical data, in particular coupling a multilayer perceptron (MLP) with the formal structure of GLE and calibrating the MLP with the data. This yields a powerful computational tool capable of describing noisy complex systems beyond the realms of statistical mechanics. It is exemplified with a number of representative examples from different fields: from a single colloidal particle and particle chains in a thermal bath to climatology and finance, showing in all cases excellent agreement with the actual observable dynamics. The new framework offers an alternative perspective for the study of nonequilibrium processes opening also a new route for stochastic modeling.

Engines for predictive work extraction from memoryful quantum stochastic processes

Quantum information-processing techniques enable work extraction from a system&amp;apos;s inherently quantum features, in addition to the classical free energy it contains. Meanwhile, the science of computational mechanics affords tools for the predictive modeling of non-Markovian classical and quantum stochastic processes. We combine tools from these two sciences to develop a technique for predictive work extraction from non-Markovian stochastic processes with quantum outputs. We demonstrate that this technique can extract more work than non-predictive quantum work extraction protocols, on the one hand, and predictive work extraction without quantum information processing, on the other. We discover a phase transition in the efficacy of memory for work extraction from quantum processes, which is without classical precedent. Our work opens up the prospect of machines that harness environmental free energy in an essentially quantum, essentially time-varying form.

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.

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.

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.

Generating constrained run-and-tumble trajectories

We propose a method to exactly generate bridge run-and-tumble trajectories that are constrained to start at the origin with a given velocity and to return to the origin after a fixed time with another given velocity. The method extends the concept of effective Langevin equations, valid for Markovian stochastic processes such as Brownian motion, to a non-Markovian stochastic process driven by a telegraphic noise, with exponentially decaying correlations. We obtain effective space-time dependent tumbling rates that implicitly accounts for the bridge constraint. We extend the method to other types of constrained run-and-tumble particles such as excursions and meanders. The method is implemented numerically and is shown to be very efficient.

Great Barrier Reef degradation, sea surface temperatures, and atmospheric CO2 levels collectively exhibit a stochastic process with memory

Quantifying ecological memory could be done at several levels from the rate of physiological changes in an ecosystem all the way down to responses at the genetic level. One way of unlocking the information encoded in a collective environmental memory is to examine the recorded time-series data generated by different components of an ecosystem. In this paper, we probe into the case of the Great Barrier Reef (GBR) which is threatened by elevated sea surface temperatures (SST) and ocean acidification attributed to rising atmospheric CO2 levels. Specifically, we investigate the interrelated dynamics between the degradation of the GBR, SST, and rising atmospheric CO2 levels, by considering three datasets: (a) the mean percentage hard coral cover of the GBR from the archives of the Australian Institute of Marine Science; (b) SST close to the GBR from the National Oceanic and Atmospheric Administration; and (c) the Keeling curve for atmospheric CO2 levels measured by the Mauna Loa Observatory. We show that fluctuating observables in these datasets have the same memory behavior described by a non-Markovian stochastic process. All three datasets show a good match between empirical and analytical mean square deviation. An explicit analytical form for the corresponding probability density function is obtained which obeys a modified diffusion equation with a time dependent diffusion coefficient. This study provides a new perspective on the similarities of and interaction between the GBR’s declining hard coral cover, SST, and rising atmospheric CO2 levels by putting all three systems into one unified framework indexed by a memory parameter \(\mu \) and a characteristic frequency \(\nu \). The short-time dynamics of CO2 levels and SST fall in the superdiffusive regime, while the GBR exhibits hyperballistic fluctuation in percent coral cover with the highest values for \(\mu \) and \(\nu \).

Extremal statistics for stochastic resetting systems.

While averages and typical fluctuations often play a major role in understanding the behavior of a nonequilibrium system, this nonetheless is not always true. Rare events and large fluctuations are also pivotal when a thorough analysis of the system is being done. In this context, the statistics of extreme fluctuations in contrast to the average plays an important role, as has been discussed in fields ranging from statistical and mathematical physics to climate, finance, and ecology. Herein, we study extreme value statistics (EVS) of stochastic resetting systems, which have recently gained significant interest due to its ubiquitous and enriching applications in physics, chemistry, queuing theory, search processes, and computer science. We present a detailed analysis for the finite and large time statistics of extremals (maximum and arg-maximum, i.e., the time when the maximum is reached) of the spatial displacement in such system. In particular, we derive an exact renewal formula that relates the joint distribution of maximum and arg-maximum of the reset process to the statistical measures of the underlying process. Benchmarking our results for the maximum of a reset trajectory that pertain to the Gumbel class for large sample size, we show that the arg-maximum density attains a uniform distribution independent of the underlying process at a large observation time. This emerges as a manifestation of the renewal property of the resetting mechanism. The results are augmented with a wide spectrum of Markov and non-Markov stochastic processes under resetting, namely, simple diffusion, diffusion with drift, Ornstein-Uhlenbeck process, and random acceleration process in one dimension. Rigorous results are presented for the first two setups, while the latter two are supported with heuristic and numerical analysis.

Multifractality through Non-Markovian Stochastic Processes in the Scale Relativity Theory. Acute Arterial Occlusions as Scale Transitions.

By assimilating biological systems, both structural and functional, into multifractal objects, their behavior can be described in the framework of the scale relativity theory, in any of its forms (standard form in Nottale’s sense and/or the form of the multifractal theory of motion). By operating in the context of the multifractal theory of motion, based on multifractalization through non-Markovian stochastic processes, the main results of Nottale’s theory can be generalized (specific momentum conservation laws, both at differentiable and non-differentiable resolution scales, specific momentum conservation law associated with the differentiable–non-differentiable scale transition, etc.). In such a context, all results are explicated through analyzing biological processes, such as acute arterial occlusions as scale transitions. Thus, we show through a biophysical multifractal model that the blocking of the lumen of a healthy artery can happen as a result of the “stopping effect” associated with the differentiable-non-differentiable scale transition. We consider that blood entities move on continuous but non-differentiable (multifractal) curves. We determine the biophysical parameters that characterize the blood flow as a Bingham-type rheological fluid through a normal arterial structure assimilated with a horizontal “pipe” with circular symmetry. Our model has been validated based on experimental clinical data.

The fluid limit of a random graph model for a shared ledger

AbstractA shared ledger is a record of transactions that can be updated by any member of a group of users. The notion of independent and consistent record-keeping in a shared ledger is important for blockchain and more generally for distributed ledger technologies. In this paper we analyze a stochastic model for the shared ledger known as the tangle, which was devised as the basis for the IOTA cryptocurrency. The model is a random directed acyclic graph, and its growth is described by a non-Markovian stochastic process. We first prove ergodicity of the stochastic process, and then derive a delay differential equation for the fluid model which describes the tangle at high arrival rate. We prove convergence in probability of the tangle process to the fluid model, and also prove global stability of the fluid model. The convergence proof relies on martingale techniques.

Non-Markovian Stochastic Processes in Physics ―Field-Theoretical Solution to the Hawkes Process

Non-Markovian Stochastic Processes in Physics ―Field-Theoretical Solution to the Hawkes Process

Analytical results for a non-Markovian process of gene expression with positive and negative feedbacks

Gene expression is a very complex process and involves many small biochemical reaction steps, resulting in a non-Markovian discrete stochastic process due to molecular memory between individual reactions. At present, this process is successfully investigated by generalized chemical master equation models. However, these models do not consider the role of feedback networks in gene expression. How the interaction between feedbacks and molecular memory affects gene expression still remains not well understood. Here, we establish generalized chemical master equation models of gene expression with positive and negative feedbacks. Assuming that the process of producing proteins follows an Erlang probability distribution, we obtain the analytical expression for this model in a steady state, as well as the measure of the noise of protein numbers. We further find that molecular memory competes with the positive feedback in suppressing the noise of the protein number. For our model with a negative feedback, molecular memory can strengthen the intensity of suppressing this noise. These interesting results imply that molecular memory are as important as the feedbacks to affect gene expression.

Reflected fractional Brownian motion in one and higher dimensions.

Fractional Brownian motion (FBM), a non-Markovian self-similar Gaussian stochastic process with long-ranged correlations, represents a widely applied, paradigmatic mathematical model of anomalous diffusion. We report the results of large-scale computer simulations of FBM in one, two, and three dimensions in the presence of reflecting boundaries that confine the motion to finite regions in space. Generalizing earlier results for finite and semi-infinite one-dimensional intervals, we observe that the interplay between the long-time correlations of FBM and the reflecting boundaries leads to striking deviations of the stationary probability density from the uniform density found for normal diffusion. Particles accumulate at the boundaries for superdiffusive FBM while their density is depleted at the boundaries for subdiffusion. Specifically, the probability density P develops a power-law singularity, P∼r^{κ}, as a function of the distance r from the wall. We determine the exponent κ as a function of the dimensionality, the confining geometry, and the anomalous diffusion exponent α of the FBM. We also discuss implications of our results, including an application to modeling serotonergic fiber density patterns in vertebrate brains.

Exponential ergodicity for a class of non-Markovian stochastic processes

The existence of an invariant probability measure is proven for a class of solutions of stochastic differential equations with finite delay. This is done, in this non-Markovian setting, using the cluster expansion method, from Gibbs field theory. It holds for small perturbations of ergodic diffusions.

Kinetic equation for particle transport in turbulent flows

With a new approach based on the ensemble average over particle state transition paths in phase space, a kinetic equation for particles transported in turbulent flows is derived. The probability density function (PDF) for particles is defined as an ensemble average of a special fine-grained PDF, referred to as the local path density operator. The kinetic equation is derived from a Taylor series expansion of the PDF in terms of the cumulants with respect to particle paths in phase space and leads to a closed expression for its diffusion terms. It shows that the random forcing of eddy fluctuations, non-stationarity of turbulence, and inertia of particles are explicitly presented in the diffusion coefficient, which could help us to understand how particles are diffused by these underlying mechanisms. The kinetic equation is applicable to non-Markovian, non-Gaussian, and non-stationary stochastic processes, while for Markovian processes, it recovers the classical Fokker–Planck equation. The macroscopic equations for particle phase are derived based on the kinetic equation and compared with the direct numerical simulation of particles transported in turbulent flows.

Volume of the set of LOCC-convertible quantum states

The class of quantum operations known as local operations and classical communication (LOCC) induces a partial ordering on quantum states. We present the results of systematic numerical computations related to the volume (with respect to the unitarily invariant measure) of the set of LOCC-convertible bipartite pure states, where the ordering is characterised by an algebraic relation known as majorization. The numerical results, which exploit a tridiagonal model of random matrices, provide quantitative evidence that the proportion of LOCC-convertible pairs vanishes in the limit of large dimensions, and therefore support a previous conjecture by Nielsen. In particular, we show that the problem is equivalent to the persistence of a non-Markovian stochastic process and the proportion of LOCC-convertible pairs decays algebraically with a nontrivial persistence exponent. We extend this analysis by investigating the distribution of the maximal success probability of LOCC-conversions. We show a dichotomy in behaviour between balanced and unbalanced bipartitions. In the latter case the asymptotics is somehow surprising: in the limit of large dimensions, for the overwhelming majority of pairs of states a perfect LOCC-conversion is not possible; nevertheless, for most states there exist local strategies that succeed in achieving the conversion with a probability arbitrarily close to one. We present strong evidence of a universal scaling limit for the maximal probability of successful LOCC-conversions and we suggest a connection with the typical fluctuations of the smallest eigenvalue of Wishart random matrices.

Manifestations of Projection-Induced Memory: General Theory and the Tilted Single File

Over the years the field of non-Markovian stochastic processes and anomalous diffusion evolved from a specialized topic to mainstream theory, which transgressed the realms of physics to chemistry, biology and ecology. Numerous phenomenological approaches emerged, which can more or less successfully reproduce or account for experimental observations in condensed matter, biological and/or single-particle systems. However, as far as their predictions are concerned these approaches are not unique, often build on conceptually orthogonal ideas, and are typically employed on an ad hoc basis. It therefore seems timely and desirable to establish a systematic, mathematically unifying and clean approach starting from more fine-grained principles. Here we analyze projection-induced ergodic non-Markovian dynamics, both reversible as well as irreversible, using spectral theory. We investigate dynamical correlations between histories of projected and latent observables that give rise to memory in projected dynamics, and rigorously establish conditions under which projected dynamics is Markovian or renewal. A systematic metric is proposed for quantifying the degree of non-Markovianity. As a simple, illustrative but non-trivial example we study single file diffusion in a tilted box, which, for the first time, we solve exactly using the coordinate Bethe ansatz. Our results provide a solid foundation for a deeper and more systematic analysis of projection-induced non-Markovian dynamics and anomalous diffusion.

White noise functional integral for exponentially decaying memory: nucleotide distribution in bacterial genomes

We utilize a stochastic functional integral approach that forms a natural framework for analyzing ubiquitous complex sequences of fluctuations with underlying non-Markovian stochastic process beyond fractional Brownian motion. We demonstrate how Hida white noise calculus, guided by mean square deviation (MSD) analysis of empirical data, allows derivation of single nucleotide occurrence probability distributions for whole genomes of four significant species of bacteria: (a) freshwater cyanobacteria Synechococcus elongatus PCC7942, 2.7 Mbp, (b) marine cyanobacteria Prochlorococcus marinus subsp. marinus str. CCMP1375, 1.8 Mbp, (c) pathogenic bacteria Staphylococcus aureus subsp. aureus NCTC 8325, 2.8 Mbp, and (d) Staphylococcus aureus ILRI Eymole1/1, 2.9 Mbp. Here, the stochastic variable is chosen to represent separation distances between succeeding identical single nucleotides where distance is defined as the number of steps through intervening bases. The stochastic parameter set takes values of nucleotide occurrence count along the genome length. The probability density function (PDF) is derived in closed form for the associated stochastic process with exponentially damped memory kernel, and is shown to satisfy a modified diffusion equation with a parameter-dependent diffusion coefficient. The PDF yields an analytical result for MSDs that match empirical plots, showing a rising nonlinear curve that flattens to a plateau starting close to 1 kb, similar to restricted diffusion. The plots exhibit compliance with Chargaff’s second parity rule for nucleotides. The same PDF describes occurrences of single nucleotides adenine, guanine, cytosine, and thymine for all four bacterial genomes considered.

Extinction Time of Non-Markovian Self-Similar Processes, Persistence, Annihilation of Jumps and the Fréchet Distribution

We start by providing an explicit characterization and analytical properties, including the persistence phenomena, of the distribution of the extinction time $\mathbb{T}$ of a class of non-Markovian self-similar stochastic processes with two-sided jumps that we introduce as a stochastic time-change of Markovian self-similar processes. For a suitably chosen time-changed, we observe, for classes with two-sided jumps, the following surprising facts. On the one hand, all the $\mathbb{T}$'s within a class have the same law which we identify in a simple form for all classes and reduces, in the spectrally positive case, to the Fr\'echet distribution. On the other hand, each of its distribution corresponds to the law of an extinction time of a single Markov process without positive jumps, leaving the interpretation that the time-changed has annihilated the effect of positive jumps. The example of the non-Markovian processes associated to L\'evy stable processes is detailed.