Alexander I. Saichev

Density fields in Burgers and KdV-Burgers turbulence

A model analytical description of the density field advected in a velocity field governed by the multidimensional Burgers equation is suggested. This model field satisfies the mass conservation law and, in the zero viscosity limit, coincides with the generalized solution of the continuity equation. A numerical and analytical study of the evolution of such a model density field is much more convenient than the standard method of simulation of transport of passive tracer particles in the fluid.In the 1-dimensional case, a more general Korteweg–de Vries (KdV)–Burgers equation is suggested as a model which permits an analytical treatment of the density field in a strongly nonlinear model of compressible gas which takes into account dissipative and dispersive effects as well as pressure forces, the former not being accounted for in the standard Burgers framework.The dynamical and statistical properties of the density field are studied. In particular, utilizing the above model in the 2-dimensional case and the ...

Quantification of deviations from rationality with heavy tails in human dynamics

The dynamics of technological, economic and social phenomena is controlled by how humans organize their daily tasks in response to both endogenous and exogenous stimulations. Queueing theory is believed to provide a generic answer to account for the often observed power-law distributions of waiting times before a task is fulfilled. However, the general validity of the power law and the nature of other regimes remain unsettled. Using anonymized data collected by Google at the World Wide Web level, we identify the existence of several additional regimes characterizing the time required for a population of Internet users to execute a given task after receiving a message. Depending on the under- or over-utilization of time by the population of users and the strength of their response to perturbations, the pure power law is found to be coextensive with an exponential regime (tasks are performed without too much delay) and with a crossover to an asymptotic plateau (some tasks are never performed).

Solution of the nonlinear theory and tests of earthquake recurrence times.

We develop an efficient numerical scheme to solve accurately the set of nonlinear integral equations derived previously in [A. Saichev and D. Sornette, J. Geophys. Res. 112, B04313 (2007)], which describes the distribution of interevent times in the framework of a general model of earthquake clustering with long memory. Detailed comparisons between the linear and nonlinear versions of the theory and direct synthetic catalogs show that the nonlinear theory provides an excellent fit to the synthetic catalogs, while there are significant biases resulting from the use of the linear approximation. We then address the suggestions proposed by some authors to use the empirical distribution of interevent times to obtain a better determination of the so-called clustering parameter. Our theory and tests against synthetic and empirical catalogs find a rather dramatic lack of power for the distribution of interevent times to distinguish between quite different sets of parameters, casting doubt on the usefulness of this statistic for the specific purpose of identifying the clustering parameter.

Probability Distributions of Passive Tracers in Randomly Moving Media

Statistical properties of fluctuations of the density of passive tracer and related random fields in a randomly moving medium are discussed. Diffusion approximation and a Gaussian velocity field model is used. We find probability distributions of density fields and of Jacobians in a chaotically compressible medium. Formulas connecting statistical characteristics of random fields in Lagrangian and in Eulerian coordinates are provided. For an incompressible medium, we analyze statistical properties of the passive scalar field’s gradient, and also statistics of the total gradient and the length of a contour carried in the chaotic flow of an incompressible fluid.

Generation-by-generation dissection of the response function in long memory epidemic processes

In a number of natural and social systems, the response to an exogenous shock relaxes back to the average level according to a long-memory kernel ~1/t1+θ with 0 ≤ θ < 1. In the presence of an epidemic-like process of triggered shocks developing in a cascade of generations at or close to criticality, this “bare” kernel is renormalized into an even slower decaying response function ~1/t1-θ. Surprisingly, this means that the shorter the memory of the bare kernel (the larger 1+θ), the longer the memory of the response function (the smaller 1-θ). Here, we present a detailed investigation of this paradoxical behavior based on a generation-by-generation decomposition of the total response function, the use of Laplace transforms and of “anomalous” scaling arguments. The paradox is explained by the fact that the number of triggered generations grows anomalously with time at ~ tθ so that the contributions of active generations up to time t more than compensate the shorter memory associated with a larger exponent θ. This anomalous scaling results fundamentally from the property that the expected waiting time is infinite for 0 ≤ θ ≤ 1. The techniques developed here are also applied to the case θ > 1 and we find in this case that the total renormalized response is a constant for t < 1/(1-n) followed by a cross-over to ~1/t1+θ for t ≫ 1/(1-n).

Generating functions and stability study of multivariate self-excited epidemic processes

AbstractWe present a stability study of the class of multivariate self-excited Hawkes point processes, that can model natural and social systems, including earthquakes, epileptic seizures and the dynamics of neuron assemblies, bursts of exchanges in social communities, interactions between Internet bloggers, bank network fragility and cascading of failures, national sovereign default contagion, and so on. We present the general theory of multivariate generating functions to derive the number of events over all generations of various types that are triggered by a mother event of a given type. We obtain the stability domains of various systems, as a function of the topological structure of the mutual excitations across different event types. We find that mutual triggering tends to provide a significant extension of the stability (or subcritical) domain compared with the case where event types are decoupled, that is, when an event of a given type can only trigger events of the same type.

Fertility heterogeneity as a mechanism for power law distributions of recurrence times

We study the statistical properties of recurrence times in the self-excited Hawkes conditional Poisson process, the simplest extension of the Poisson process that takes into account how the past events influence the occurrence of future events. Specifically, we analyze the impact of the power law distribution of fertilities with exponent α, where the fertility of an event is the number of triggered events of first generation, on the probability distribution function (PDF) f(τ) of the recurrence times τ between successive events. The other input of the model is an exponential law quantifying the PDF of waiting times between an event and its first generation triggered events, whose characteristic time scale is taken as our time unit. At short-time scales, we discover two intermediate power law asymptotics, f(τ)~τ(-(2-α)) for τ<<τ(c) and f(τ)~τ(-α) for τ(c)<<τ<<1, where τ(c) is associated with the self-excited cascades of triggered events. For 1<<τ<<1/ν, we find a constant plateau f(τ)=/~const, while at long times, 1/ν</~τ, f(τ)=/~e(-ντ) has an exponential tail controlled by the arrival rate ν of exogenous events. These results demonstrate a novel mechanism for the generation of power laws in the distribution of recurrence times, which results from a power law distribution of fertilities in the presence of self-excitation and cascades of triggering.

Hierarchy of temporal responses of multivariate self-excited epidemic processes

Many natural and social systems are characterized by bursty dynamics, for which past events trigger future activity. These systems can be modelled by so-called self-excited Hawkes conditional Poisson processes. It is generally assumed that all events have similar triggering abilities. However, some systems exhibit heterogeneity and clusters with possibly different intra- and inter-triggering, which can be accounted for by generalization into the “multivariate” self-excited Hawkes conditional Poisson processes. We develop the general formalism of the multivariate moment generating function for the cumulative number of first-generation and of all generation events triggered by a given mother event (the “shock”) as a function of the current time t. This corresponds to studying the response function of the process. A variety of different systems have been analyzed. In particular, for systems in which triggering between events of different types proceeds through a one-dimension directed or symmetric chain of influence in type space, we report a novel hierarchy of intermediate asymptotic power law decays ∼ 1/t1−(m+1)θ of the rate of triggered events as a function of the distance m of the events to the initial shock in the type space, where 0 < θ < 1 for the relevant long-memory processes characterizing many natural and social systems. The richness of the generated time dynamics comes from the cascades of intermediate events of possibly different kinds, unfolding via random changes of types genealogy.

Time-Bridge Estimators of Integrated Variance

We present a set of log-price integrated variance estimators, equal to the sum of open-high-low-close bridge estimators of spot variances within

Most Efficient Homogeneous Volatility Estimators

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