Stochastic Aggregation: Rate Equations Approach

We investigate a class of stochastic aggregation processes involving two types of clusters: active and passive. The mass distribution is obtained analytically for several aggregation rates. When the aggregation rate is constant, we find that the mass distribution of passive clusters decays algebraically. Furthermore, the entire range of acceptable decay exponents is possible. For aggregation rates proportional to the cluster masses, we find that gelation is suppressed. In this case, the tail of the mass distribution decays exponentially for large masses, and as a power law over an intermediate size range.Comment: 7 page

Self-Excited Multifractal Dynamics

We introduce the self-excited multifractal (SEMF) model, defined such that the amplitudes of the increments of the process are expressed as exponentials of a long memory of past increments. The principal novel feature of the model lies in the self-excitation mechanism combined with exponential nonlinearity, i.e. the explicit dependence of future values of the process on past ones. The self- excitation captures the microscopic origin of the emergent endogenous self-organization properties, such as the energy cascade in turbulent flows, the triggering of aftershocks by previous earthquakes and the "reflexive" interactions of financial markets. The SEMF process has all the standard stylized facts found in financial time series, which are robust to the specification of the parameters and the shape of the memory kernel: multifractality, heavy tails of the distribution of increments with intermediate asymptotics, zero correlation of the signed increments and long-range correlation of the squared increments, the asymmetry (called "leverage" effect) of the correlation between increments and absolute value of the increments and statistical asymmetry under time reversal

Endogeneous Versus Exogeneous Shocks in Systems with Memory

Systems with long-range persistence and memory are shown to exhibit different precursory as well as recovery patterns in response to shocks of exogeneous versus endogeneous origins. By endogeneous, we envision either fluctuations resulting from an underlying chaotic dynamics or from a stochastic forcing origin which may be external or be an effective coarse-grained description of the microscopic fluctuations. In this scenario, endogeneous shocks result from a kind of constructive interference of accumulated fluctuations whose impacts survive longer than the large shocks themselves. As a consequence, the recovery after an endogeneous shock is in general slower at early times and can be at long times either slower or faster than after an exogeneous perturbation. This offers the tantalizing possibility of distinguishing between an endogeneous versus exogeneous cause of a given shock, even when there is no ``smoking gun.'' This could help in investigating the exogeneous versus self-organized origins in problems such as the causes of major biological extinctions, of change of weather regimes and of the climate, in tracing the source of social upheaval and wars, and so on. Sornette, Malevergne and Muzy have already shown how this concept can be applied concretely to differentiate the effects on financial markets of the Sept. 11, 2001 attack or of the coup against Gorbachev on Aug., 19, 1991 (exogeneous) from financial crashes such as Oct. 1987 (endogeneous).Comment: Latex document of 14 pages with 3 eps figure

Revisiting random deposition with surface relaxation: approaches from growth rules to Edwards-Wilkinson equation

We present several approaches for deriving the coarse-grained continuous Langevin equation (or Edwards-Wilkinson equation) from a random deposition with surface relaxation (RDSR) model. First we introduce a novel procedure to divide the first transition moment into the three fundamental processes involved: deposition, diffusion and volume conservation. We show how the diffusion process is related to antisymmetric contribution and the volume conservation process is related to symmetric contribution, which renormalizes to zero in the coarse-grained limit. In another approach, we find the coefficients of the continuous Langevin equation, by regularizing the discrete Langevin equation. Finally, in a third approach, we derive these coefficients from the set of test functions supported by the stationary probability density function (SPDF) of the discrete model. The applicability of the used approaches to other discrete random deposition models with instantaneous relaxation to a neighboring site is discussed.Comment: 12 pages, 4 figure

A hybrid memory kernel approach for condensed phase non-adiabatic dynamics

The spin-boson model is a simplified Hamiltonian often used to study non-adiabatic dynamics in large condensed phase systems, even though it has not been solved in a fully analytic fashion. Herein, we present an exact analytic expression for the dynamics of the spin-boson model in the infinitely slow bath limit and generalize it to approximate dynamics for faster baths. We achieve the latter by developing a hybrid approach that combines the exact slow-bath result with the popular NIBA method to generate a memory kernel that is formally exact to second order in the diabatic coupling but also contains higher-order contributions approximated from the second order term alone. This kernel has the same computational complexity as NIBA, but is found to yield dramatically superior dynamics in regimes where NIBA breaks down---such as systems with large diabatic coupling or energy bias. This indicates that this hybrid approach could be used to cheaply incorporate higher order effects into second order methods, and could potentially be generalized to develop alternate kernel resummation schemes

Unbounded solutions of the nonlocal heat equation

We consider the Cauchy problem posed in the whole space for the following nonlocal heat equation: u_t = J * u - u, where J is a symmetric continuous probability density. Depending on the tail of J, we give a rather complete picture of the problem in optimal classes of data by: (i) estimating the initial trace of (possibly unbounded) solutions; (ii) showing existence and uniqueness results in a suitable class; (iii) giving explicit unbounded polynomial solutions

Probability & incompressible Navier-Stokes equations: An overview of some recent developments

This is largely an attempt to provide probabilists some orientation to an important class of non-linear partial differential equations in applied mathematics, the incompressible Navier-Stokes equations. Particular focus is given to the probabilistic framework introduced by LeJan and Sznitman [Probab. Theory Related Fields 109 (1997) 343-366] and extended by Bhattacharya et al. [Trans. Amer. Math. Soc. 355 (2003) 5003-5040; IMA Vol. Math. Appl., vol. 140, 2004, in press]. In particular this is an effort to provide some foundational facts about these equations and an overview of some recent results with an indication of some new directions for probabilistic consideration.Comment: Published at http://dx.doi.org/10.1214/154957805100000078 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org